VCJE‘ LIBRARY Michigan State University This is to certify that the dissertation entitled SEQUENTIAL ESTIMATION OF PARAMETERS IN A FIRST ORDER AUTOREGRESSIVE MODEL presented by THARUVAI NARAYANASNAMI SRIRAM has been accepted towards fulfillment of the requirements for BhID degree in 5mm ° «XML Major professor . . x4. LKJLJ) 3% 7— M. ( MS U is an Affirmative Action/Equal Opportunity Institution 0-12771 RETURNING MATERIALS: IVIESI_J Piace in book drop to LJBRARJES remove this checkout from an your record. FINES wiii be charged if book is returned after the date stamped beiow. SEQUENTIAL ESTIMATION OF PARAMETERS IN A FIRST ORDER AUTOREGRESSIVE MODEL by THARUVAI NARAYANASWAMI SRIRAM A DISSERTAION Submitted to Michigan State University in partial fulfilment of the requirements for the degree of DOCTOR OF PHILOSPHY Department of Statistics and Probability 1986 ABSTRACT SEQUENTIAL ESTIMATION OF PARAMETERS IN A FIRST ORDER AUTOREGRESSIVE MODEL by THARUVAI NARAYANASWAMI SRIRAM In this thesis two sequential estimation problems concerning the parameters in a first order. non—explosive. autoregressive model are considered. The first is the sequential point and fixed width interval estimation of the location parameter. The second is the sequential point estimation of the autoregressive parameter when the location is assumed to be known. For the location problem. sequential estimators based on sample mean are proposed using certain stopping times. The sequential point estimator is shown to be asymptotically risk efficient as the cost per observation tends to zero. The sequential interval estimator is shown to be asymptoti- cally consistent and the corresponding stopping rule is shown to be asymptotically efficent as the width of the interval tends to zero. For the estimation of the autoregressive parameter. sequential estimator based on least squares estimator is proposed using certain stopping time. The sequential point estimator is shown to be asymptotically risk efficient as the estimation error tends to infinity. Furthermore. a second order expansion to the expected stopping time is also ii obtained. Finally. a thorough analysis of the 'regret' is also carried out as the estimation error tends to infinity. To my parents and my wife iii W I wish to express my sincere thanks to Professor Hira L. Koul for his continuous encouragement. guidance and especially his patience during the preparation of this dissertation. I would also like to thank Professors James Hannan, Joseph Gardiner and Habib Salehi for serving on my committee. My special thanks go to Professor James Hannan for his careful reading of this dissertation and for giving useful suggestions. which helped greatly in the final presentation of this thesis. I would also like to thank Professor Michael Woodroofe for many helpful discussions. I would like to thank my wife Vidya for her moral support and encouragement during the preparation of this thesis. Finally. special thanks go to Miss Sharon Carson for her excellent typing of the manuscript. iv Chapter TABLE OF CONTENTS Introduction and Summary Sequential Estimation of the Mean of a First Order Autoregressive Process 1.1. Introduction 1.2. Proofs Sequential Estimation of the Auto- regressive parameter in a First Order Autoregressive Process 2.1. Introduction 2.2. Risk Efficiency 2.3. A Second Order Expansion for the Expected Stopping Time. Asymptotic Normality and Uniform Integrability of Standardized Stopping Time . . . . . 2.4. Regret Analysis Bibliography Page 13 28 28 30 40 55 72 CHAPTER 0 INTRODUCTION AND SUMMARY The sequential point and fixed width interval estima- tion has seen a prolification of literature ever since the fundamental papers of Robbins (1959) and Chow and Robbins (1965). See. for example. the book by Sen (1982). the monograph by Woodroofe (1982) and detailed references given in Chow and Martinsek (1982) and Martinsek (1983. 84. 85). Two basic problems of estimation are the following. The first problem is to determine the sample size that minimizes the risk Rn (expected loss) for a suitably defined loss function. Ln' The second problem is to construct a confidence interval for a parameter 9 of prescribed width 2d and coverage probability 1 - a. for O < a < 1. In either case. the best fixed sample size procedure (BFSP). say no. possessing the desired property generally depends on the underlying nuisance parameter(s). Therefore. the sample size cannot be specified in advance to solve these problems. In order to overcome this dependence on nuisance parameter(s) it is customary to define a sequential sampling rule (often called stopping time) similar to the rule considered by Robbins (1959). Essentially. this rule. say t. samples observations sequentially. updates a suitable estimator of nuisance parameter(s) in BFSP and and stops sampling as soon as the number of observations exceed that of the estimated BFSP. 1 Thus one is led to solve these problems using sequential procedures. In the point estimation problem. the performance of a stopping rule t is usually measured by (i) the risk efficiency: Rt/Rn . and 0 (ii) the regret: Rt - Rn . 0 where Rt is the risk due to using the stopping time t. In the interval estimation problem. the performance of t is usually measured by (i) the consistency: P[O € It]' and (ii) the efficiency: E(t/no). where I is the fixed width interval estimator using t. t The problem of sequential point estimation of the mean of independent. identically distributed (i.i.d) observations. with unknown variance and loss equal to linear combination of squared error and sample size. has been considered by Robbins (1959). Starr (1966). Starr and Woodroofe (1969) and Woodroofe (1977) for the normal case; by Starr and Woodroofe (1972) and Woodroofe (1977) for the gamma case; by Vardi (1979) for the poisson case: by Ghosh and Mukhopadhyay (1979). Chow and Yu (1981). Chow and Martinsek (1982) and Martinsek (1983) for the unknown distribution case. In all of these papers the sample mean is used to estimate the population mean. In some of the papers the penalty for estimation error is A units when- ever the cost c for each observation is taken to be one unit and vice versa. Accordingly. the asymptotic results are obtained as A 9 w or c a O. For the Normal case. Robbins (1959) introduced a stopping time based on updated versions of sample variance and obtained some numerical and Monte Carlo results which suggested the boundedness of the regret. Using the same stopping rule. Starr (1966) established that Rt/Rn » 1 as O A 4 m (asymptotically risk efficient). and a much stronger result Rt - Rn = 0(1) as A 4 fl (bounded O regret) has been estblished by Starr and Woodroofe (1969). Furthermore. Woodroofe (1977) has given second order expansions for the expected stopping time and has also shown that R — R = 1/2 + 0(1) as A a m. t no For the Gamma scale and Poisson mean problems. bounded regret has been obtained by Starr and Woodroofe (1972) and Vardi (1979). respectively. using stopping rules which differ from the rule considered in the normal case and which only makes sense in these special cases. In the gamma