VllHlllWllllIWIHIWIIIIIHIWIIIHIIHIWHIWHI THS_ {THESIS l , \ LIBRARY Michigan State University ~ “ .w‘. Q... ”'0‘”. This is to certify that the dissertation entitled STRONG LAWS OF LARGE NUMBERS AND LAWS OF THE ITERATED LOGARITHM IN BANACH SPACES presented by ANANT P. GODBOLE has been accepted towards fulfillment of the requirements for PhoD. degree in Stat'iSt'lCS and Probability ngmw Major professor V. Mandrekar Date June 19. 1984 :WSU is an Aflirmatiw Action/Equal Opportunity Institution 0-12771 MSU LIBRARIES “ RETURNING MATERIALS: Place in book drop to remove this checkout from your record. FINES will be charged if book is returned after the date stamped below. STROIG LAHS or LARGE MERS AID LAHS OF THE ITERATED LOGARITM IN BANACH SPACES By Anant P. Godbole A DISSERTATION Submitted to Michigan State University in partial fulfillment of the requirements , .for the degree of DOCTOR OF PHILOSOPHY Department of Statistics and Probability 1984 ABSTRACT STRONG LAWS OF LARGE NUMBERS AND LAWS OF THE ITERATED LOGARITHM IN BANACH SPACES By Anant P. Godbole The validity of many Limit Theorems of Probability Theory is intimately connected with the geometry of the underlying Banach Space. This is especially true of the Strong Law of Large Numbers (SLLN). In this thesis, Cotype g Banach Spaces are characterized as those in which a certain condition is necessary for the SLLN to hold. Also, Logtype 9 spaces are characterized as those in which another condition is sufficient for its validity. The best results are obtained for a Hilbert Space. The results on the SLLN are formulated in terms of the validity of a SLLN for real-valued random variables, necessary and sufficient conditions for which have been obtained by Nagaev. It is shown that the above results are the best of their kind. In addition, Laws of the Iterated Logarithm are proved for certain classes of random variables taking values in an arbitrany separable Banach Space. To the memory of my father. ii ACKNOWLEDGEMENT I wish to express my sincere thanks to Professor Joel Zinn for his guidance and encouragement in the preparation of this dissertation, for teaching me most, if not all, of the theory and techniques that I have employed, for his patience in dealing with my somewhat erratic work patterns and for supplying me with a seemingly endless stream of ideas to work on. I would like to thank Professor V. Mandrekar for his constant help and advice and Professors R. V. Erickson, H. L. Koul and J. Shapiro for serving on my committee. Professors E. Giné, M. Marcus and G. Morrow reviewed my work and made invaluable suggestions. To them go my thanks. Finally, I wish to thank the Department of Statistics and Probability, M.S.U. and the Department of Mathematics, Texas A&M University for their more than generous financial support during my years in Graduate School, and Jan Want for her superb typing of the manuscript. CHAPTER I. II. III. IV. TABLE OF CONTENTS Introduction and Preliminaries . . . . . . Necessary Conditions for the SLLN. . . . . Sufficient Conditions for the SLLN . . . . The Bounded Law of the Iterated Logarithm. Bibliography. . . . . . . . . . . . . . . . . . . iv PAGE 11 19 47 53 CHAPTER I INTRODUCTION AND PRELIMINARIES He will consider a sequence {In}:=1 of symnetric and independent (but not necessarily identically distributed) random variables defined on some probability Space (9', ff, P') and taking values in a real separable Banach Space B equipped with the norm I . I. Recall that a vector valued random variable X is said to be symmetric if the probability distributions of X and -x are the same. Consider also a sequence {en}:-1 of independent random variables each assuming the values +1 and -I with probability l/2. Such a sequence is called a Rademacher sequence. We will assume throughout that the sequence {en} is defined on another probability space (n',‘[", P“) and is independent of the sequence {Xn}. He will often consider the sequence {enxnihsl defined on the product space (a, E" P) 3 (a. x a“, E. x in, P. x P"). n 2 2 Let Sn 8 Z X. s = E 'Sn' , and denote the set of integers j=1J " {2"+1,...,2"*1} by I(n). The function LL(-) is defined by LLx . max(1,log(log x)). Throughout, C will denote a generic constant whose value will usually be unspecified. We need to introduce notation and terminology from Probability Limit Theory on the one hand and from Banach Space Theory on the other. Let us first examine the probabilistic side of the coin. He shall say that the sequence {Xn} satisfies (a) (b) (C) and (d) (1.1) and (1.2) The Strong Law of Large Numbers ({xn} e SLLN) if lim ISnI/n . 0 a.s. n” The Heak Law of Large Numbers ({Xn} e NLLN) if lSnI/n + O in probability as n + o. The Bounded Law of the Iterated Logarithm 2 1/2 n) ({xn} e BLIL) if lim sup nsnu/(zsfi LLs < . a.s. n + - (By Kolmogorov's zero or one law, {xn} c BLIL iff there exists a constant A e [0,») such that 2 1/2 _ n) - A a.s.). lim sup 'Sn'/(2‘§ LLs n + o The Compact Law of the Iterated Logarithm ({Xn} e CLIL) if there exists a non-random, compact, synmetric and convex set D: B such that P(d(sn/(zs§LLs§)1/2 , D) + 0) = 1 P(C{Sn/(Zs: LLs§)1/2} = D) = 1 Here, d(x,A) - inf Ix-yl and C(An(w)) denotes the set of all cluster ycA points of the (random) sequence {An}. The set D in the CLIL is called the "limit set“. The above definitions of the BLIL and CLIL differ from the original 2 n " 2 a Z EIle . We shall 3’1 definitions of Kuelbs [22,24] who defines 5 _see that the two formulations are similar if B is a Hilbert Space, or more generally, a type-2 space. Suppose that {Xn} c SLLN. It follows from the triangle inequality that lim anI/n = o a.s. This trivial necessary condition for the SLLN n” shows that one may, without loss of generality, assume that Ixnl - o(n) a.s. while proving strong laws. Since {Xn} is a sequence of symmetrically distributed random variables, it is easy to see that {Xn} and {enxn} are equidistributed. It follows that {Xn} c SLLN iff {enxn} e SLLN. An application of Fubini's theorem shows that {enxn} c SLLN iff {en(-)xn(n)} e SLLN for almost all m e n. This elementary device of Kahane [21] will be used repeatedly in what follows. He shall denote Xn(o) (for a fixed m e a) by xn whenever there is no possibility of confusion. Let us next consider some basic notions from Banach Space'Theory. Let B be an arbitrary real separable Banach space. For p.: 1 we will denote by Lp(B) the equivalence class of B-valued random variables X with IXIp = (EIXIp)1/p < a. L°(B) will denote the equivalence class of B-valued random variables, with distance do(X,Y) = E(lX-YI/1+IX-Yl). Then Lp(B) is a Banach space for p 3_1, while L°(B) is a Fréchet space. A B-valued random variable x is said to be Gaussian if (f1(X),...,fn(X)) has an n-dimensional normal distribution for each f1,...,fn e B*, n l l. The notions of typg_and 223122 are fundamental to Banach space theory and were formulated by Maurey, Hoffmann-Jorgensen and Pisier in a series of papers in the early and mid seventies (see, for example [19, 33, 34, 38]. To motivate the definitions, let us begin with the parallelogram law in Hilbert spaces: Ix1+x212/2 + lxl-xZIZ/Z = lel2 + lx2I2(x1,x2 e H) which can be rephrased probabilistically as 2 2 2 (1.3) Elelx1+czx2l = lxll + lle and thus by induction on n as n n (1.4) El 2 c.x.l2 . Z Ix.