case, Woodroofe (1977) has also obtained second order expansions to expected stopping time and to the regret. For the general case. Ghosh and Mukhopadhyay (1979) and Chow and Yu (1981) have proposed sequential procedures using a stopping rule of the type considered by Robbins (1959). They have shown. under certain growth rate conditions on the initial sample size. that their procedures are asymp- totically risk efficient as c a O. Chow and Martinsek (1982) have shown that the regret of these procedures is bounded as A 9 N. Furthermore. Martinsek (1983) has obtained a second order expansion for the regret of these procedures. Martinsek (1983) has also constructed an example where the regret can take arbitrarily negative values. as the distribution of the observations varies (even among symmetric distributions). This shows that for some (nonnormal) distributions and large A. it is better to use sequential procedure when the variance is unknown than to use BFSP when the variance is known.. In effect. the sequential procedure does better by being sensitive to characteristics of the distribution other than the variance. As mentioned above. all of the above papers use the sample mean to estimate the population mean. even though sample mean is not necessarily the best estimator. To improve things even further. Martinsek (1984) has proposed sequential procedures for choosing an estimator (from the class of trimmed means) as well as sample size. when the distribution of the observations is symmetric about the parameter of interest. At this point. it should be mentioned that results of asymptotic risk efficiency. in symmetric distribution case. have also been obtained by Sen (1981) and Jureckova and Sen (1982) using M-. L-. and R- estimators of location. rather than the sample mean. In an elegant work by Sen and Ghosh (1981). asymptotic risk efficiency has been established for sequential estimation of estimable parametric functions using U-statistics based on i.i.d observations. This result. indeed. generalizes the results of Ghosh and Mukhopadhyay (1979) and Chow and Yu (1981). In a paper by Finster (1983). independent obser- vations are generated via a general linear model with normal errors and the slope parameter is estimated sequentially using the least squares estimator. Using the normality of errors and independence in the model. he shows that the regret of his sequential estimator is bounded. As a last reference we cite a paper by Lai and Siegmund (1983), where. for a first order. non-explosive. autoregressive process with unknown autoregressive parameter B e [-1.1]. it is shown that if the data are collected according to a particular stopping rule. then theleast squares estimator of B is asymptotically normally distributed uniformly in B. It should. perhaps. be pointed out here that Lai and Siegmund (1983) do not discuss the sequential point estimation of B. In this thesis we consider the problem of sequential estimation of parameters in a first order autoregressive model. More precisely. consider the first order. non-explosive. autoregressive process .(0.1) X1 - u = B(xi-l - u) + 81. i = 1. 2.... where -¢ < u < m. IBI < 1 and 81. 82.... are i.i.d. according to some unknown d.f. F. with E81 = O and O < E8? = 02 < w. The initial state X0 is a random variable (r.v.). which is assumed to be independent of 2 0 considered in this thesis deals with sequential estimation ((1. i 2 1}. with 0 < EX < w. The first problem of the location parameter p. The second problem deals with the sequential estimation of the autoregressive parameter B when u is known. Chapter 1 discusses the sequential estimation of p in (0.1). Section 1.1 discusses the sequential point estima- tion of u. using the sample mean R and the loss function that is squared error plus cost per observation. Sequential fixed width confidence interval for u is also given in this section. Theorem 1.1 gives the asymptotic risk efficiency of the sequential point estimator and the asymptotic normality of the standardized sequential point estimator. Theorem 1.2 gives the asymptotic consistency and efficiency of the sequential fixed width interval estimator. The proofs of these theorems are given in Section 1.2. Chapter 2 discusses the sequential point estimation of B in (0.1) when u is known. In Section 2.1. the parameter 5 is estimated sequentially using the least squares estimator subject to a loss function that is squared error plus sample size. Here the cost per observation is assumed to be one. but there is a penalty for estimation error. In Section 2.2. Theorem 2.1 gives the asymptotic risk efficiency of the sequential estimator of B. In Section 2.3. Theorem 2.2 gives a second order expansion for the expected stopping time and Theorem 2.3 gives the asymptotic normality and the uniform integra- bility of the standardized stopping time. Section 2.4 contains the ‘regret' analysis of the proposed sequential estimator of B. Although the proofs of the above theorems use methodologies similar to those of Sen and Ghosh (1981). Chow and Yu (1981). Chow and Martinsek (1982) and Woodroofe (1982). it should be noted that. due to lack of independence between observations. none of the results in these papers can be directly applied to prove these theorems. However. in these models we can exploit the basic structure to provide us with some martingales and reverse martingales. which enable us to obtain uniform integrability of various stopped sums that are so crucial for the analysis. Here we also use some of the almost sure convergence results of Lai and Siegmund (1983). Finally. we conclude this introduction by listing some of the intermediate results which we obtain as a by-product of our analysis and which are interesting in their own way: (a) In Chapter 1. we obtain (non—sequential) Lp-rates of convergence for the sample mean and the autocorrelations. (see Lemma 1.2). which are extremly crucial for Theorem 1.1 and which. I believe. are not available in the literature. (b) In Chapter 2. we obtain the uniform integrability of the standardized least squares estimator. which is one of the key tools for all of the theorems in this chapter (see Lemma 2.2). We also obtain some results which guarantee that the standardized least squares estimator satisfies the well known Anscombe's condition. which inturn enables us to establish the asymptotic normality of the standardized and stopped least squares estimator. (See Lemma 2.5. Corollary 2.1 and the equation (2.2.36). (c) In Section 2.3. we state and prove a lemma which is a slight extension of Theorem 2 of Hagwood and Woodroofe (1982) and Theorem 3 of Lai and Siegmund (1979) (see Lemma 2.7). (d) In Section 2.4. we obtain Lp - boundedness of certain standardized and stopped martingale and stationary sum. (See Lemma 2.15 and 2.16). We also obtain the first and second moments of certain stopped martingales (see Lemma 2.17) which are similar to the Lemma in Chow and Martinsek (1982). CHAPTER 1 SEQUENTIAL ESTIMATION OF THE MEAN OF A FIRST ORDER AUTOREGRESSIVE PROCESS 1.1 INTRODUCTION Consider the first-order. non-explosive. autoregressive model (1.1.1) X1 - u = B(X1_1 - u) + 81. i = l. 2..... where —@ < u < 0. IBI < 1 and £1. 82,... are i.i.d. according to some unknown d.f. F. with E81 = O and O ( E8? = 02 < w. The initial state X0 is a random variable (r.v.). which is assumed to be independent of 2 0 Note that the model above can be written as {81. i 2 l}. with O ( EX < N. i 1 1 1-3 (1.1.2) X — p = B (X - u) + 2 B 8 . i = 1.2..... i 0 3-1 j POINT ESTIMATION OF E. Given a sample of size n. one _ -1 n wishes to estimate u by the sample mean Xn = n 2 X . i 1 subject to the loss function. (1.1.3) Ln = a(Mn-u)2 + on. a > O. c > 0. where c is the cost per observation. The object is to minimize the risk in estimation by choosing an appropriate sample size. Now. by Proposition 1.1 proved below. 1 (1.1.4) E(Rn-u)2 = n-laz/(l-B)2 + o(n- ). as n a w. 10 Therefore. if a and B are known. the risk. -1 1 (1.1.5) R = E Ln = n a02/(1-B)2 + cn + o(n- n ) is approximately minimized by the BFSP -1/2 a (1.1.6) 1’2a/(1-13) 30 no with corresponding minimum risk (1.1.7) Rn a 2cno. 0 However. if either a or B are unknown the BFSP cannot be used. and there is no fixed sample size procedure that will achieve the risk Rn . For this case. we now describe a O sequential procedure for choosing a sample size whose risk will be close to Rn for small 0. 0 Let m (23) be an initial sample size. h(>0) be a suitable constant to be defined later on and define the stopping rule N. in analogy with by no. (1.1.8) N = inf {n 2 m: n 2 cul/zal/2 [3n A —h lll-Bnl + n ]}. where. A n-l n 5 = gl 2. and that 2 h e (O.LE%%l-) ghere h is as in (1.1.8). Then. s c 4 0. (1.1.14) N/n0 4 1 a.s.. (1.1.15) BIN/no - 1| » 0 (1.1.16) «N (RN- u) ”a x(o.a2/(1—p)2) and. (1.1.17) RN/Rno » 1. HE M 1.2. (FIXED-IIQIE INTEREAL ESTIMATION). gage; e it s f The re 1.1. as d 4 0 (1.1.13) T/k » 1 a.s.. O 13 (1.1.19) P[u 6 IT] 4 l - a and (1.1.20) E(T/ko) 4 1. ghere k0 is as in (1.1.12). REMARK: The stopping time T also appears in Subramanyam (1984). where (1.1.8) and (1.1.19) are proved. under somewhat different conditions. However. (1.1.20) is a new result. 1.2. [30028. The following representation and observation will be repeatedly used in the proofs below. First. from (1.1.2). 11 (1.2.1) in-p = [(xo-u)B(1-B“) + Jiltjtl-B“ Secondly. (1.1 2). Elrllp < m. Elxolp < w for p > 1. and - +1 3 )J/n(1-B)- the Minkowski inequality yield 0 Before proving Theorems 1.1 and 1.2 we state and prove (1.2.2) nxi — pup g ux - pup + uzlup/(l-lpl) < m. 1 2 1. a proposition which justifies (1.1.4). [Roroslrgon 1.1. F a1 |p| < 1. all real u an a l n 2 l. 2 2 + n‘152(1-n“)2E(x0 - u) + n'102(1-52)‘1(1-b“)tn nil-pizniin-u12 = a n+2 - 23 - 52]. l4 [ng{. Square the representation (1.2.1) for in - u and use the independence of X0 and {81. i 2 1} to get the required result. 0 For the sake of completeness. we state the following inequalities and a result which will be repeatedly used in this thesis. A. Bgrkhglde; Ineguality. (See Chow and Teicher. 1978. Theorem 11.2.1.) If p 6 (1.”) and f = (fn; n 2 1} is an Lp-martingale. 3/ 3 2/(p-1)]-1 and BD = 18p l2/(p-1)1/2 then with Ap = [18p Apusnu)"p g urn"p g BpNSn(f)flp . n 2 1/2 where Sn(f) = 1:1(f1-f1_1) and f0 = 0. D B. MarginkIeIIQg-Zygmugd iggguaIity. (See Chow and Teicher. 1978. Corollary 10.3.2.) If (xn; n 2 1) are i.i.d. with 2x1 = o. Elxllp < m and p/2 n p 2 2. then El 2 Xilp = 0(n ) as n 4 m. 0 i=1 C. 's a ma ine u it . (See Stout (1974) Corollary 3.3.2.) Let (TR; k 2 1} be a martingale or a nonnegative sub- martingale. If p > 1 and EITkIp < w for each 15 k 2 1. then E max ITkIp S (‘ETQP EITnIp for each n 2 1. n 1$k$n P D. Result. (See Chow. Robbins. Siegmund (1970). Example 2.1(d).) Let Y1. Y2. be i.i.d. with EIY1I( n. Then _ n Yn = 2 Yiln is a reverse martingale with respect to i=1 9n = 0{ n. Yn+1.... }. U E. a im t the e erse martin al (Tk; k 2 n} (See Chow. Robbins and Siegmund (1970). display (4.39) and Stout (1974). (second) Lemma 3.3.1.) Let Y Y be i.i.d. with EIYIIp < w for some 1. 2.... p > 1. Then E sup ITkIp 5 (BET)p EITnIp for each n 2 1. D an The proofs of Theorems 1.1 and 1.2 depend on a series of four lemmas. the first of which will be stated without proof. In the first three lemmas all the limits are as n 4 m. LEMMA 1.1. I; f maps Rk intg Rm and f I; ggntigugug at a. s 6 (0.“) and {Xn} is a geguence of random k-vectgrs such that P[uxn - a" 2 5] = 0(n“s) fgr eyery 5 > o, 16 P[ur(xn) - f(a)N 2 e] e 0(h‘s) {gr eyegy e > o. n In order to state the next two lemmas. we need to n introduce the following notations. Let 8n = n.-1 2 8d J=1 and 3n = a{Xo. £1. £2..... 8“} for n 2 1. Define for k = 0,1. -1 n-k - - Ck.n = n 1:1 (Xi-xn)(xi+k-xn)’ and (1.2.3) x -l n-k Ck.n = n 1:1 (xi- ”)(Xi+k- “)° Note that (1 2 4) E = c /c and 32 - 0 (1-32) ' ' n 1.n O.n n ' O.n n ' Lemma 1.2 below gives Lp rates of convergence of in to u. and of the autocovariances C and C to O.n 1.n 02/(1-B2) and B02/(l-B2). respectively. Lemma 1.3 below A deals with rates of convergence in probability of B to B n and 02 to 02. n LEMMA 1.2. It Eltllzp < m and Elxol2p < a. then ’1/2). ii p 2 1. '1’2). ii p 2 2. (1.2.5) "in-pa - 0(n 2p - (1.2.6) uc - 02/(1-32)Np = 0(n and (1.2.7) "C1.n' B02/(1-B2)Np = 0(n O.n '1’2). ii p 2 2. 17 Prggf: Assume w.l.o.g. that u = 0 and 02 = 1. From (1.2.1) and the Minkowski inequality - -1 - -1 n n-j+1 (1 (3)"an2p g n uxou2p + nznu2p + lulu2p n 3:1IBI = 0(n-1,2). by the Marcinkiewicz-Zygmund (M-Z) inequality (see Inequality B above) and IBI ( 1. As for the assertion (1.2.6). use the definitions (1.2.3) of C0 n and G; n to write - 62/(1-62) e c; 2 (1.2.8) c - 62/(1-62) - in . O.n Now. use (1.1.2) to write .n n i c; - 62/(1-52) = n'1 2 [621x2 + 2 szi’Jlaz — 02(1-B2) .n 0 J 1:1 1:1 1 (1.2.9) + 2 2 62"J‘k ejak + 2xop‘( 2 pi'J: )3. 1$j 0. (1.2.20) P[|co n - 02/(1-B2)I > e] = 0(a‘P/2). (1.2.21) P[|CLIn - a 62/(1-52)| > e] = 0(n'2’2). and (1.2.22) P[I;:/(l-Bn)2 - 62/(1-6)2| > e] = 0(n'p’2). lhglg 3: and En arg as in (1.2.4). ELQQL: Lemma 1.2 and the Markov inequality yield (1.2.20) and (1 2.21). By (1.2. 4). 3,2,(1-1'5n)“2 = co, n[1- (c (20 n121/[1-(c co nn2. Hence (1.2.22) follows from (1.2.20). (1.2.21) and Lemma 1.1 applied to the function f(x.y) = x[1-(y/x)2]/[1-(y/x)]2 continuous at (02/(1-B2). B02/(1-B2)). D The next lemma gives a rate on the tail behavior of the stopping rule N. which is crucial for the proof of Theorems 1.1 and 1.2. Before we state the lemma we need to introduce the following notations. Let = (a/c)1/2. n1 = [bl/(1+h)], (1.2.23) r12 = [no(1-€)] and 113 = [no(l+€)] for 0 < e < 1. Henceforth all unidentified limits are taken as c 4 0. 21 L555; 1.4. Agsgmg the gggggtgggs g; Ihggzgm 1.1. Then r9; eygzy e > o gng to; s = p/2. (1.2.24) P[N g n2] = 0(e(2'1)/[2(1+h)]). Eng (1.2.25) 2 P[N > n] = 0(c n2n3 (s-1)/2). [£221. From (1.1.8). N 2 n1. Moreover. P[ N 3 n2 ] g P[ 3n/Il-Bnl g b-ln for some n1 5 n g n2 ] S P[ ;:/(1-Bn)2 S b-zng for some nl S n S n2] S P[ 3:/(1-Bn)2 S (1-€)202/(1-B)2 for some nl S n S n2 ] (by (1.1.6)) (1.2.26) 3 2 P[I 3i/(1-Bn)2-a2/(1-p)2|2e(2-e)a2/(1-p)2 J. n=n1 Now use Lemma 1.3 and (1.2.23) to get (1.2.24). As for (1.2.25). it follows from (1.1.8) that. for n 2 n3 P[N > n] g P[3n/|1-§n| > b‘ln - n‘h] (1.2.27) 3 P[3n/|1-Bn| - a/ll—BI > h’1(n3-no) - ugh] Choose 'c' small enough so that ea/(l-B) - {(1-B)c1/2/[aal/2(1+e)]}h > 06/2(1-B). Then P[N > n] g P[3n/|1-§n| - a/Il-BI > a€/2(1-B)] 22 2 s PtlSfi/Il-Enl2-a2/(1-212I > a 6214(1-5121. Now argue as for (1.2.24) to conclude (1.2.25). 