|2 j31 J J j'l J He now generalize (1.4) by replacing the squares by pth moments and the equality by an inequality: A Banach space B is said to be of type p (1 §_p 5_2) if there is a constant A . Ap(B) e (0,-) such that P for any finite sequence {x343}.l in B, n 2 Ix.lp. n Eli e.x.lp_<_A i=1 J 3.1 J J P Each Banach space is trivially of type 1, and it can be shown using the Kolmogorov three series theorem (and an alternative definition of the type of a space) that no non-trivial Banach space can be of type p for p > 2. A Banach Space of type p is automatically of type p' for each p' < p, so that we may talk of {pIB is of type p} which is an interval that need not, in general, be closed above. In particular, there exist spaces of type p' for each p' < p that are not of type p. (See Pisier [41] for examples of such spaces.) Among the classical Banach Spaces, C[O,1], co, L" and spaces of measures are of type 1 (and no better) and the Lp spaces (l‘§_p < .) are of type min(2,p) (and no better). Hoffmann-Jorgensen and Pisier [19] proved an important result connecting the two sides of the aforementioned coin. They Showed that a Banach Space B is of type p (1.: p.5 2) iff each sequence {Xn} of independent, zero mean, p-integrable B-valued random variables satisfying the condition 2 EIlep/jp < . also satisfied the SLLN. 3'1 Their SLLN improved previous strong laws of Beck [3,4] and Noyczynski [47], just as the notion of type generalized previous notions of B-convexity and Ga spaces due to Beck [3,4] and Mourier and Hoyczynski [35,47] respectively. Let us turn next to the definition of cotype. A Banach Space B is said to be of cotype g (2 _g q < a) if there exists a constant Aq . Aq(B) e (0,~) such that for each finite sequence {xj}2=1 in B, n 0 Eli e.x.lq:A X lx.|q jgl J J QJ.1 J A cotype a Space is one in which n sup I.Z ejle.iiA. sup Ix.l {ej}e{-1.1}" “'1 191" for some constant A”. Each Banach Space is trivially of cotype a and it can be shown that a non-trivial Banach Space cannot be of cotype q for q < 2. A space of cotype q is also of cotype q' for q' > q, but {q'B is of cotype q} need not be closed below. Recently, Ledoux has generalized the examples of Pisier to construct spaces of type 2 - c and cotype 2 + e (for each s > 0) that are not of type 2 (or cotype 2). The Spaces C[O,1], co and L" are, predictably, of cotype o; the Lp Spaces are of cotype max(2,p) (and no better). While a considerable amount of research has been done relating the cotype of a Space to the validity of a central limit theorem in that Space, such a link has not, to the best of my knowledge, been made for the Strong Law of Large Numbers. We deal, in Chapter II, with this question. Kwapien [26] proved that a Banach space is both of type 2 and of cotype 2 iff it is isomorphic to a Hilbert space. we need to define another class of Banach spaces: 8 is said to be _ of (LLn)p'1-type p or Simply of logtzpe p (1 i p i 2) if there exists a constant A; a A;(B) such that for each finite sequence {x3};1 in B, n n Eiz ejx 'p 5 A;(LLn)p'1{ p Ix.| i=1 3 J-l J These spaces have been studied before: Pisier [41] characterized logtype-2 spaces (he did not call them that) as those in which each sequence {Xn}:1 of i.i.d. random variables with EX 8 0 and ram2 < a obeyed the CLIL. He also Showed (Lemma 4 in [39]) that a logtype p Space is of type r for each r < p. We Shall study the relationship between the geometry of the under- lying Banach space (as manifested in its cotype, type or logtype) and the validity of a SLLN for independent symmetric random variables taking values in that space. He study necessary conditions for the SLLN in Chapter II and sufficient conditions in Chapter III (Chapter IV deals with the BLIL for certain classes of B-valued random variables; no link is made with the geometry of B). Most of the results on the SLLN are expressed in terms of the validity of a real valued SLLN, and would not be of much use unless one could find necessary and sufficient criteria for the validity of the latter. Such criteria were obtained by Nagaev [36] and later generalized by him and Volodin [37,46] to cover the case of an arbitrary stabilizing sequence {bn} (bn + a). For completeness, let us state the basic result of Nagaev. Theorem 1.1 (Nagaev [36]) Let {Xn} be a sequence of independent, symmetric real valued random variables. Then {X } e SLLN iff for each s > O, n (1.5) nZIP(IXH' > e n) < o and (1.6) E exp(-ehn(e)2n+1) < a (e > 0). n=1 Here f5(h,e) . E[exp(hxj)]I('Xj"£_j e) and hn(e) is the solution of the differential equation d _ n+1 ¢n(h9€) 8 j621(n)[alfi fj(h,€)]/fj(h’e) .. £2 provided sup ¢n(h,e)‘3_52n+1. h Otherwise, hn(e) = a. hn(e) is well defined by the monotonicity of ¢n(h,e) in h. The conditions of Theorem 1.1 are, to say the least, complicated. This is only to be expected. The problem of finding necessary and sufficient conditions for the SLLN is a long-standing one (see Chung [8] for a discussion of the problems involved). Moreover, Prokhorov [42] expressed the belief that criteria in terms of the moments of the individual summands were probably impossible. Nagaev proved that this was indeed the case by exhibiting two sequences {xn} and {Yn} having the same moments up toany given order s < a but such that {Xn} e SLLN and {Yn} ¢ SLLN. Hhile (1.6) is complicated, it can certainly be verified. Moreover, we shall see that the real-valued SLLN'S that do arise can often be verified or disproved by other relatively simple means (such as by direct calculation). The utility of our results should not, there- fore, be gauged by the fact that they m1ght_be hard to verify, but rather by the fact that they often yield a conclusion when all other _ SLLN'S are inconclusive. It should be pointed out that we will need to verify the generalizations of Theorem 1.1 (Nagaev and Volodin) rather than Theorem 1.1 itself. Also, Nagaev's conditions nay be reexpressed more simply in terms of a standard minimization in Markov's inequality. See [36] for details. In Chapter II, we consider necessary conditions for {Xn} to satisfy the SLLN. He show (Theorem 2.1) that cotype q spaces (2 5_q < .) are n . precisely those in which the condition l/nq Z Ileq + 0 a.s. is a i=1 necessary condition for the SLLN to hold, for each independent symmetric sequence {xn}. He also show that the above necessary condition can be expressed in terms of the individual nonents if anI _<_ Cn a.s. (n 3. 1). Examples are given to show that the necessary condition is the best of its kind. Sufficient conditions for the validity of a B-valued SLLN have been studied by Beck [3,4], Beck, Giesy and Warren [5], Hoyczynski [47], Hoffmann-Jorgensen and Pisier [19], Kuelbs and Zinn [25] and Heinkel [15,16,17]. we first obtain an exponential inequality for Rademacher sequences in B and use it to prove a SLLN for random variables in type p spaces (1 §_p.§_2). This result is improved in Theorem 3.14, which characterizes logtype p Spaces as those in which each independent symmetric sequence {xn} satisfying (LLn)p'1/np.§1 Ilep + 0 a.s. also J. satisfies the SLLN. Examples are given to Show that the above result is best of its kind and that it may be used in situations where all other relevant SLLN'S are inconclusive. Furthermore, the sufficient 10 conditions of Theorem 3.14 may be expressed in terms of the individual moments if anI 5_Cn/LLn a.s. (n 3_1). For B a H, a Hilbert Space, the necessary and sufficient conditions n almost coincide: {Xn} c SLLN if LLn/n2 Z Ile2 + 0 a.s. and only if i=1 2 '3 2 1/n ) Ix.l + O a.s. For B = H we also prove (Theoreh 3.22) an i=1 extension of the Hoffmann-Jorgensen and Pisier Theorem. We do not consider non-symmetric random variables, but they may easily be studied using an elementary result of Kuelbs and Zinn [25] which states that {Xn} c SLLN iff {X3} e SLLN and {xn} c NLLN. Here, x: is the symmetrized version of Xn and is defined by S 3 _ n Xn(w.n) Xn(m) th) where X5 is an independent copy of Xn(n.1 1). A symmetrized version must always exist, at least on the probability Space (a x o, f_x E, PxP). In Chapter IV we treat the Bounded law of the Iterated Logarithm. A BLIL is proved for B'valued Rademacher sequences (Theorem 4.3). This is related to a theorem of Kuelbs [24]. Similarly, a BLIL is proved for independent Gaussian sequences using an inequality of Fernique [11,12]. This result is related to a theorem of Carmona and Kono [6]. 