0 Before we proceed to prove Theorem 1.1 we quote a definition and a lemma from Woodroofe (1982). which are not only useful here but also in Chapter 2. DEFIMITION 1.2. A sequence En. n 2 1 of random variables is said to be uniformly continuous in probability (u.c.i.p.) if for every 6 > 0 there is a 5 > 0 for which (i) P{ max OSkSnB The u.c.i.p. condition is often called Anscombe’s condition. lfn+k - Enl > e} < e for all n 2 1. Ngtg 1. En 4 £(<¢) a.s.. as n 4 0. implies (i). LEMMA 1.4V. (See Woodroofe (1982). p. 10). If Yn’ n 2 1. and Zn. n 2 1. are u.c.i.p.. then so is Y + Z . n 2 1. If in addition Y . n 2 1. and Z . n 2 1. n n n n are stochastically bounded (tight). and if p is any continuous function on R2. then 0(Yn.Zn). N 2 l. is u.c.i.p.. In what follows. A = [n2 < N < n3]. B = [N S n2] and F and F denote the indicator and the complement of a set F. respectively. D = [N 2 n3]. Also. 1 ERQOE QF TMEOREM 1.1. From Lemma 1.3 and the Borel- Cantelli Lemma. 01 n » 662/(1-52) a.s.. and 23 C 4 02/(1-B2) a.s.. as n 4’”. From this and (1.2.4) O.n (1.2.28) Bn 4 B a.s.. and a: 4 02 a.s.. as n 4 m. Therefore it follows that N < w a.s. Also. since N 2 n N 4 m a.s.. Hence from (1.2.28) 1. (1.2.29) B" 4 B a.s. and 3: 4102 a.s.. But by definition -1/2 1/2 “ A c a aN/Il-BNI S N. and -1/2 1/2 “ c a c -h N g aN_1/I1-BN_1| + (N-l) + m. Therefore from (1.2.29) and (1.1.6) N/no 4 1 a.s.. As for the Ll-convergence (1.1.15). observe that (N/no - 1)- S 1. Therefore by the dominated convergence theorem and (1.1.14) E(N/no - 1)” e 0. Moreover. since (N - no)+1B = 0 we have + + (n3-n0) P(D) + 3) no + 1 E(N/no - 1) = E(N/no - 1) 1A + 33-E(N-n S 2€ + 0(1). by (1.2.25) and (1.2.23). Since 6 is arbitrary. (1.1.15) follows. As for (1.1.16). recall (1.2.1). By example 1.8 of Woodroofe (1982) {4; En' n 2 1} is u.c.i.p. By the Markov inequality n-j+1 n-j+lg p/2 n P(In-1/2 2 1 1:1 5 "MS tjl > e) g e'p "M8 p1 IIM'u" 'Cb n n 1 J ( w. 24 n since N 2 Bn-J+1 ‘JHP S K (constant). by the Minkowski i=1 inequality. Therefore by the Borel-Cantelli lemma n 11-1,2 2 fin-J+l fj 4 O a.s. as n 4 m. i=1 Also. (1(1-13“)(1-.3)"1(xo - u)/\/I-i- .. o a.s. as n 4 co. From these. Note 1 following Definition 1.2 and Lemma 1.4W it follows that (1.2.30) {¢;(Mn - u). n 2 1} is u.c.i.p.. The required result now follows from (1.1.14). (1.1.11) and the Anscombe's theorem (see Woodroofe (1982). Theorem 1.4). It remains to show that N is asymptotically risk efficient. Assume without loss of generality that u = 0 and 02 = 1. Write -2 RN/Rno _ (a E xN + c E N}/2cno. From (1.1.15). it suffices to show that, -2 (1.2.31) a E XN/cno 4 1. In order to establish (1.2.31) it suffices to show that -2 (1.2.32) a E XN IB /cn0 4 O and -2 (1.2.33) a EXN lAUD/cno 4 1. In what follows we will often use the fact that -1/2 2 (cn0)-1 = 0(c ) and cno = 0(1). 25 Now. clearly from (1.2.1) -2 -2 2 (1.2.34) xn g 4(1-B) 2[(xo /n)2 + 1n + (J21 a As for (1.2.32). by the Holder inequality and N 2 nl n-j+1‘J/n)2]. we have -2 -2/p (cno)' 2(xo /N)l B; (eno)' «X0 "2 {2(3)} = ux -1/2) c1/(1+h) [0(c(p- 2)/4(l+h))](p-2)/p 13:10.10 0" °(° (1.2.35) 4 0. since h < [(p-2)2/2p]. By the Holder inequality. the maximal inequality for the reverse martingale (in. n 2 n1} (see Result D and E above). the M-Z inequality and (1.2.24). —1 -2 -1 — 2 1-2/ (cno) E max In 1B 5 (cno) n max anupo(P(B)) P nlSnSn2 nlSnSn2 S (cno)'1(;§T)2 "inlu: {P(B))(P‘2)’P -1/2 = 0(c )O(nIl)°O(c(p-2)2/4p(l+h)) (1.2.36) 4 0. Furthermore by the Schwarz inequality and IBI ( 1. for all n 2 1. n n _ _ n In-l 2 Bn~j+l ‘le S (n 1 2 52(n j+l))(n 1 2 :3). J31 J=1 1:1 11 (1.2.37) 3 n'1(1-132)"1(n"1 2 :2). i=1 Therefore. N (I—B2lccno) 1 E(N 1121 B" 3+1: :1)2 13 1 1 -1 2 S (cno)_ n; E max (n 2213 nlSnSn2 j=§12J 26 n S (c:no)-1 n11" max no1 2 (in l2{p(3)}1’2/P nlSnSn2 j=1 p n1 2 _2_ -1/2 1/2(l+h) -1 2 (13-2) /4p(1+h) s (p_2) 0(c )c un1 1:1 :an/20(o ) (1.2.38) 4 O. by the Holder inequality. the maximal inequality for the n 2 82. n 2 n ). the Minkowski J=1 J 1 inequality. (1.2.24). Combining (1.2.34) with (1.2.35). reverse martingale (n-l (1.2.36) and (1.2.38) proves (1.2.32). As for (1.2.33). since n3 2 (a/e)(1-s)’2 (by (1.1.6)) and P(AUD) 4 1 (by Lemma 1.4). it follows from (1.1.16) and (1.1.14) that 2 2 N AUD’220 ”a 21' where 22 is a chi-square r.v. with one degree of freedom. Furthermore. (1.2.39) a i l 2 1n-2 0 AUD S (220)- 2 (1.2.40) . 4 O. -1 2 (cno) E(Xo/N) 1 2x Also. since 32 1 /cn g (on2)‘1( 2 z IVE )2 N AUD 0 2 1_1 1 o N .J' 2 :01 2‘, 0 ()(m i 20) we have that -2 (1.2.41) {8" lAUD/cno} is 0.1. by Lemma 5 of Chow and Yu since (N/no} is 0.1. (by (1.1.15)). Finally. by (1.2.37) and arguments similar to 27 (1.2.38) we have. for s ) 1. N (1_B2)-s(cno)-s E(N-l 2 N—j+1‘ J l J)2s 1nun ’S ‘S 1;le Elall2s(cno) n2 V\ (1.2.42) 0(1). Combining (1.2.34) with (1.2.40). (1.2.41) and (1.2.42) we have that -2 (1.2.43) {a XN lAUD/cnO} The required result now follows from (1.2.39) and (1.2.43). is U.I. Hence the theorem. 0 ZMQQE_QE_IM§QM§M 1.2. Since the stopping times N and T are similar. arguments given for assertions (1.1.14). (1.1.15) and (1.1.16) also show that T/ko 4 1 a.s.. E(T/k0) 4 1 a.s.. as d 4 O. and ¢T(iT-u)/w ”a N(O.1). as d 4 0. where 1 = a/(l-B). Therefore. P[p€IT] = P[\/TI)-(T-ull'r g dVT/w] 4 1 - a. as d 4 0. Hence the Theorem. 0 CHAPTER 2 SEQUENTIAL ESTIMATION OF THE AUTOREGRESSIVE PARAMETER IN A FIRST ORDER AUTOREGRESSIVE PROCESS 2.1 IN ODUCT ON. In this chapter we consider the sequential estimation of B in the model (1.1.1) when p is assumed to be known. and hence taken to be 0. For convenient reference later on. we restate this modified model here. Thus one observes a process (X1) such that (2.1.1) X1 = BX1_1 + 8 1 = l. 2..., i I where B. X0 and 81. 82.... are as in (1.1.1). Given a sample of size n. one wishes to estimate B by the least squares estimator n n A 2 B = 2 X _ X )/( 2 X _ ) n i=1 i 1 1 i=1 i 1 n n 2 (2.1.2) = B + ( 2 Xi_181)/( 2 Xi_1). 1:1 1:1 subject to the loss function -12 2 c 2 (2.1.3) Ln - A n 1:1 X1_1(Bn-B) + n. where A ()0) is the cost of estimation error. The object is to minimize the risk in estimation by choosing an appropriate sample size. Recall from Theorems 4.2 and 4.3 of Anderson (1959) that for 1/2 2 c 2 (2.1.4) i21Xi_l (Bn-B) 4d N(0.a ). as n 4 m. 28 29 Moreover. by Lemma 2.2 below. under certain regularity n A conditions. { 2 Xf_1(Bn-B)2: n 2 l} is uniformly i =1 integrable (U.I). Therefore. from this and (2.1.4) B A (2.1.5) B 2 X? 1(fi -B)2 = 02 + 0(1), as n 4 w, i=1 - n Now. if a and B are known. then the risk (2.1.6) Rn = E Ln = A n'1 02 + n + 0(n-1) is approximately minimized by the BFSP 1/2 (2.1.7) no 2 A a with the corresponding minimum risk (2.1.8) R 2 2A1/2a. no However. if a is unknown then the BFSP cannot be used. and there is no fixed sample size procedure that will achieve the risk Rn . O For this case. we define a stopping rule t. in analogy with no. by (2.1.9) t = inf{n 2 nA: n 2 A1/2 3“} where 11A is an initial sample size depending on A and *2 -1 n . 