11 CHAPTER II NECESSARY CONDITIONS FOR THE SLLN The following is the main result of Chapter 11. Theorem 2.1. The following are equivalent: (2.1) B is of cotype q (2 £_q < .) (2.2) Each sequence {Xn} of independent symmetric B-valued random n variables satisfying the SLLN also satisfies 2 IXqu/nq + 0 a.s. i=1 .Erggf: He will first Show that (2.1) implies (2.2). Assume that (2.1) holds and let {xn} be any independent symmetric sequence satisfying the SLLN. It follows that {anxn} e SLLN for almost all m e 9. We need to n prove that 2 Ix.I/nq + 0. i=1 3 Kahane [21] proved that a Rademacher series 2 lJejx that converges 3‘1 in probability also satisfies El 2 ejlep < . for each p > O. Motivated i=1 by this result, we define the Fréchet space (E,I-IE) and the Banach space (F"°'F) by E = {x a (x1,x2,....) e B" : X ‘jxj converges in probability} 331 dE(x.y) = E(I I ej yyj)I/[l + I 2 ej (xj -yj Ill) 3'1 i=1 12 F 3 {x=(x1’x2’...) 8 Ba : E'jél ijj'q < G} leF a (El X e.x.Iq)1/q. J31 J J Kahane's theorem implies that E C F. The injection E + F is clearly a linear operator with closed graph (Since the probability and Lq limits of a sequence must coincide) and is therefore continuous (in particular at the origin)(see Theorem 4, page 57 of Dunford and Schwartz [50]). It follows that for each 5 > 0, there is a 6 > 0 such that (El X cJ j 1'1 < e whenever E(I X e. x. I/(1+I _X e x. I)) < 6. Since {c x } satisfies j-l P1 jj n n J J n n the HLLN by hypothesis, we have (1/nl X 1eJxJ. .I)/(1+1/nl X eJxJ .I) + 0 in J=1J=1 probability. The bounded convergente theorem now proves that n E[(1/nl X c .x. JI)/(1+1/nl X e x. JI)] < o(n > N) so that J=1 ‘JJ J=1JJ n . 1/n(EI X eJxJ 'q)1/q < e(n > N). By the cotype q inequality, J81 n X Ix.Iq/nq + 0, as asserted. J-1 3 Conversely, suppose that {xn} C B, and define the sequence {Xn} by n X = n c anx (n > 1). By assumption, X quxJIq/nq + 0 if J81 X j e .x J/n + O a. s. and thus (by Kronecker' 5 Lemma, which is valid in J=1 JJ 13 any Banach space) if the series X .x converges almost surely. jal eJJ Kahane [21] and Ito and Nisio [20] have Shown that the a.s. convergence X eJ xJ is a consequence of its convergence in Lq(B). Hith this in J-lJ mind, we define the Banach Spaces (E,I-IE) and F,l-IF) by E = {x=(x1,x2,...) c B. : JX 1eJxJ converges in Lq(B)} IxIE =(EI X i:.leq)1/q J-l JJ ° X quxJ lq/nq + 0} J=1 F = {x-(xlx2,...) e B " 1/4 leF = sup( X quxJIq/nq) . n j-l The above discussion shows that E c F. Assume that lxn-XIE + 0 and Ixn-yIF + 0. It can be easily shown that ng-xJI + O and ng-yJI + O, for any j. It follows that xJ - yJ, j = 1,2,... so that x a y. The injection E + F thus has a closed graph and is, by the closed graph theorem, continuous. Hence there exists a constant C < . such that sup( X quxJ.Iq/nq)1/q < C(EI X c .x qu)1/q nJ=1J=1JJ 14 ' In other words (keeping in mind that C denotes a generic constant), for n any n and for {xJ}J81 in B, n n (2.3) X qux lq/nq _<_ c El X x. Iq. J=l J J=1 JJ Fix N 3_1 and define, as in Hoffmann-Jogensen and Pisier [19] y. = O 1.5.J.i N J a xJ_n N < j.: N + n. we have by (2.3), "4’" N4.“ 2 JquJ Iq < C(N+n)q EIX XeJy.Iq jal jal J so that X ("+J)qlx lq < C(N+n)qu X e Jx. Iq jsl j j-l jj and thus n X (N+j)qli Iq/(N+n)q < c El X eJx. .l q(n, N > 1). J'1 jaI Choosing N a n yields (1/2)q X Ix. lq < c El X c x. lq, J=1J J1 JJ proving that B is of cotype q. This completes the proof. » The'necessary condition in Theorem 2.1 can be verified using the criteria of Volodin and Nagaev [37,46] but is far less appealing than an n individual moment condition such as X EIxJIq/nq + 0. When is such a J=1 15 condition necessary for the SLLN to hold? Corollary 2.3 below provides an answer to this question. He will need the following Lemma 2.2. Let {Xn} be a sequence of independent symmetric random variables taking values in an arbitrary separable Banach space 8. Assume that Ile _gj (j 3 1) and that {xn} e HLLN. Then EISn/nlq + o for each q _>_ 1. 3522:; The proof is an obvious modification of Lemma 2.3 in Kuelbs and Zinn [25]. For B a R and q - 2, a proof may be found in Stout [43] (Theorem 3.4.2) or in Loeve [28] (Corollary 1, page 253). Fix 0 < e < 1 and q.1 1. Since {xn} e HLLN, there exists an no such that sup P(l$nl‘3_ne) §_1/8 . 3Q. An application of Hoffmann- n>n -0 Jorgensen's inequality (Theorem 3.1 in [18]) yields, for any A > 0 A A/3 f qtq'lPUSnl Z. nt)dt . q - 3‘1| f tq‘lpusnu _>_ 3nt)dt o 0 /3 A/3 A _<_ q - 3"[4] tq'lPZUSnI 3 nt)dt + f tq‘lpmn 2. nt)dt] o o A NB 5 q - 3°[4e + 1/2 . 3q f tq'lpusnl 3 nt)dt +1 tq'lpwn l nt)dt o o A/3 _<_ 8q 3q 6 + 2q 3"; tq'1P(Nn 3 nt)dt o for n.: "0’ where N . max IX.I. Since Nn 5.n and A is arbitrary, we n 1513p have 1 EISn/nlq _<_ 8q :4:q e + 2q 3H Pm" 3 nt)dt . 0 . 16 For any t > 0, however, PM": nt) _g P( max ISKI _>_ nt/2) 5_ 2P(ISnl Z. nt/Z) + 0 1§k§n (By Lévy's inequality). The dominated convergence theorem now implies the result. Corollary 2.3. Let {xn} be a sequence of independent symmetric random variables with values in a cotype q Banach space (2 5.9 < .), Assume that (2.4) lle _gj a.s. (.131) (2.5) {xn} e HLLN Then n . X Elleq/nq + 0. 3'1 Proof: Immediate from Lemma 2.2 and the (alternative) definition of a cotype q space. Notice how the above conclusion was obtained merely by altering the blanket assumption lxnl - o(n) a.s. to Ix“: §_n a.s. (n_>_ 1). Remark 2.4. The proofs of Theorem 2.1 and Corollary 2.3 show that if {Xn} is any sequence of indepdendent symmetric B-valued random variables (B is arbitrary) satisfying the SLLN, then n 2 2 EEIJX1 ejle /n + 0 for almost all {xj}. If Ixnn'5_n a.s. (n 3_1) then EISn/nl2 + 0 as well. 17 Lemma 2.5. (Due to Prokhorov; for a proof, see Stout [43], page 159). Let {Xn} be a sequence of independent symmetric B-valued random variables. Then {xn} e SLLN iff (5 )/2n + o a.s. 2 n+1'gn Lemma 2.6. (Prokhorov [42]). Let {Xn} be a sequence of independent real-valued random variables satisfying for some C < o, (2.6) 'Xn' 1Cn/LLn a.s. (n 31) (2.7) E(Xn) . o (n 3_1). Then {xn} e SLLN iff X exp(-e/A(n)) < . for each s > o, where n-l A(n) . X exg/4". 361(0) Examples 2.7 and 2.8 below show that the necessary conditions in Theorem 2.1 and Corollary 2.3 are the best possible of their kind and that they are not, in general, sufficient conditions. Example 2.7. Let on be any sequence of real numbers increasing to + n. Then there exists a sequence {Xn} of independent Symmetric real-valued random variables satisfying the SLLN but such that n 4 /nq X X. q + - a.s. .. 3.1: J and n ¢n/flq 321 Elleq + a 18 To see this, let a n 1/2q X2n eznz /(¢2n) xK = 0 (K t 2n for any n) Then n 3 1/2q IS n-l-l'sznl/2 1/(¢2n) + 0 2 for each m e n, so that {xn} e SLLN by Lemma 2.5. On the other hand, i n/2nq Z lleq = <¢<2”))1’2 + ~. 2 jeI(n) Such an example exists in any Banach Space B, since R C 8. Example 2.8. The condition n X IX.Iq/nq + 0 a.s. M J is not sufficient condition for the SLLN in any Banach space. In fact, for each sequence on . o(LLn) there exists a sequence of independent symmetric real random variables satisfying ? + 0 a.s. 2 n (¢n/n ).X XJ J-l but failing the SLLN. To see this, let xn a en(n/LLn)1/2 (n 3.1). (2.6) and (2.7) are clearly satisfied, but A(n) 3_C/LL2"‘3_C/log n. Hence 2 exp(-e/A(n)) = ~ n-l if e §.C, so that the SLLN fails by Lemma 2.6. On the other hand, 2 " 2 2 " . . (¢n/n ) 2 x3 = (an/n ) X J/LLJ : c an/LLn + o a.s. J-l j=1 19 CHAPTER III SUFFICIENT CONDITIONS FOR THE SLLN The study of SLLN's in separable Banach spaces was initiated by Mourier [35], who proved that an i.i.d. sequence {Xn} satisfies the SLLN iff Elxll < a. Subsequent work (in the non-identically distributed case) was done by Beck [3,4], Beck, Giesy and Warren [5], Noyczinski [47], Hoffmann-Jorgensen and Pisier [19], Kuelbs and Zinn [25] and Heinkel [15,16,17]. As with all limit theorems in Banach spaces, these results fall into two natural categories, with restrictions being placed either on the probability distributions of the sequence {Xn} or on the geometry of the underlying Banach space. He will start by considering the first class of results. The