2 a = n 2 (X -B X ) . The proposed sequential point n i=1 i n i-l A estimator of B is Bt and its risk is E Lt It will be proved in Section 2.2 that this sequential procedure is asymptotically risk efficient. In Section 2.3 a second order approximation for the expected stopping time is obtained. In Section 2.4 the regret Rt - Rn is O analysed. 30 2.2 glgx grfizgxgncx. Ignoggn 2.1. Asgnm . Lgn_ s > 2. thnt Elzll48 < m. 15le48 < a nng El1/(1f+...+¢§)|2s < a £9; gnmg nggitive intgggr N. In ndditinn. 1; 11A in (2.1.9) is such that Allah") S 11A = GUI/2) LLLQ 7! € (0.3-2). 21:11.. a. A)» a. (2.2.1) t/Al/za » 1. a.s. (2.2.2) B t/Al/za » 1 nnd (2.2.3) Rt/Rno » 1. The proof of Theorem 2.1 depends on a series of lemmas. the first two of which give the uniform integrability of n 2 . 2 ( 2 X1_1(Bn-B) ; n 2 1}. which was extremly crucial to 1:1 compute E Ln. Before we state the lemmas. we need to introduce the following notations. Let. n n 2 D = 2 X 8 . I = 2 X _ n i=1 i-l i n i=1 i 1 (2.2.4) I = I /n and c = D2/(nI ) n n n Note that 3 - B = D /I n 11 (B - p)2 = c and n n n' n n n' (2.2.5) A _ n 02 = n 1 2 If - Cn i=1 Henceforth all unidentified limits are taken as A a w. 31 Lanai 2.1. L: Elallp < m nnd Elxolp < w fnr P 2 2. than. 1/2 (2.2.6) nun"p - 0(n ) as n a on- finnng. The required result follows from (1.2.17) and (1.2.18). 0 LEgui 2.2. LL 8 2 1. and Elall4s < m. EIX and EIl/(f?+...+8§)l28 < w f9; snme ngsitiye integer M. tnen (2.2.7) sup £|1n(§ -p)2|8 < w. n21 n flgnce. {IIn(Bn - p)2|q. n 2 1} is U.I. 19; all q < s. Znnnfi. From (2.2.5). “ 2 2 (2.2.8) In(Bn-B) - Dn / In' Now. use (2.1.1) to get. for any e > 1, e-1 e-1 2 2 2 8 = 2 (X - BX _ ) 1:1 1 i=1 1 1 1 2 (2.2.9) S (1+IBI) 1) Using the above inequality. we will first show. for any fixed m 2 1. that 25 (2.2.10) E :gpIk/Imk+ll < w. Then use this to show that for M in the assumption. (2.2.11) B sup In/Inl28 < w. n2M+1 By the Schwarz inequality 2 2 (2.2.12) "Du/Inns S IIDn/nllzslln/Inll2s From Lemma 2.1. 32 2 (2.2.13) lan/nll28 = 0(1). as n 4 W. Therefore. the required result will follow from (2.2.11). As for (2.2.10). it follows from (2.2.9) and the Jensen's inequality that for each m 2 1. 2 k k/Imk+1 s (1+IBI) k/(ilei'm) 2 -1 k (2.2.14) g (1+|p|) k 2 (1/B1 m). =1 ’ im 2 where B1 m = 2 ‘J ' j=(1-1)m+l Observe that for each fixed m 2 1. (l/Bi m; i 2 1} forms an i.i.d sequence. Now. by Result D (see Section 1.2) k (2.2.15) {k 1 2 1/31 m. 9k m; R21) is a reverse martingale. i=1 2 O -1 e where 9k = a{£ 2 1/B : 2 2 k}. Therefore. an .m i=1 i.m application of maximal inequality for reverse martingale (see Inequality E Section 1.2) yields. -1 k 23 23 (2.2.16) E sup k 2 l/B s (-—:—) k21 i=1 i.m 28 1 2s 2s Ell/Bl’ml ( 0°. for m 2 M. by assumption. As for (2.2.11). let n 2 N + 1. Then there exists k 2 1 such that k! + l S n S (k+1)M. Now. use the fact that In is I in n to get n/In g (k+1)M/I kM+1 (2.2.17) 5 2M sup(k/I R21 Use this along with (2.2.14) and (2.2.16) to get (2.2.11). kH+1)° Hence the lemma. 0 33 L355; 2.3. Agsume ghnt Elall2s < m 11th s 2 1 an 22;; nA = 0(11’2). 12g; (2.2.18) (It/Allzals; A 2 1) i; U.I.. znnng. Assume w.l.o.g. 02 = 1. Define another stopping time t' by (2.2.19) c' = inf{ n 2 nAi A'1n2 2 n_li¥1‘f ). Then. by Lemma 2 of Chow and Yu (1981) applied to (If: i 2 1). with an. a. bn and A in Chow and Yu (1981) replaced by n2. 2. O and A-1 respectively. we have that 1’2 : A 2 1} is U.I.. (2.2.20) (Iva Now. since t' 2 t we have the required uniform integra- bility. D The next lemma gives a rate of convergence on the tail behavior of the stopping time t. Lfiflfin 2.4. Asgnme tne ngngnt ggnditign 9f Lemma 2.2. Then in; A1/2(1+n) S 11A and € 2 0. (2.2.21) P(t < (l-€)A1/20) = 0(A’(s'1)/2(1*")). nng (2.2.22) P(t > [(1+€)Al/2a] + 1) = 0(A-(s—l)/2(1+n)). finnni. Assume w.l.o.g. 02 = 1. Let s > 1. e < 1 and 6 = 1 - (1-e)2. Then by the definition of t in (2.1.9), n P(t < (1-e)A1/2) = p(n'1 2 1n2 a? - cn g A 1-1 2 for some nA S n < (1-€)A1/ ) 34 -1 n 2 2 S P(n 2 81 - C S (1-6) for some nA S n) i=1 n 1 n 2 = P(n 2 (1-8 ) + C 2 6 for some n S n) 1_1 i n A -1 n 2 S P(n 2 (1-81) 2 6/2 for some nA S n) i=1 + P(Cn 2 5/2 for some nA S n) (2.2.23) S P(V 2 nA6/4) + 2 P(C 2 6/2) n2nA n + 2 - 6/4) . Then by Theorem 4.13 of n with V = sup 2 (1-81 n21 i=1 Chow. Robbins and Siegmund (1971). E v‘"1 < 9. Therefore by the moment inequality and 11A 2 A1/2(l+n) (s-l) _ - _ P( v 2 nA6/4) g (g) nA(s 1) I V(3 1) dP {V > nA6/4} (2.2.24) e 0(A-(s-l)/2(l+n)). An application of the moment inequality yields 8 (2.2.25) 2 P(C 2 6/2) S (3) 2 BC3 n 5 n n2nA n2nA 2 s l s s (5) 1 2 -;-) sup Ewen) . n2nA n n21 = 0(n;(s-l)), 35 by Lemma 2.2. The required result now follows from (2.2.23). (2.2.24) and (2.2.25). As for (2.2.22). recall the definition of t' in (2.2.19) and the observation that t' 2 t a.s.. Let N": [(1+€)A1/2] + 1. Then NO. P( t' > N" ) g P((N")’1 2 1f 2 A’1(N")2) i=1 -1 N" 2 2 g P((N") 2 :1 > (1+6) ) i=1 -1 N" 2 2 g P((N") 2 (81-1) > (1+6) - 1). i=1 Let h = (1+6)2 - 1. Then. for V defined above with '5’ replaced by '2h' and 1 - 8? replaced by 8? - 1. the moment inequality implies N" 2 P( 2 (ti-l-h/Z) ) N"h/2) S P(V ) N"h/2) i=1 {v > N"h/2). = 0(A-(s-l)/2). by another usage of the Chow. Robbins and Siegmund theorem. The following lemma and corollary deal with the well-known Anscombe's condition. These are not only used in the proof of Theorem 2.1 but also used in the next section for a deeper analysis of our stopping time t. 36 LEHHA 2.5- Assums_ths_mndsl (2-1-1). 1292 {Du/V5. n 2 1}. in gtgnhagticaiiy hgunged ang u.c.i.p.. finnni, Stochastically boundedness follows from Lemma 2.1 and the Markov Inequality. To prove u.c.i.p.. observe that for k 2 0. n 2 1 2213-- 22 g -l D - D + (1 - VE7TEIET) 25 Vn+k 4; V5 n+k n n If k S n5. then the Ll-norm of the second term on the right is bounded by [1-(1+6)-1/2 DD Ju33u1 . n which tends to zero as 6 4 O. uniformly in n 2 1. by Lemma 2.1. For the first term. the Doob's maximal inequality applied to the martingale {Du} (see Inequality C. Section 1.2.) and the independence of X and 8k for j < R J yield ¢E 16 2 P{ max In - D | > e ——) g -—— EID - D | OSkSnG n+k n 2 €2n n+[n6] n n+[n6] = 12-— 02 2 EX?_1 G n i=n+1 0(6). as 6 4 O. uniformly in n 2 1. by (1.2 2). That Dn/JE. n 2 1 is u.c.i.p. now follows easily. 0 37 QQBQLLARI 2-1- Asgsms_2hs;mgggl (2-1-1)- 1229 ( Dn/Ill2. n 2 1}. 19 u.c.i.p. nnd stgchasticnily bgunged. n finnnfi. From Lemma 2.5 above. we have Dn/JH. n 2 1. is u.c.i.p.. Moreover. from the proof of Theorem 4.1 of Lai and Siegmund (1983) we have (2.2.26) in » 62/(1-82) a.s.. as n a m. and therefore is u.c.i.p. and stochastically bounded. Hence the corollary follows from the Lemma 1.4V of Chapter 1. D Before we proceed to prove Theorem 2.1 we need to introduce the following notations. Let N' = [(1—€)A1/20]. N" = [(1+€)A1/2a] + 1 (2.2.27) F={N' S t SN"). EROQE OF IEEQBE! 