results of Kuelbs and Zinn (Theorems 3.2 and 3.3 below) fall into this category, but may easily be restated as statements about the geometry of B. Theorem 3.1 below is due to Beck, Giesy and Harren. It is a result that (a) is valid for each Banach space and (b) is in terms of the moments of the individual summands. Ne will see subsequently how difficult it is to prove a non-trivial result at this level of generality without making additional assumptions (Kuelbs and Zinn hypothesize, fbr example, that the NLLN is satisfied). Theorem 3.1. (Beck, Giesy and Harren [5]). Let {xn} be a sequence of independent B-valued random variables with EXn = 0 (n.3 1) and satisfying either 20 " 2 1/2 (3.1) X EIxj I 2/j2 . and lim X (EIx.I ) /n = 0 jzl n+0 jsl J or n (3.2) lim X ess sup Ixj I/n = 0 nee jsl Then {Xn} s SLLN. Furthermore, (3.1) and (3.2) are the best possible in the sense that weakening either yields a result that is no longer true for all Banach spaces. Theorem 3.1 follows as a corollary of Theorem 3.3 below (Kuelbs and Zinn). In fact, both (3.1) and (3.2) imply that n X IX.I/n + 0 a.s. in J which is a completely trivial sufficient condition for the SLLN. (3.2) obviously implies that n X IX.I/n + 0 a.s. J=1 3 Assume that (3.1) holds. We need to prove that {IxnI} e SLLN It suffices, therefore, to show that {IxnIS} e SLLN and {anl} e NLLN. Note that X E(IX. I S) 2/J'2 <2 X EIX. l 2/j2 J=1 J-l so that {anlS} e SLLN by Kolmogorov's real-line SLLN. Also [I P( X IXj I > he) < 1/ne X EIXJ . < l/ne 2 (EIXJ l2)1/2 j=1 j=1 j=1 proving that {Ian} e NLLN. 21 Theorem 3.2. (Kuelbs and Zinn [25]). Let {Xn} be a sequence of independent B-valued random variables satisfying (3.3) Xn/n + 0 a.s. (3.4) for some p a [1,2] and for some r e (0,»), X A(n,p)r < . , where A(n,p) = 1/2np X EIX.Ip. n=1 jc (n) J (3.5) {xn} e NLLN Then {x } e SLLN. n Kuelbs' and Zinn's next result extends Lemma 2.6 (the sufficient part) to the B-valued case: Theorem 3.3. (Kuelbs and Zinn [25]). Let {Xn} be a sequence of independent B-valued random variables such that (3.5) holds and (3.6) lle 5 Cj/LLj a.s. for some C < a, (j.: 1). (3.7) X exp(-e/A(n)) < a for all e > 0, where A(n) a A(n,2) = 1/4n X EIX.I2 n81 jeI(n) Then {xn} e SLLN. Remarks 3.4. The HLLN hypothesis of Theorem 3.3 may be difficult to verify unless one assumes, for example, that B is a type-p space in which case it may be replaced by n (3-3) 1/np X EIx.Ip + 0 (n + .) J i=1 which is a condition in terms of the individual moments. Notice, however, how stringent (3.8) is for p a 1. This just reaffirms the fact that norm and/or moment assumptions will not yield useful SLLN's if B is 22 of type 1 (and no better). (If B = 2', for example, and one defines xn = anenen, where {an} C R and {en} is the canonical basis of z', n then it is easy to see that {xn} e SLLN iff X lle/n + 0 a.s.) Notice J=1 also that if B is a type-2 space, then (3.5) is automatically implied by (3.4) or (3.7) and need not be hypothesized. If B is of type p, 1.3 p < 2, however, then (3.5) is crucial and may not be omitted even if {Xn} is a symmetric sequence. In other words, the NLLN hypothesis of Theorems 3.2 and 3.3 is not merely a “desymmetrization” assumption. To see this, consider, for p s [1,2), the 2p valued sequence {Xn} defined by xn(u.k) = en(w)(n/(LLn)°>1’21{n}(k) 1 is arbitrary. We have, for any m, lle s (j/(LLj)°)1/2 so that (3.6) is satisfied. Also, 2" 2 2" a a-J (LLn/n ) X EIX.I a (LLn/n ) X j/(LLj) .3 C/(LLn) + 0 i=1 J J'1 so that (log n) A (n) + 0. It follows that (3.7) holds. Notice, however, that {Xn} i SLLN, since for any m, n n '-21 xj/n- - n‘ltjz1 (J/(LLJ)“)p’ZJl/P.;,en‘ltnp’2*1/(LLn)°p’211/P + ., A similar example may be constructed to show that (3.5) is a crucial hypothesis in Theorem 3.2 as well. We next consider the second class of results on the SLLN; ones in which conditions are imposed on the Banach space B. The basic result in this direction is due to Hoffmann-Jorgensen and Pisier who obtain an 23 analog of the classical Kolmogorov-Chung SLLN (Theorem 3.5). Heinkel has obtained an improvement of Theorem 3.3 for Hilbert-space valued random variables (Theorem 3.6 below). Theorem 3.5. (Hoffmann-Jorgensen and Pisier [19]). The following are equivalent (3.9) Bis of type p (1 1p: 2). (3.10) Each sequence {Xn} of independent zero-mean B valued random variables with X EIXjIp/jp < a satisfies the SLLN. J=1 Theorem 3.6. (Heinkel [16]). Suppose {Xn} is a sequence of independent centered random variables with values in a 2-uniformly smooth Banach space 8 (see [16] for a definition). Assume that (3.6) holds and that (3.11) X exp(-e/r(n)) < o for each e > 0, n=1 where r(n) s 1/4n X sup E(f,x.)2 . jeI(n) ngerr J n (3.12) 1/n2 X EIXJIZ + o. 3'1 Then {xn} e SLLN. Heinkel also constructs a sequence of sz-valued random variables satisfying the hypotheses of Theorem 3.6 but not of Theorem 3.3, proving ' that (3.7) is not a necessary condition for the SLLN to hold, even in a cotype 2 space and even if (3.6) holds. Zinn [49] has given another example. 24 There are three major differences between the present investigation and the work of the above authors. Most of the results are stated in terms of the validity of a real-valued SLLN. They are shown to be the best possible of their kind. Finally, no HLLN hypothesis is made in the symmetric case. Convergence in probability need only be hypothesized, therefore, to be able to conclude that {Xn} obeys the SLLN if {xi} does. Also, the results resemble Theorems 3.2 and 3.5 in that no specific hypothesis is made on the magnitudes of the norms of the Xn's and because one obtains characterizations of certain classes of Banach spaces through the validity of a SLLN. There is also a strong similarity with Theorems 3.3 and 3.6 (See Remark 3.12). The first group of results depend on an exponential inequality (Lemma 3.7). They are not the best possible (and are, in fact, improved later in this chapter) but are included because the nature of their proofs is quite revealing (see Remark 3.18) and also because Lemma 3.7 plays an important role in Chapter IV on the (bounded) law of the iterated logarithm. Lemma 3.7. Let B be a separable Banach space. Consider the Rademacher n series "X1 5 x", where {xn} c B. Set Sn . X e and s 2 2 .x n 3’1 J j n a ElSnl . Then there exists a constant M a M(B) such that for each a > 0, (3.13) P(ISnI/sn > e) 5 3 exp(-e2/M2). 25 ‘25221; He shall use a basic result of Kwapién [27] which states that for any sequence {xn} in an arbitrary Banach space B, the almost sure convergence of X e x. implies that E[exp al X e.x.l2] < . for each j'l J J jal J J a > 0. Let the Banach space (E,I-IE) and the Orlicz space (F,I-IF) be defined by E a {x a (x1,x2,...) 2 8° : .Xl ejxj converges in L2(B)} J8 I2 1/2 IxIE = (El 5 I) jél J 1 F = {x 8 (x .x ....) e B. = E eXP(°| X 8 X '2) < ° for 93¢" 0 > 0} 12 i=1“ leF s inf{t > 0 : E exp(l X ejlezltz)‘g e} 1'1 The fact that both E and F are Banach Spaces is well known and may be verified by a tedious but routine calculation. Kwapien's theorem asserts that E c F. Suppose that lxn-XIE + 0 and Ixn-ylF + 0. It then follows that EI X ej (xn -x )I2 + 0 and El X e (xn-y )I2 + 0 so that . 321 J J jsl J J J x - y. The closed graph theorem now implies that there is a constant M < . such that (3.14) iflf{t > 0 2 E EXPUSIz/tz) 1 e}: "(EISI2)1/2 where S - X J J In other words, for each n, J81 E exp(ISI2/M2EISI2+n‘1) 5.e. An application of Fatou's lemma yields 26 (3.15) E exp(ISI2/M2EISIZ) _<_ e so that for each n, 2 2 2 2 2 P(ISnl/sn3§) = P(ISnI /M Sn2§ /M ) §_E(exp ISnlz/Mzsg) exp(-e2/M2) _<_ 3 exp(-eZ/M2). by (3.15). This proves (3.13). Remark 3.8. Kahane [21] showed that E exp(aISI) < . for each a 5-00 whenever S is an a.s. convergent series; Kwapien's theorem is obtained by using Kahane's result together with an additional argument. Marcus and Pisier [31] generalize Kwapien's result and also show how it nay be proved directly. The Orlicz norm of Lemma 3.7 was first used by Pisier [40]. It must be pointed out that Lemma 3.7 is implicit in the work of the above authors and has merely been retrieved from Kwapien's theorem. The following result of Kahane is a consequence of Kwapien's result and generalizes the classical Khinchin inequalities. It shows that the Lp norms (1 5_p < .) are all equivalent on the linear span generated (in B) by the Rademacher random variables. It nay also be interpreted as a converse Holder inequality on this subspace. Lemma 3.9. (Kahane's inequalities [21]). For each p and q satisfying 1 5_p §_q < a, there exists a universal constant Kp q such that for each. Banach space 8 and for each finite sequence {x3};1 in B, n n q l/q P l/P 3'1 27 The next lemma is a fundamental result of Hoffmann-Jorgensen. It provides conditions under which the almost sure and Lp-convergence of a series of independent B-valued random variables are equivalent: Lemma 3.10. (Hoffmann-Jorgensen [18]). Let {xn};.1 be a sequence of independent B-valued random variables so that Sn converges a.s. to S, and let 0 < p < a. Then the following are equivalent. (3.16) 5n + s in Lp(B). (3.17) s s 19(3). (3.18) n . s:plSnI 5 19m ). (3.19) N =- sgplxnl e Lp(R ). (3.20) {Sn}:.1s a bounded subset of Lp(B) . The next proposition gives 3 sufficient conditions for the SLLN. The first two are in terms of the a.s. convergence of a series of real random variables, while the last is in terms of the validity of a real-valued SLLN. Proposition 3.11. Let {Xn} be a sequence of independent symmetric 2 B-valued random variables. Let tau and Th” (or simply t2 n and T", when there is no possibility of confusion) denote the Quantities 2 E I e.X.(m)I and X e x.(m) , respectively. Then ‘ je§(n) J J 381(0) J J 28 (a) {xn} e SLLN if' (3.21) X exp(-e4n/t:m) < o a.s. for each a > 0. n-l If, in addition 8 is of type p (1 §.p §_2) then (b) {xn} e SLLN if (3.22) X exp(-e4"/[ X lXj(m)Ip]2/p < o a.s. for each s > 0 n=1 jeI(n) and (c) {xn} e SLLN if n (3.23) [(LLn)p/2/np] { llep + o a.s. J'l Proof: We will show that {anxn(m)} e SLLN for each m satisfying (3.21). By Lemma 2.5 and the Borel Cantelli Lemma it suffices to show that X P(ITnI > Zne) < «- n-1 for each a > 0. Fix 6 > 0 and apply Lemma 3.7 to get P(lTnlz2ne) a P(lTnl/tn12ne/tn) §_3 exp(-4"c2/t:M2). By (3.21) the last series is summable for each s > 0. This proves Part (a). Part (b) follows immediately from (a), Kahane's inequality and the definition of type, since t2 . El X e.x (w)l2‘§_K§’2(El X c.X.(m)Ip)2/p "“’ Jenn) JJ 3cm)“ 2 2 2/ 5.xp’2A gp(XlXj(m)lp) P. 29 To prove (c), notice that by (3.23) E exn(-e4"/[ IX. w lp Z/p = n31 je§(n) J( ) 1 X exp(-e{2np/(LL2n)p/2 X 1x.(e)up}2/PL12") n-l jeun) J ' -2 < C X n < .. - and In other words 3.23 implies 3.22. This completes the proof of (c). Remarks 3.12. The sufficient condition in part (c) above may be verified by the criteria of Volodin and Nagaev [37.46]. One may attempt similarly to check the conditions in (a) and (b) using Kolmogorov's three-series theorem, but it is more convenient to use Hoffman- Jorgensen's result (Lemma 3.10). Assume that (3.21) holds. Since (3.19) is obviously satisfied we must have (3.17). In other words, (3.24) nX1 E exp(-e4"/t:m) < o (e > 0). Conversely, if (3.24) holds, the series X exp(-e4n/t:m) converges in n21 L' for each s > 0 (since it is a positive term series) and thus almost surely, fbr each s > 0 (by Lévy's theorem). (3.21) and (3.24) are thus equivalent. Similarly, (3.22) is equivalent to (3.25) X E exp[-e4"/(. X llep)2/p] < a (e > 0) n=1 JeI(n) so that (3.21) and (3.22) hold iff just ggg_of the series in Kolmogorov‘s three-series criterion converges. Unfortunately, (3.24) and (3.25) cannot be thought of as being computationally easy 30 substitutes for (3.21) and (3.22). It is not clear, therefore, how (a) and (b) of Proposition 3.11 may be verified, even though they are formulated in terms of the almost sure convergence of a series of independent real random variables. In Proposition 3.13 below, we show when the sufficient conditions of (b) and (c) are equivalent. The rates in (c) are not the best possible (this is painfully obvious for p . 1) unless p . 2. He shall obtain the best rates in Theorem 3.14. Assume therefore that p . 2. Part (b) of Proposition 3.11 states that a Rademacher sequence {anxn} satisfies the SLLN if (3.26) X exp(-e4n/ X Ix.l2) < a (e > 0) , n-1 je n J which is exactly what the sufficient conditions of Kuelbs and Zinn (Theorem 3.3) and Heinkel (Theorem 3.6) reduce to for such a sequence. Suppose one were trying to prove the SLLN for an arbitrary sequence {Xn} of independent symmetric random variables with values in a type 2 space, using the criteria of Kuelbs and Zinn or Heinkel. Suppose also that one chose to prove the SLLN for {enxn(m)} (for almost all m) instead of for {xn}. (3.7) and (3.11) would then coincide with (3.22), the sufficient condition of Proposition 3.11(b). This is not to suggest that these 3 conditions are equivalent; they are not. The above remark provides a clue, however, as to how they might be related. Notice also that (3.22) implies the SLLN even if the boundedness hypothesis (Ile §_Cj/LLj) of Kuelbs, Zinn and Heinkel is not satisfied. 31 Proposition 3.13. Let A(n,p) = X( )IXjIp/an form a decreasing jeI n sequence of real numbers, and suppose that X exp(-e/[A(n,p)]2/p) < . n=1 for each s > 0. Then (log n)p/2 A(n,p) + 0 (thus (b) and (c) of Proposition 3.11 are equivalent if A(n,p) forms a decreasing sequence for almost all m e n, a condition that most sequences {Xn} would satisfy). 'frggf; Let{an} be any decreasing sequence of real numbers. We will prove that an = o(1/log n) if X exp(-e/an) < . for each s > 0. n=1 The proof is along the lines of Proposition 6.7 in Dudley [9]. Let an a an/log n. We will show that “n + 0. Suppose on the contrary that lim sup “n a 26 > 0, so that an > 6 for arbitrarily large values n+9 1/2 of n. Choose such an n and suppose that [n ] 5_j 5_n. Then a. a ajlog j 3_anlog j . (an/log n)log j-Z-“n/z . J Thus n n -€/a. n -Ze/an X exp(-e/aj) = X j J 3_ X n j-[n1’23+1 j-[n1’2]+1 i=[n1/2]+1 n -26/5 2- 2 n : (n-[n1/2])/n28/6 . jsan/2]+1 It is clear that X exp(-e/an) cannot converge for each s > 0. This n81 proves the result. 32 The fbllowing example shows that Proposition 3.13 is false in general. Define a symmetric independent real-valued sequence {xn} by K X 8 6K 22 [Kl/2 2 K (K31) 2 +1 K 2 +1 for any K). Xj = 0(3 t 2 Then for any m e n and for each K, K 22K+1 A(2K,p) 3 1/22 “D 2 K llep Z. C/Kp/g j=22 so that lim sup (log n)p/2 A(n,p) is strictly positive. Note also that n + o (11(2K,p))2/p g C/K. Suppose now that n ¢ 2K for any K. Then 2£+1.i n .£.2£+1'1 for some I, so that 2n+1 > 22t+1 and 2n+1 < 22£+l+1. It follows that A(n,p) - 0 and thus '2' exp(-e/[A(n.p)]2/p) - '2' exp<-e/u<2".p)12’p) 5 f exp(-eK/C) <- n81 K81 K81 for each s > 0. There are several directions in which one may hope to improve Proposition 3.11(c). One possibility would be to widen the domain of its validity and another would be to improve the rate in (3.23) (which is intolerably bad for p - 1). Finally one may want to obtain a characterization of certain Banach spaces through the validity of a SLLN. Some of this is accomplished in the following theorem, which is the main result of this chapter. 