2.1. Assume w.l.o.g that 02 = 1. From Theorem 4.2 of Lai and Siegmund (1983) we have that a: 4 1 a.s.. as n 4 w. Use this, the definition of the stopping time t in l/2(1+n) (2.1.9). the fact that t 2 A and similar arguments as in the proof of Theorem 1.1 to get t ( w a.s. (2.2.28) t 4 m a.s. and t/All2 4 l a.s.. 1/2 Moreover. because {t/A ; A 2 l} is U.I (by Lemma 2.3). we have that E(t/Al/2) e 1. 38 Therefore it only remains to show that t is asymptotically risk efficient. For this write. 1/2 Rt / Rno _ {A E Ct + Et}/2A . From (2.2.2) it suffices to show (2.2.29) A”2 E 6t 4 1. For this. clearly it suffices to show that. (2.2.30) AV2 E 8t 1_ 4 0. FA and (2.2.31) A1/2 E ct 1F 4 1 Consider (2.2.30). Now. N'-1 1/2 1/2 A E ct 1{t < N.) s A 2 E c 1{t = n} “=11 A N'-l g All2 2 ucnuS Pl’l/s(t=n n=n A N C S A1/2 ( 2 E 0:)1/3 P1—1/s{t<(1_€)A1/2} n=n A 0 l/s g A“2 sup unc u ( 2 n’s) Pl’l’s(r < (l-€)A1/2} n 3 n21 n=nA = A“2 0(n;(s'l)/s) 0(A’(s’1)2/23(1+")) (by Lemma 2.2) 4 0. since 0 < n < s - 2. Similar but simpler arguments yield 1/2 (2.2.32) A E c o. t 1{r > N") 2 39 As for (2.2.31). it suffices to show (2.2.33) (AV2 CtlF. A 2 1) is U.I. and 1/2 2 (2.2.34) A CtlF 4a 11 For (2.2.33). use In is I in n. the Holder inequality and choose A large so that 11A 2 R + 1 to get A“2 E 0:1F g A8/2 E max 08 N'SnSN" “ s/2 S -A-- EIN’II .I8 max D28 28 N n (N!) NOSHSNN As/2 8 25 S --—— fl sup(n/I ) H n max D H (N.)2s n2M n 2s N'SnSN" n 45 A9/2 9 $ (N715; 0[(N") J (2.2.35) = 0(1). by the Doob’s maximal inequality for the martingale {Dn} (see Inequality C. Section 1.2). Lemma 2.1 and (2.2.11). As for (2.2.34). it follows from (2.1.4). Corollary 2.1. (2.2.1) and the Anscombe's theorem that (2.3.36) c ot *a |z|. where 2 ~ N(0.1). Therefore by (2.2.1). the Slutsky theorem and the fact that 1F 4 l a.s. we have the required result. Hence the theorem. 0 40 2.3. A Second order expansion for the expected stopping time. Asymptotic normality and Uniform integrability of standardized stopping time. One of the main results of this section is Theorem 2.2. where we obtain a second order expansion to the expected stopping time EtA. Since the proof of this theorem has its roots in the non-linear renewal theory developed by Lai and Siegmund (1977. 1979) and Hagwood and Woodroofe (1982). we shall give a brief description of their papers. In the i.i.d set up. Lai and Siegmund (1977. 1979) have developed the non-linear theory to analyse stopping times that stop the first time a perturbed random walk exceeds a boundary. Under the condition that the perturbation sequence is slowly changing (see below). besides some other regularity conditions. Lai and Siegmund (1979) have obtained a second order expansion to the expected stopping time. as the boundary tends to infinity. Later. Hagwood and Woodroofe (1982) have shown that the second order expansion obtained by Lai and Siegmund (1979) holds under an alter- native set of regularity conditions. For an excellenct survey of these results. see Woodroofe (1982). In view of the existing literature mentioned above. we first show that our stopping time t also stops the first time a perturbed random walk exceeds a boundary. Here the perturbation sequence is also shown to be a slowly changing sequence. Once this is done. although many of the results 41 and methodologies developed by Lai and Siegmund (1977. 1979) and Hagwood and Woodroofe (1982) can be used. yet their main theorems which guarantee a second order expansion cannot be directly applied. However. a careful analysis of their proofs yields an extension of their main theorems. This extension is stated and proved as a Lemma below. Using this Lemma we prove Theorem 2.2. Before we proceed with the analysis of stopping time t we shall give a definition of a slowly changing sequence (see Woodroofe (1982). Chapter 4). 22212111911124. Asequence {E slowly changing if: n' n 2 1) is said to be (1) (En; n 2 l) is u.c.i.p (see Definition 1.2) and (ii) nm1 max I51] 4 0 in probability. as n 4 4. lSiSn N2££.2- En/n 4 0 a.s. as n 4 4 implies (ii). See the Definition 1.2 for a sufficient condition for (i). Now. for a deeper analysis of our stopping rule t defined in (2.1.9). write (2.3.1) c = inf{ n 2 nA= 2n 2 Al/za ) where Z = n(a/; ). n II By Taylor’s theorem (2.3.2) (Oz/0:)1/2 = 1 - (1/2)(0fi/02-1) + (3/8)A;5/2(;:/02-1)2. 42 where An is a random variable lying between 1 and oi/az. A Now substitute the representation (2.2.5) for a: in the second term of the R.H.S of (2.3.2) to get n (2.3.3) 2 = n - (n/2) n’1 2 (12/62-1) - c /02 n i=1 i n + (3/8)>.'5/2n(32/62—1)2 I1 I1 =Sn+ En, “ 2 2 where s e 2 [1-(1/2)(8 la —1)]. n i=1 i and /2 2 -5 “2 2 2 En - (nCn/20 ) + (3/8))\n n(an/a -1) . Next lemma shows that the En. n 2 1. is slowly changing. This fact enables us to use the nonlinear renewal theory to analyse our stopping time t. Before we state the Lemma we need to introduce some notations. Let /2 2 n(; /02 n 2 -5 (2.3.4) L n e nCn/2a . L n = (3/8)An - 1) Note that En = L n + L Lfififin 2.6. Asgune the nndel in (2.1.1). Ihen (L . n 2 1) nnn (L2 n’ n 2 1} are s wl chan in . 1.n e en . (En. n 2 1) is a 8 s1 wl chan in Ennni. From Lai and Siegmund (1982) (see Theorem 4.2 and its proof) CD 4 0 and a: 4 02 a.s.. as n 4 4. Therefore. 11-1 L 4 0 and n-1 L 4 0 a.s.. as n 4 w, 1.n 2.n since A 4 l a.s. n 43 By Corollary 2.1 and Lemma 1.4W (2.3.5) {L . n 2 1} is stochastically bounded and 1.n u.c.i.p.. From the representation (2.2.5) for 3:. example 1.8 of Woodroofe (1982). (2.3.5). An 4 1 a.s. and Lemma 1.4W it follows that (2.3.6) {L2.n‘ n 2 l) is stochastically bouned and u.c.i.p.. The required result now follows immediately. a ngnigx. Consider the linear model. where (X1. 1 2 l} are i.i.d. (£1. 1 2 1} are i.i.d N(O.02). (x 1 2 1) is indpendent of (11. 1 2 1). Using 1. the least squares estimator of B. the normality of (81} n A 2 and the given independence. one can write 2 (Y1 - BXi) 1:1 as a sum of (n-l) i.i.d Chi-square r.v.‘s. Using this representation. Finster (1983) investigated his sequential point estimator of B by expressing the defining variable of his stopping time as a sum of a random walk and a slowly changing sequence of r.v.‘s. In our problem. we neither need to assume any form of the underlying distribution nor do we have the independence between {X i > 1) and {81. i 2 1}. Nevertheless. by i-l' exploiting the model (2.1.1) and the representation (2.2.5) 44 for a: we are able to express the defining variable Zn in (2.3.1) as a sum of a random walk and a slowly changing sequence of r.v.‘s as evidenced by Lemma 2.5. Corollary 2.1 and Lemma 2.6. This indicates that the normality assumption in Finster (1983) was unnecessary to obtain a second order expansion for the expected stopping time. D In what follows. references to Lemmas. Theorems and assumptions in Woodroofe (1982) will be followed by 'W’. Before we state the main theorem we need to introduce some more notations. Let 1 = inf{n 2 nA: S 2 Al/za} and n (2.3.7) 2 p — E STD/(2 E 310). where To is same as T with 'Allza‘ replaced by '0'. unie 3. The finiteness of p follows from Theorems 4 1 ( m. 2.3V. 2.4V and Corollary 2.2V. if E: Innnngn 2.2. (Secnng 9rder gxnansign) Assume the gnnnn d.f. F is nnn arithngtic. Under the cgnditions 9f 1229229 2-1. 