33 Theorem 3.14. Let {Xn} be a sequence of independent symmetric random variables taking values in a real separable Banach space B. Assume that p-1 p " p (3.27) For some p e [1.2], (LLn) /n X Ile + 0 a.s. i=1 (3.28) {enxn(m)} e HLLN for almost all m. Then {Xn} e SLLN. In particular, the following are equivalent (3.29) B is of log type-p. (3.30) Each sequence {Xn} of independent symmetric B valued random variables satisfying (3.27) also satisfies the SLLN. Proof: The proof makes use of Theorem 3.3 (Kuelbs and Zinn) and a truncation argument. Let Xj . Yj + 23, where Yj - XjI(lle§j/LLJ) and 23 = XJI(lle > j/LLj). Note that {Yn} and {Zn} are both independent symmetric sequences. By Lemma 2.5, {Zn} 3 SLLN iff X )Zj/z" + 0 a.s. je (n We have 1 z./2"l < 1/2n Ip'1(nx.u > j/LLj)Ix.l 3301) J ‘ Jeifin) J J _<__C(LLz")P‘1/2"p X IX.Ip + o a.s., jeun) J by (3.27). Let us next consider the sequence {Yn}. It is clear that {Yn} e SLLN iff {enxnI(lxnl 5 n/LLn)} e SLLN for almost all m. Choose an m for which (3.27) and (3.28) hold. We need to verify that (3.5) through (3.7) hold for the sequence {anxn1('xn'-3 n/LLn)}. (3.6) is obviously satisfied. Let yn denote xnI(lxnl'§_n/LLn). We have 34 L12"/4" X ijl2 = LLz"/2"p X uy.upnyj|2‘p/2"(2‘PJ jean) jean) J 5_CLLz"/2"p - 1/(LL2")2'p X ly.Ip= JeI(n) J C(LLz")p'1/2"p X ny.lP;C(LL2")p'1/2"p X IX.Ip + o, Jenn) J 361(0) by (3.27). Hence X exp(-c4n)/( X ijl2)§_Ce X n"2 < a n=1 jeI(n) n-l so that (3.7) holds. (3.27) implies that lxnl §_n/(LLn)p'1/p if n is large enough. (3.28) asserts that {enxn} e NLLN. He may thus apply n Lemma 2.2 to conclude that 1/n E I X ejle + 0. Kahane's contraction i=1 principle [21] or Hoffmann-Jorgensen's comparison principle (Lemma 4.1 n in [18]) now show that 1/n El X ejyjl + 0 so that (3.5) holds. Another 3'1 proof of the last fact may be given by using Lévy's inequality. This proves the first part of the theorem. If B is of logtype-p then n _ n _ _1 n P(l X e x.I > ne) < (ne) pEI X c.x.lp < C(nc) p(LLn)p X Ix.Ip + 0 . J J - . J 3 - , 3 J 1 J 1 j 1 for each m satisfying (3.27), so that (3.28) holds. It follows that (3.29) implies (3.30). He turn next to the converse proposition. This is proved by using yet another closed graph argument. Consider the Rademacher sequence {enxn}, {xn} c 8. By hypothesis, {enxn} e SLLN if n (LLn)p‘1/np X Ilep + 0. Kronecker's lemma and an argument similar to 3'1 35 the one in Theorem 2.1 now show that n . _1 1/an1 X e.x.lp + o if X (LLj)p IX.Ip/jp < . . -,1 J J -=1 J J J Define the spaces (E,l-IE) and (F,I-IF) by E . {x e B“ : X (LLj)p'1Ix.lp/jp < .} #1 J IXIE = E Z (LLJ)p'1Iijp/jp]1/p 3‘1 a n F a {x e B : 1/np E I X e.x.lp + 0} j']. JJ le = sup 1/n(El X +:.x.lp)1/p F n j=1 J J (E,I-IE) and (F,l-IF) may easily be seen to be Banach spaces with E C F. Suppose that lxn-xlE + 0 and Ixn-ylF + 0. It follows easily that lxg-le + 0 and lxg-yjl + D for each j. It follows that x . y. The closed graph theorem implies that there exists a C < a such that n O sup 1/n(El X e.x.lp)1/p‘§_C( X (LLj)p’1|x.up/jp)1/p. fl jal J J 3.31 J In other words, for each n 3_1 ll n _1 (3.31) El X c.x.lp < c nP( X (LLj)p Ix.lp/jp). j']. J J "' j’]. J Fix ".1 1 and define ”=00531N) = xj_N(N < j §_N + n). 36 By (3.31), for each n N+n N+n E: X e .y. 'p < C(N+n)p( X (LLj)p 11x. Ip/jp) j.1 J j j_1 so that n EI X e x. I" < C(N+n)p( X [LL(N+j)]p'1 Ix. lp/(N+j)p) 3-1 JJ 3'1 c{(N+n)p[LL(N+n)]P'1/(N+1)p}XI1 llep. J8 Setting N . n, we obtain " p +1 p- -1 p El X ejx .I < c 2p (LLn) X Ix. I i=1 3'1 3 which proves that 8 is of logtype-p. (Pisier [39] has shown that this implies that B is of type r for each r < p.) No assumptions are made on the magnitudes of the norms of {xn} in Theorem 3.14. Theorems 3.2 (Kuelbs and Zinn) and 3.5 (Hoffmann- Jorgensen and Pisier) fall in this category. We would like to show next that our result may be used in situations when both the above theorems are inconclusive. Example 3.15. Define the sequence {Xn} by 3/4 x a e n - 2"/(LL2") (n . 1,2,...) 2 2n XJ. = 0 (j at 2n for any n). 2n n . Then,jX1E|lep/jp=X 1/(LL2‘J)3p/4 + a for each p e [1.2], so that 1'1 Theorem 3.5 is inconclusive. Also, 37 A(n,p) = 1/2np X E|x.)P a (LL2")'3p/? It follows that X A(n,p)r = . 351(0) J n=1 for each p s [1,2] and r s (0,0). No conclusion may thus be reached using Theorem 3.2. On the other hand, for any p s [1,2] and n e I(K), K+1 n 2 (LLn)p'1/np X )x.'P < (LL2K*J)p’1/2Kp X Ix )9 3-1 J " 3-1 " K+1 . . = (LL2K+1)p'1/2Kp X ZJp/(LLZJ)3p/4‘5_C(LLZK)p/4'1 + o J'1 a.s., so that {Xn} e SLLN. Theorem 3.14 does not, however, contain either of the two theorems; to see this, let X x (n a 1,2,...) 3 E an an an Xi = 0 (j e an for any n) and xn = an/nz. He then have ' C O -2 jélg'lep/Jp ' n§lfilxanlp/an ' "£1" p < . K for each p e [1,2]. Also it is clear that A(n,p) = 0 (n t 22 ) and K ~ ~ 2 ,p) a K'2p so that X A(n,p) < o for each p e [1,2]. However, n81 A(2 a n n (LLan)p'1/a: X lx.'p < C 2n(p-1)/ap X aP/j2p + . a.s. . J - n - J J31 J31 Similar examples may be constructed to show that for B . fl! , (3.27) is not comparable to any of Teicher's [45] sufficient condi- tions for the SLLN. 38 Theorems 2.1 and 3.14 may now be combined (with p = q = 2) to obtain the following. Corollary 3.16: A sequence {Xn} of independent symmetric random variables with values in a Hilbert space H (or more generally a cotype n 2, logtype-2 space) satisfies the SLLN if LLn/n2 X Ile2 + 0 a.s. J31 . 2 “ 2 and only if 1/n X lle + 0 a.s. Furthermore, the condition 3'1 n o(n)/n2 X Ile2 + 0 a.s. is neither necessany nor sufficient for the 3'1 SLLN for any function ¢(n) + a, ¢(n) = o(LLn). (This may be seen from Examples 2.7 and 2.8.) Example 3.17. We showed in Example 2.7 that the rate in Theorem 2.1 could not be improved for any q.: 2. He would like to show next that the rate in Theorem 3.14 is also the best possible, in the following sense: For each p s [1,2], and for each sequence o(n) + a, there exists a sequence {xn} of independent symmetric real random variables n satisfying the condition [(LLn)p'1/np¢(n)] X |lep + 0 a.s. but failing 3'1 the SLLN. Such a sequence may be defined by xj . sjzn/LLZ" j = 2" + 1,...,2"+[LL2"] n s 1,2,... X. = 0 Otherwise 39 Then for any K s 1(n), and p a [1,2], p-l p K p n+1p-1 up n "*1 3 3'9 3p [(LLK) /K M10121 Ix” _<_ [(LL2 ) /2 3(2 )].X1[u.2 12 /(LL2 ) 3= J. 5_ o(LLz'J“1)"'1/2np - 2("“1)F’/(LI.2""1)"'1 - 1/¢(2") + o as n + ., 0n the other hand, while (2.6) and (2.7) of Prokhorov's Lemma 2.6 are satisfied, we have Mn) a 1/4" - [LL2"] . 4"/(LL2")2 _>_ C/l’og n so that X exp(-c/A(n)) = a for 8 small enough. Hence {Xn} t SLLN. n-l Remark 3.18. He have seen how the necessary and sufficient conditions for the SLLN ”almost“ coincide if B is a Hilbert space. If B is arbitrary, however, necessary and sufficient conditions that are close to one another cannot be formulated in terms of the validity of a real valued SLLN. Observe, however, that Remark 2.4 and Proposition 3.11(a) n together imply that a B-valued SLLN holds if LLn/n2 Es. X ijj(m)|2-+ O i=1 n a.s. and only if 1/n2 EE' X erj(w)l2 + 0 a.s. These conditions are i=1 not, however, easily verifiable since there is no general technique of 2 estimating EISnI for an arbitrary sequence of B-valued random variables. (The two-sided estimates obtained by Giné and Zinn [14] may be expressed in terms of the individual summands only if B is a Hilbert space.) 40 He saw in Corollary 2.3 how the necessary condition of Theorem 2.1 could be expressed in terms of the individual moments if we assumed that lxhl‘§_n a.s. (n 3_1). We show now that the sufficient condition of Theorem 3.14 may be similarly rephrased if Ixnn‘5_c n/LLn a.s. (".1 1). The following Lemma of Loéve generalizes Prokhorov's result (Lemma 2.6). Lemma 3.19. (Loéve [28]). Let {Yn} be a sequence of independent zero mean (real) random variables such that 'YnI-fi C bn/LLbn a.s. (n.: 1) where bn + . and 1 < Clibzml/bznéc2 < a for some constants 2 n C and C 1 2. Set Tn = S n+1 - Sznlbn, t . ET: . 1/b2 X EYZ. Then 2 2" jeI(n) J Sn/bn + O a.s. iff the series X exp(-e/t:) < a for each 3 > 0. n=1 Proposition 3.20. Condition (3.27) may be expressed in terms of the moments of {xn} if lxnl g Cn/LLn a.s. (n 3_1). In fact, (3.27) holds iff -1 " (3.32) [(LLn)p /nP] X ulep + o in probability 3'1 and (3.33) X exp[-e 22"”/{(LI.2")2"'2 X E(IX.Ip-lxtlp)2}] n=1 jeI(n) J J < o, for each s > O I where Xj is an independent copy of Xj(j 3 1). (Here, (3.32) may be - expressed in terms of the individual moments by the classical degenerate convergence criterion.) 