1/2 (2.3.8) B tA - A a = p - (1/2) - (318)2(1f/62-1)2 + 6(1). n; A 4 a. The proof of this theorem will follow from a series of lemmas. The following lemma is a slight extension of Theorem 2 of Hagwood and Woodroofe (1982). which appears as Theorem 4.5 in Woodroofe (1982). For convenience. we will refer to Theorem 4.5 of Woodroofe (1982) and assume all of the relevant material from Chapter 4 of Woodroofe (1982). In order not to confuse with our notations we will add a ~ to all the notations used in Woodroofe (1982). nggn 2.7. Assume the conditions (4.10)W. (4.12)W and (4.16)' nf Theggen 4.5V. Agnnmg. instead of (4.11)W. that ~ ~ (2.3.9) {A = en + L1.n + L2.n on An. n 2 1. gnezg inn ggig An. n 2 1 nng ggngtanig 8n. n 2 1. r as in Thggzen 4.5V. nng Li n nng, Lé n fgr n 2 l are sncn h r ach i = 1. 2 (2.3.10) ( max IL; nI; n 2 1} L. U.I. OSkSn ' m ~ (2.3.11) 2 P(L; n < - n6) ( w £nn_§nnn G. 0 < 6 < p n=l ' (2.3.12) Li.n 4a L1 as n 4 w. I; L1.n nng L2.n‘ n 2 1 e s c an n . F t e d.f. su an SD i§ nnn-nritnentig. nnd Ear-[M3]. 21.29.99. 94" (2.3.13) E ta = (1/u)(a + p — EN — ELi - ELé) + 0(1). a 2122;. A careful analysis of the proof of Theorem 4.5W shows that we only need to prove analogues of Lemmas 4.3V and 4.4V in order to conclude (2.3.13). That is. we need to show that 46 I E; d? = 3N + E(Li) + E(Lé) + 0(1), and B t a a a (2.3.14) 1 Rad? = p + 0(1). as a a w. B a N ~ ~ N N ~ m ~ N where Ba = (N S ta S N". Tb ) N'} n n~ Ak with N' _ k=N ° [GlNa]. N" = [€2Na] + 1 and b = a - eNa. As for (2.3.14). write (it - 3 )1~ = (f; - r; )1~ + L' ~ 1~ + L' ~ 1~ t Na Ba c N B I’Na B 2’"a Ba a a a a a Since 58. n 2 1 is slowly changing. the Anscombe's theorem and t /N 4 1 a.s. implies (EL -EL )1~ d O in a a c N B a a a Probability. as a 4 fl. Also. (E; - 8“ )1~ is dominated t a B a a by H; = max IEi-en I. which is U.I. by condition ~O ~09 a Nagksfla (2.3.10). Moreover. (2.3.10) implies {Li E 1~ ; a 2 0) is a Ba U.I. for each i = 1, 2. Therefore. ((f; - E; )1~ ; a 2 0} t N a a is U.I. Consequently. since P(Ba) = 0(1/Na) we have the first assertion (2.3.14). by (2.3.12). The other asser- tion in (2.3.14) uses the uniform integrability of M; and follows by arguments similar to the proof of Lemma 4.4V. 0 Note 4. In the application of Lemma 2.7 we will take ~ the set A = O. the sample space. for all k 2 1. u = 1. k a=A a. §'=N'.fi"=N".z =0. L' 1.n = L ’ N 1.n a 47 REQAKK. Had we assumed that (Ll.n'L2.n) 4d (L1.L2). Lemma 2.7 would have directly followed from Theorem 4.5V. Thus the main observation here is that one needs only the weak convergence of the marginals. and not those of the Joint distributions. besides of course the other assumptions. in order for (2.3.13) to be valid. We use this Lemma to prove Theorem 2.2 below where we readily have the weak convergence of the marginals and where the Joint convergence is not at all apparent. Finally. this Lemma could be further extended to the case where ~ ~ En = en + L1n + L2n +...+ Lkn on An. n 2 1. for k > 1 and finite. and where each Lin satisfies conditions similar to (2.3.10) - (2.3.12) of Lemma 2.7. D Note that the proof of Lemma 2.7 assumes the results such as Lemma 4.5V. and weak and the moment convergences of fia' Therefore. in order to prove Theorem 2.2 we need to establish these in the present context. The next lemma is similar to Lemma 4.5V. and therefore only a sketch of its proof will be given. In what follows all unidentified limits are taken as A 4 m. 48 LE!!A 2.8. angr tne nnngitinng 91 Ingnren 2.2. £ st dP = Et + 0(1). nhnnn_ F is ng in (2.2.27). ELQQL. Since En 2 O we have that t S T (see (2.3.7)). Therefore. by Note 4 the set F is same as the set Ea defined in Woodroofe (1982) (see also (2.3.14). Also. by Lemma 2.4 /2 (2.3.15) p(f) = 0(1‘1 ). Use this. (2.2.2) and arguments similar to the proof of Lemma 4.5V to get the required result. D Before we state the next lemma. let (2.3.16) w = n32 n n and note that the relation 5n = Bn-l + (xn - fln-lxn-l) xn-llIn 2 ‘ 2 yields Wn = wn-l + (1 - Xn_l/In)(Xn- pn_lxn_l) . Therefore (2.3.17) Vn S wn+1 for all n 2 1. The next lemma essentially verifies (2.3.10) of Lemma 2.7. Lang; 2.9. Aggnng thnt Blzll8 < m. Elxol8 < m and EIl/(lf+...+8:)l4 < 0 f9; sgme nnsitiye integer M. Then (2.3.18) { max L1 n1F; A 2 1 ) in U.I. NOSnSNN o and (2.3.19) { max L2.n1F; A 2 l } ;_ U.I. N'SnSN" 49 nnnng F in ng in (2.2.27). C e t ( max Ean; A 2 1) in U.I. N'SnSN" 2 21nn1. Assume w.l.o.g. a = 1. Argue as for (2.2.33) to get (2.3.18). Now for (2.3.19). -5/2 max L2.n1F S (3/8) max A 1 N'SnSN" N'SnSN" “ F 0 max n(;:/02-1)21F. N a SDSN" where. by the definition of An (see (2.3.2)) and (2.3.17) max h;5l2 1F S 1 + max (n/Wn)5/21F NOSDSNOO NoSnSNN S 1 + 25/2 max (n-l/Wn_1)5/21F N'SnSN" 5/2 g 1 + 0(1) [(N'— 1)/wN._l)] 1F. By the definition of t 1/2 1/2 F C ( (N’- 1) < A [WN._1 /(N'- 1)] ) = G. Therefore. [(N'- 1)/wN._1]5’21F g [Al/2/(N'— 1)]5 1G (2.3.20) = 0(1), Therefore. in order to show (2.3.19). it suffices to show 50 (2.3.21) { max n(;:/02-1)21F; A 2 1 ) is U.I. N 0 SnSNn By example 4.3 of Woodroofe (1982) applied to (ti/02} we have n ( max n‘1[ 2 (sf/a2-1)]2; A 2 1 ) is U.I. N'SnSN” 1:] Moreover. by the Schwarz inequality E max n0: 3 (N')'3 usup (n/In)fli n max Du"; N'SnSN" azn N'SnSN" = 0(A‘3’2)0(N")2 » O. by the Doob's maximal inequality (see Inequality C. Sec. 1.2) and Lemma 2.1. Use the above arguments and (2.2.5) to get (2.3.21). 0 In order to state the next two lemmas. denote the "overshoot" of the stopping time t (see (2.3.1)) by (2.3.22) 0 = zt - A1/2a. The next two lemmas give the asymptotic distribution of U and the moment convergence of UlF. L235; 2.10. Snnnggg that the errnr distributinn F in nnn;n;15nmnnin, Ihen U has a limiting distrinuginn H. 12222 1 H{dr) = T To P(S 2 r) dr. r ) 0. To 51 Ingng To 1; ng in (2.3.7). Zrnnf. Follows from Theorem 4.1V. 0 Lfififln 2.11. Unge; the assumntigns 9f Lemmas 2.9 and 2.10. I U d? = p + 0(1). F 12222 9 13.23.13 (2-3 7)- finnnfi. The proof uses Lemma 2.9 and 2.10 and follows from similar arguments given in the proof of Lemma 4.4V. 0 ZBQQE_Q£_Ifi§QRE! 2.2. Assume w.l.o.g. a2 = 1. The proof will follow from Lemma 2.7. once we verify all the regularity conditions in it. By (2.1.4) (2.3.23) Ll.n ea (1/2)af = L1. Therefore. by CLT. the Slutsky Theorem and An 4 1 a.s. we have 2 2 2 (2.3.24) L .n *a (3/8)al 3(11 - 1) = L2 2 The verification of rest of the conditions of Lemma 2.7 follows from Note 4. (2.3.15). Lemma 2.9 and Lemma 2.6. Hence the theorem. D The next theorem gives asymptotic normality and uniform integrability of standardized stopping time t. 52 Before we state the next theorem let 1/2 (2.3.25) a = A a and t” = (t-a)/¢E. This notation will be repeatedly used in Section 2.4. Iflfigflfifl 2.3. (Asymptotic normality and Uniform integrability) 11, Eltll4 ( u. then (2.3.26) c” 43 x(o.a2). nhgng a2 = (1/4) V(:f/a2). Aggunn gng ngngng cgnniginng 9f [negren 2.1. L; 11A in (2.1.9) in gngn ghn; A1’2(1*") g nA s o(A1/2). 11:; n e (O.(s-2)/s). Inga (2.3.27) (It*|s: A 2 1) is U.I. The proof of Theorem 2.3 depends on a technical lemma. Lfififin 2.12. Lg; b > 0. defing fnr 1 > 0. x 2 0. 87(3) = X3/b2 - x - 1(x-b). Tngn (1) 32(X) 2 0 £21_all, x 2 0. (ii) 11 1 G (0.2) then there exgsts 0 < 6 < 1 such tnnt in: n11 x € [(1-€)b.b]. g1(x) S 0. 