41 . n . Proof: Assume that (LLn)p'1/np X llep + 0 a.s. for some p s [1,2]. 3'1 By the result of Kuelbs and Zinn (see the Introduction), this is equivalent to (3.22) and p-1 p " p ' p (3.34) (LLn) /n X (lle -Ile ) + O a.s. 3'1 We have C'jp/(LLJ)p'1 LL(Jp/(LLJ)p'1) ' lllep-IX3IPI‘§_2ij/(LLj)p < so that Loéve's result aplies to (3.34). It follows that (3.27) is equivalent to (3.32) and (3.33), as claimed. Notice that E(IXle-Ixslp)2 a 2 Var(lle ) §_2EIle29 so that (LL2")29'2 X E(ux.uP-nx'np)2/22"P < jen J J - 2(I.L2")2P"2 X 2p 2np JBI(MEIXJJ /2 .5 C(LLz")2J"2 X Elx.|2(2"/L12")ZP‘2/22"p JeI(n) J = 04(0). so that (3.33) is a slight improvement of the Kuelbs-Zinn condition. One the other hand, (3.32) may do worse than their HLLN hypothesis. 42 Next, we proceed in a somewhat different direction to obtain a SLLN for Hilbert-space valued random variables. One may use the following example to motivate what follows: Define the sequence {Xn} of 22- 1/2 valued random variables by Xn = enen(n/LLn) , where {en} is the usual 2 basis of t . Does {xn} e SLLN? Theorem 3.14 as well as the results of Hoffmann-Jorgensen and Piser, Kuelbs and Zinn, and Heinkel are all inconclusive. To see this, note that -|2/Jz - X 1/j LLj = e, X Elx . J J31 3'1 1/4n X EIX.I2 a 1/4n X sup E(f,X )2 ch(n) J jeI(n) mil J = 1/4n X j/LLj _>_ C/log n 381(0) and n LLn/nZX muzlc > o a.s. i=1 J 2 " 2 0n the other hand, 1/n X Ile ‘5_C/LLn + O a.s., so that no negative j-l conclusion may reached using Theorem 2.1. The fact that {Xn} e SLLN may however be deduced from Theorem 3.22 below, which is a generaliza- tion to the Hilbert space setting of Teicher's [45] extension of the classical Kolmogorov SLLN. It extends the Hoffmann-Jorgensen and Pisier theorem in exactly the same way (for B . H). Teicher's result is stated next. It is really the first in a hierarchy of successfully stronger (and more complicated) results. 43 Theorem 3.21. (Teicher [45]). Let {Xn} be a sequence of mean zero, independent (real) random variables satisfying 1 (3.35) X Exf/i4 1X EX? < . i=2 j=1 J n 3'1 J There exist constants Cj such that “ ° 2 2 4 (3.37) JX1P(|xj' > cj) < and lecj EXj/j < . Then {x } e SLLN. l‘l Egorov [10] showed that (3.35) and the condition Xn = o(n) a.s. were sufficient for the SLLN to hold, but we shall see that such an extension will not be possible in the case of Hilbert Space-valued random variables. Theroem 3.22. Let {xn} be a sequence of independent symmetric random variables with values in the real separable Hilbert space H equipped with the inner product (-,-). Assume that 2 " 2 (3.38) 1/n X Ix.l + 0 a.s. 3-1 J and . _ _ . _4 k-l 2 (3.39) X x X (xj,xk) < . a.s. k=2 j=1 Then {xn} e SLLN. Before we prove the theorem, we shall need to state a basic lemma of Chow [7]. Lemma 3.23. (Chow [7]). Let {Y E"; n 3.1} be a martingale n, difference sequence and suppose that 0 < aj + a, where aj is f3_1 measurable for each j 3_1. Then (3.40) X E( Y p.§ ._ )/ap < a a.s. for some 0 < p < 2 n implies that X Y /a + 0 a.s. J-IJ" Proof of Theorem 3.22. He will show, as before, that {enxn(o)} e SLLN for almost all m. Choose any m satisfying (3.38) and (3.39). Denote Xn(m) by xn(n‘3.1). He have (3.41) lelxl+...+enxnl2 a (elx1+...+enxn,e1xl+...+enxn) f 2 E ( a Ix.l + 2 c c. x ,x.) j‘]. J 1,j'11J1J i 0 such that sup P(ISnI > La") 5_1/24, where n a 2 2 1/2 an (21:n Llan) . 48 . Then there exists a constant A e [0,o) such that lim sup IS I/a a A a.s. "-1). n n Notice that our definition of the BLIL (see Introduction) and that of Kuelbs are similar if 8 is a Hilbert space. He shall first prove a BLIL for Rademacher sequence {enxn}, {xn}c: 8 under weaker hypotheses than (4.1) and (4.2). The method of proof relies on a result of Marcus and Zinn [32]. They generalize a Theorem and a construction of Volodin and Nagaev [46]. Let us describe this construction: Given an increasing sequence {bn}, fix C > 1 and consider the intervals (0,C], (C,Cz], . . . . From these, discard the ones for which t t +1 k CK+1] = a and label the rest (C r,C r {bn}fl (c . 1 (r - 1.2....) in tr+1 t such a way that tr < t In other words, (C r,C ] is the rth r+1° interval having a non-empty intersection with the sequence {bn}. Let t t +1 n = sup(n : bn e (C r,C r r J} and consider the sequence {bn}' It is clear that tr 3_r. He shall, following Marcus and Zinn, call {"r} the Volodin-Nagaev (NV) subsequence determined by {bn} and C. Lemma 4.2. (Marcus and Zinn [32]). Let {Xn} be a sequence of independent B-valued random variables. Let C > 1 and {bn} + a be arbitrary. Assume that for some a > O, (4.3) lim P(ISnl > a b") = 0 . M 49 (4.4) For the NV subsequence {"r} determined by {bn} and C, X P(IS -S I > Zab ) < .., r=1 nr nr-1 nr Then (4.5) lim sup 'Sn'lbn-i {(4C+[2/C-1])a} a.s. n + - He are now ready to state our BLIL for Rademacher sequences. Our relaxing of (4.1) in no way contradicts the classical examples of Marcinkiewicz and Zygmund [30] since we do not wish to insist that we obtain A - 1. Theorem 4.3. Consider the Rademacher sequence {enxn}, where {xn}<: B. Assume that n Si 2 El X e x I2 + c as n + 0. Then there exists A e [0.0) such 3=1JJ that 3 A a.s. n 2 2 1/2 l;m+sup IJX1 ejij/(an LLsn) 2 Proof: Let b" = (25}: LLsn)1/ 2 (n 3 1). Theorem 2.5 in Hoffmann- 2 Jorgensen [18] shows that {sn }, and hence {bn}’ are increasing sequences. By Kolmogorov's zero-or-one law, we need to show that lim sup lSnI/bn < a a.s. Let {"r} be the NV subsequence based on {bn} _ n + a and e. Let A > 0 be arbitrary (we will choose a specified A later). By Lemma 3.7, 3 2 1/2 P(ISnI > 1 bn) P('Sn'/sn > x(2LLsn) ) _<_ 3 exp(-zxZLLs§/M2) + 0, so that (4.3) holds for each x > 0. Also, 50 (4.6) P(IS -S I > 21b ).§ nr nr-l nr P(ISn l > 1 bn ) + P(ISn l > A b ) = r r r-1 n r-1 P(ISn l/sn > x(2LLs§ )1/2) + P(ISn l/s > 1(2LLs2 )1/2).3 r r r r-1 nr-l nr-1 6 exp(-212 LLsg /M2) . r-1 t t +1 Recall that "r a sup{n : (Zstz‘LLstz‘f/2 e (e r,e r )}. It is clear 2 2 1/2 tr 2 2 that (Zs LLs ) > e so that log 2 + log 5 + LLLs > 2t . It n n - n n - r r r r r follows that log sfi 3-tr 3'r. (4.6) now yields r P(ISn -Sn I > 21bn ).g 6 exp(-212 log(r-1)/M2) r r-1 ' r proving that (4.4) holds for A > M/f2’. It follows that M lim sup IS l/b i — (4e-r2/e-1) a.s. n + 0 n n 1/2 He turn next to the Gaussian case. Laws of the Iterated Logarithm for independent B-valued Gaussian random variables have been proved by Mangano [29] and Carmona and Kono [6]. He have, for example, the following. Theorem 4.4. (Carmona and Kono [6]). Let {Xn} be a sequence of independent Gaussian random variables with values in a Banach space 8. 1/2 Assume that bn + a and that Sn/bn converges in distribution to a (Gaussian) random variable X. He then have 51 (4.7) P(d(sn/(2bn¢(bn))J/2.o) + 0) = 1 and (4.8) P(C{Sn/(2bn¢(bn))1/2} e o) e 1 for each “admissible” function a. .352225? In the above theorem, D is the unit ball of the Reproducing Kernel Hilbert Space determined by L(X), and C(A) denotes the set of cluster points of the set A. An admissible function a is defined as follows: Given a sequence {bn} as in Theorem 4.4 define the sequence {nK} by n1 = inf{n Z 1 l bn s o} smfln31|%3ebn }u32). x-1 "k A strictly positive, increasing function p on (0,.) is said to be admissible if o(bn ) a log K (K 3_2). K 2 If we take bn s s", the function LLx clearly need not be admissible in general. Carmona and Kono show, however, that LLx is 2 admissible if Cln §_sn‘§_C2n for some C1, C2 e (O,-). He shall prove a BLIL for independent B-valued Gaussian sequencs under less stringent conditions than those of Theorem 4.4. Also, we will prove the result for the (usually) inadmissible but more natural function LLx. The following basic inequality is due to Fernique. 52 Lemma 4.5. (Fernique [11,12]). Let X be a 8-valued Gaussian random variable. Then there is a constant N a N(8) such that for each 2 > 0, 2)1/2 (4.9) P(IXl > e(Ele ) 5_3 exp(-e2/N2). Theorem 4.6. Let {Kn} be a sequence of independent B-valued Gaussian n random variables such that s: . El X x.I2 + o as n + -. Then there i=1 exists A e [0,~) such that lim sup lSnl/(Zsfi LL53)“2 = A a.s. n + . ‘Erggf: Exactly the same as that of Theorem 4.3. The results of Kwapien and Fernique yielded BLIL's for Rademacher and Gaussian sequences respectively. One may, in the same way, use the following result of Kuelbs (see DeAcosta [1,2] for related results) to prove BLIL's for other classes of variables. Lemma 4.7. (Kuelbs [23]). Let B be a cotype 2 Banach space with {xn} c B. Let {Yn} be a sequence of i.i.d. real-valued random variables such that EY1 a O and E(exp(BY§)) < a for some 8 > 0. If S s X ijj converges a.s. then E(exp BISIZ) < a . 3‘1 ‘ Proposition 4.8. Let {Yn}, {xn} and B be as in Lemma 4.7. Assume 2 n n n . El X Y.ij2 + a and let Sn - X Y x.. Then there Assume that s jsl J jal JJ exists A e [0,») such that lim sup ISfil/(ZsfiLLs n + o 2 1/2 _ n) - A a.s. Proof: Exactly the same as that of Theorems 4.3 and 4.6. BIBLIOGRAPHY 1) 2) 3) 4) 5) 6) 7) 3) 9) 10) 11) 12) BIBLIOGRAPHY A. OeACOSTA, Exponential Moments of Vector Valued Random Series and Triangular Arrays, Ann. Probability, Vol. 8, 1980, pp. 381-389 0 A. OeACOSTA, Strong Exponential Integrability of Sums of Independent B-Valued Random Vectors, Probability and Mathematical Statistics, Vol. 1, 1981, pp. 133-150. A. BECK, A Convexity Condition in Banach Spaces and the Strong Law of Large Numbers, Proc. Amer. Math. Soc., Vol. 13, 1962, pp 0 329- 3340 A. BECK, On the Strong Law of Large Numbers, Ergodic Theory, F. B. Hright, ed., Academic Press, New York, 1963. A. BECK, D. P. GIESY and P. HARREN, Recent Developments in the Theory of Strong Law of Large Numbers for Vector Valued Random Variables, Theor. Probability Appl. 20, 1975, pp. 127-134. R. CARMONA and N. KONO, Convergence en loi et Lois du Logarithme Itéré pour les Vecteurs Gaussiens, Z. Hahrscheinlichkeitstheorie und Verw. Gebiete, Vol. 36, 1976, pp. 241-267. V. S. CHOH, A Martingale Inequality and the Law of Large Numbers, Proc. Amer. Math. Soc., Vol. 11, 1960, pp. 107-111. K. L. CHUNG, The Strong Law of Large Numbers, Proceedings of the 2nd Berkeley Symposium on Statistics and Probability, 1951, pp. 341-352. R. M. DUDLEY, The Sizes of Compact Subsets of Hilbert Space and Continuity of Gaussian Processes, J. Funct. Anal., Vol. 1, 1967, pp. 290-330. V. A. EGOROV, On the Strong Law of Large Numbers and the Law of the Iterated Logarithm for Sequences of Independent Random Variables, Theor. Probability Appl., Vol. 15, 1970, pp. 509-513. X. FERNIQUE, Intégrabilité des Vecteurs Gaussiens, C. R. Acad. Sc., Paris, Series A, Vol. 270, 1970, pp. 1698-1699. x. FERNIQUE, Régularité des Trajectoires des Fonctions Aléatories Gaussiennes, ecture otes n athematics, Vol. 480, Springer Verlag, Berlin, 1975, pp. 1-96. 53 13) 14) 15) 16) 17) 18) 19) 20) 21) 22) 23) 24) 25) 26) 27) D. GILAT, Every Non-negative Submartingale is the Absolute Value of a Martingale, Ann. Probability, Vol. 5, 1977, pp. 475-481. E. GINE and J. ZINN, Central Limit Theorems and Heak Laws of Large Numbers in Certain Banach Spaces, 2. Hahrscheinlichkeitstheorie und Verw. Gebiete, Vol. 62, 1983, pp. 323-354. B. HEINKEL, On the Law of Large Numbers in 2-Uniformly Smooth Banach Spaces, 1983, to appear in Ann. Probability. B. HEINKEL, Une Extension de la loi des Grands Nombres de Prokhorov, Preprint, 1983, University of Strasbourg. 8. HEINKEL, The non i.i.d. Strong Law of Large Numbers in 2- Uniformly Smooth Banach Spaces, Preprint, 1984, University of Strasbourg. J. HOFFMANN-JORGENSEN, Sums of Independent Banach Space Valued Random Variables, Studia Math., Vol. 52, 1974, pp. 159-186. J. HOFFMANN-JORGENSEN and G. PISIER, The Law of Large Numbers and the Central Limit Theorem in Banach Spaces, Ann. of Probability, Vol. 4, 1976, pp. 587-599. K. ITO and M. NISIO, On the Convergence of Sums of Independent Banach Space Valued Random Variables, Osaka J. Math., Vol. 5, 1968, pp 0 35-48 0 J. P. KAHANE, Some Random Series of Functions, 0. C. Heath and Company, Lexington, Massachusetts, 1968. J. KUELBS, A Strong Convergence Theorem for Banach Space Valued Random Variables, Ann. Probability, Vol. 4, 1976, pp. 744-771. J. KUELBS, Some Exponential Moments of Sums of Independent Random Variables, Trans. Amer. Math. Soc., Vol. 240, 1978, pp. 145-162. J. KUELBS, Kolmogorov's Law of the Iterated Logarithm for Banach Space Valued Random Variables, Illinois J. Math., Vol. 21, 1977, pp. 784-800. J. KUELBS and J. ZINN, Some Stability Results for Vector Valued Random Variables, Ann. Probability, Vol. 7, 1979, pp. 75-84. S. KHAPIEN, Isomorphic Characterizations of Inner Product Spaces by Orthogonal Series with Vector Valued Coefficients, Studia Math., Vol. 44, 1972, pp. 583-595. S. KHAPIEN, A Theorem on the Rademacher Series with Vector Coefficients, Probability in Banach Spaces, Lectures Notes in Math. 526, pp. 157-158, Springer-Verlag, Berlin. 54 28) 29) 30) 31) 32) 33) 34) 35) 36) 37) 38) 39) 40) 41) N. LOEVE, Probability Theory, 3rd. Edition, D. Van Nostrand Co., Inc., Princeton, New Jersey, 1963. G. C. MANGANO, On Strassen-type Laws of the Iterated Logarithm for Gaussian Elements in Abstract Spaces, Z. Hahrscheinlichkeitstheorie und Verw. Gebiete, Vol. 36, 1976, pp. 227-239. J. NARCINKIEHICZ and A. ZYGMUND, Remarque sur la Loi du Logarithme Itere, Fund. Math., Vol. 29, 1937, pp. 215-222. N. B. MARCUS and G. PISIER, Random Fourier Series with Applications to Harmonic Analysis, Princeton University Press and University of Tokyo Press, Princeton, New Jersey, 1981. N. B. MARCUS and J. ZINN, The Bounded Law of the Iterated Logarithm for the Weighted Empirical Distribution Process in the Non-I.I.D. Case. To appear in Ann. of Probability, May, 1984. B. MAUREY, Espaces de Cotype p (O < p.3 2), Sem. Maurey-Schwartz, 1972-73, Ecole Polytechnique, Paris. 8. MAUREY and G. PISIER, Séries des Variables Aléatoires Vectorielles Independentes et Proprietés Géométriques des Espaces de Banach, Studia Math, Vol. 58, 1976, pp. 45-90. E. NOURIER, Eléments aléatoires dans un espace de Banach, Ann. S. V. NAGAEV, On Necessary and Sufficient Conditions for the Strong Law of Large Numbers, Theor. Probability Appl., Vol. 17, 1972, pp. 573-581. S. V. NAGAEV and N. A. VOLOOIN, On the Strong Law of Large NMmbers, Theor. Probability Appl., Vol. 20, 1975, pp. 626-631. G. PISIER, Type des Espaces Normes, Sem. Maurey-Schwartz, 1973-74, Ecole Polytechnique, Paris, pp. III-1-III-11. G. PISIER, Sur les espaces qui ne contiennent pas de l: uniformement, Sem. Naurey-Schwartz, 1973-74, Ecole Polytechnique, Paris, pp. VII-1-VII-19. G. PISIER, Sur la Loi du Logarithme Itere dans les Espaces de Banach, Probability in Banach Spaces, Lecture Notes in Math 526, pp. 203-210, Springer-Verlag, Berlin. G. PISIER, Le Theoreme de la Limite Centrale et la Loi du Logarithme Itere dans les Espaces de Banach, Sem. Maurey-Schwartz, 1975-76, Exposes III, IV and Annexe 1, Ecole Polytechnique, Paris. 55 42) 43) 44) 45) 46) 47) 48) 49) 50) YU. V. PROKHOROV, Some Remarks on the Strong Law of Large Numbers, Theor. Probability Appl., Vol. 4, 1959, pp. 204-208. H. F. STOUT, Almost Sure Convergence, Academic Press, New York, 1974. H. F. STOUT, A Martingale Analogue of Kolmogorov's Law of the Iterated Logarithm, Z. Hahrscheinlichkeitstheorie und Verw. Gebiete, Vol. 15, 1970, pp. 279-290. H. TEICHER, Some new Conditions for the Strong Law, Proc. Nat. Acad. Sci. U.S.A., Vol. 59, 1968, pp. 705-707. N. A. VOLODIN and S. V. NAGAEV, A Remark on the Strong Law of Large Numbers, Theor. Probability Appl., Vol. 23, 1978, pp. 810-813. H. A. HOYCZYNSKI, Random Series and Law of Large NUmbers in Some Banach Spaces, Theor. Probability Appl., Vol. 18, 1973, pp. 371- 377 o ‘ V. V. YURINSKII, Exponential Bounds for Large Deviations, Theor. Probability Appl., Vol. 19, 1974, pp. 154-155. ' J. ZINN, Private Communications. N. DUNFORD and J. T. SCHWARTZ, Linear Operators, Vol. I, John Riley and Sons, New York, 1957. 56