21921. Observe that g1(b) = 0 for every 1 > 0. Moreover. g;(x) = 3x2/b2 - 1 - 1. From this it readily follows that g1 is strictly convex on (0.") and it has a unique minimum at x = b¢(1+1)/3. 53 Assertion (i) follows since g2(x) 2 g2(b) = 0 for all x 2 0. For (ii). note that g1(bV(1+1)/3) < g1(b) = 0. Now let 6 = 1 - V(l+1)/3 to get the required result. 0 zggg£_9§_xn§ggzg 2.3. Assume 02 = 1 w.l.o.g.. The assertion (2.3.26) follows from Lemma 4.2V. since (En. n 2 1) is slowly changing and En/JH a 0 in probability (see (2.3.5) and (2.3.6)). As for (2.3.27). it suffices to show “+3. (2.3.28) ( [(t ) ] . A 2 1 } is U.I. and (2.3.29) ( [(t”)']s; A 2 1 } is U.I. Consider (2.3.28). By Lemma 2.12 (i) applied with x = t - 1 and b = a we have 2(t - 1 - a) g (c - 1)3/a2 - (c - 1) t-l g 2 (a? - 1). 1:1 by (2.1.9). (2.2.5) and since CD 2 0. By the cs-inequality t-l [(c")*]8 g csz's 2 (z? - 1)/JE + cs(1/JE)S. i=1 Now. by Lemma 2.3 and Lemma 5 of Chow and Yu (1981) t (2.3.30) (I 2 (a? - 1)/¢Z|s; A 2 1) is U.I. i=1 54 2 t p. 148. Lemma 2). Therefore by (2.2.2) Moreover. Ell - 1|s = 0(Et) (see Chow and Teicher (1978). (2.3.31) Elsi - 1|3/(JE)s 4 0. Hence. (2.3.28) follows from (2.3.30) and (2.3.31). It only remains to show (2.3.29). Now. O u f s xs'l P(t g -x. t < (1-€)a}dx. Et ¢E Now use (2.2.21) to conclude J. g as xs'lr(c g (1-€)a}dx = as’2~0(A'(s’1)’2(1*")) (2.3.32) 4 0 (since n ( [(s-2)/s]). From these it follows that [(t”)-]81[t < (1-€)a] is U.I. As for the Uniform integrability of [(t*)-]8 on the set [t 2 (1-€)a]. it suffices to show that [(t*)-]S on the set [(1-€)a S t S a] is U.I.. because (t*)— = O for t > a. Let 1 E (0.2). Then by Lemma 2.12 (ii) 1(a - t)1F. S (t - t3/a2)lF. where F' = ((1-€)a S t S a). Therefore. (t - a)-1F. S 1-1(t - t3/a2)lF. t -1 2 55 By the Schwarz inequality -s/4 E(tct/VE)‘1 g A (N')'° "sup(n/In)fls umax D "23 n2H 2s DSNX n 43 F. (2.3.34) 4 0. by the Doob's maximal inequality (see Inequality C. Section 1.2) and Lemma 2.1. The required result now follows from (2.3.33). (2.3.34) and (2.3.30). Hence the theorem. 0 2.4. Regret Analysis. The main purpose of this section is to analyse the regret Rt - Rn of the sequential procedure Bt' To that 0 effect. use (2.3.25) to write R - R = A E c + E c - 2a 1'. no t = A E (Ct - 02t'1) + l‘::(t-a)2t"1 (2.4.1) s A E 1'1 121(03 - a21t) + E at'1(c*)2. As for the second term above. it follows from (2.3.26). (2.2.1) and the Slutsky theorem that a t-1(t*)2 4a a2 If. as A 4 w. Moreover. by the Holder inequality. (2.1.9) (2.3.27) and (2.2.21) -1 * 2 fi 2 . 1-1/p a E t (t ) 1':t < ".1 S a nAflt fl2p{P(t < N )} . _ 0(A1/2)A-1/2(1+n) 0(A-(p-1)2/2p(1+n)). 0(1). as A 4 w. Also. by (2.3.27) (a c'l(c")21[t > ".1. A 2 1} is U.I.. 56 Therefore. E a t-l(t*)2 = a2 + 0(1). as A 4 w. Of course. all this is valid under the conditions of Theorem 2.1 with the proviso that n < [(p-1)2/p]. As for the first term in (2.4.1). use the Taylor’s theorem to write 1 -1 2 _ 2 A E c‘ 1t (0t 0 It) s 0’1 5(03 - azlt) - 0‘2 2(03 ~ 021t)(A-1t1t - 0) + E(Df - azlt))\;3(A'1cIt - 0)2 (2.4.2) = (a) + (b) + (c). say. where At is a r.v. lying between 6 and A-ltIt with 0 = a4/(1-52). To analyse (a). (b) and (c) further. we will prove a series of lemmas. These lemmas are of interest in their own way. Moreover. these will yield. under enough moment conditions. that (a) = 0. (c) = 0(1) and (b) = -A1/ 2 2(Df - 021t)XE + 0(1). as A a m. In what follows assume further that the r.v.‘s X0 and X1 have the same distribution. Note that this assumption implies (Xi. i 2 0) is stationary. LEfiEA 2.13. Asgnmg ghe mnnent cgnggtigng 9f Lgnmn 2.2. 11 mA 13 (2.1 9) 1s snch that A1’2(1+") g nA 1232 n e (0. (s-1)/s) and s > 1. nnnn ((t/a)'s; A 2 1) is U.I. 57 2229;. Observe that (a/t)s1[t Z N'] = 0(1). Moreover. [t < N.] g as n;8 P(t < N') S 0(A8/2) A-8/2(1+")0(A—(s-1)/2(1+n)) E(a/t)‘1 4 0 as A 4 w. by (2.2.21) and since n < (s-1)/s. Hence the lemma. D In fact the extra assumption of stationarity will be needed only for the next lemma and Lemma 2.16 below. LElgn 2.14. Agsnng tne nnggl in (2.1.1). 11, r 2 1 and altllr < a 231 EIXOI’ < m. thgn gg; gyeny p > 0 -1 n r -1 P(sup n 2 IX1_1I )3) S B n I IXOIr dP. n21 i=1 (sup n-1 2 IX1_1Ir)B} n21 i=1 H2:22!22.1£ p > 1 331 EI‘llrp < w and EIXOIrp < “. than 1 n r p 2 p rp (2-4 3) E :§¥(n 1:1|X1.1| ) S (p_1) EIXOI - [1991. Since (IXilr. i 2 0) is also stationary. the assertions follow from the Doob's maximal inequality for stationary averages (see Doob (1953). Chapter X. Lemma 2.2. p. 466) and (second) Lemma 3.3.1 of Stout (1974). D The following lemma is a sequential analogue of Lemma 2.1. Henceforth all unidentified limits are taken as A 4 w. 58 ngEA 2.15. 1;_ p 2 1 ang Elxol4p < m and Bltll“? < ., sass -1/2 H a Dt up = 0(1). where Dn is as in (2.2.4). Ennnf. Assume w.l.o.g. 02 = 1. Let B = {t S N"). By the Doob's maximal inequality for the martingale (Du) (see Inequality C. Section 1.2) and Lemma 2.1 we get a'“2 n 0t 13 u g a"1’2 n max D u p nASnSN" -1/2 " 1/2 = (% M) 0[(N ) 1 (2.4.4) = 0(1). By the Schwarz inequality -1/2 -1/2 (2.4.5) a H Dt 13 up S H a t 1B H2pll (D t/t)lBII 2p Now. by the Minkowski inequality -1/2 x + - 1/2 H a t 13 u2p g u (t ) 13 "2p + #3 {P(B)} 9. S ( 1 + 6.1) fl (t*)+1§ "2p (since on B. t - a 2 ea) (2.4.6) 4 0. by (2.3.28) and because P(B) 4 0 (by (2.2.22)). By the Schwarz inequality and the Minkowski inequality I(D tlt)lBll 2p S H sup IDn Inll 2p “ZN" Tl S H sup Ii/205u: (n-1 2 8f)l/2II 2 n2N" n2N" 1= -1 p n g 2"1 n sup in + sup (n’1 2 a?) "2 n2N" n2N" 1=1 P n g 2"1 u sup in "2 + 2'1 n sup (n'1 2 a?) n 21121!" p n2N" 1:1 2 2 4(22 _1)[IIXOII4p + H81fl4p] (2.4.7) < w. 59 by Lemma 2.14 and the maximal inequality for the reverse n -1 2 If . n 2 1) (see Inequality E. Section i=1 1.2). The required result follows from (2.4.4). (2.4.5), martingale {n (2.4.6) and (2.4.7). 0 L535; 2.16. 1; p 2 2. and Elxol4p < w. Elzll4p < m. then. a’“2 "(l-62) It - tazflp = 0(1). where In is as in (2.2.4). £nnn£. Assume w.l.o.g. 02 = 1. Squaring the model in (2.1.1) and summing over 1 yields 2 2 2 n 2 (2.4.8) (1-B )In - X0 - Xn + 23 Dn + 12181. Now. clearly -1/2 2 (2.4.9) a "XONP 4 0. Therefore. in view of (2.3.30) and Lemma 2.15 it only remains to analyse a-1/2HXEHP. For this. first observe that t+l t a-lIthzp = a 1 2 IX1_1I2p - 2 IX1_1I2p 1:1 1:1 4 O a.s.. -1 t 2p 2p by (2.2.1) and t 2 X1_1 4 E(Xo I3) a.s.. where 3 is a 1:1 a-field of invariant events (see Breiman (1968). Theorem 6.28 for non-sequential a.s. convergence). Moreover. t+1 a’llx |2p g a'1 2 Ix t i=1 12? i-l 60 n S ((t+1)/a) sup n"1 2 IX n21 i: 2 n S 2-l((t+1)/a)2 + 2-lsup [n.1 2 IX IZP] i-l n21 1:1 and therefore U.I. by Lemmas 2.3 and 2.14. From these we have (2.4.10) EIthzp = 0(a). Thus. for p 2 2 (2.4.11) a'l’2 nxfnp = 0(1). The lemma now follows from (2.4.8) and the Minkowski inequality. D The next lemma is similar to Lemma in Chow and Hartinsek (1982). LEEEA 2-17. Assume.£hs.ms§sl (2 1-1)- L2; t 22 231 sn-ggagggg ggnnnjng time such that Et < w 321 EIDtI < m. Inga (1) EDt = 0. (11) ED: = a BIt (iii) (D: - 02 In. in. n 2 1) is a mean zero martingale. LL. 1n nggitinn. B(8?) < m then 61 t t 2 2 2 2 4 2 2 2 (iv) E(Dt - a It) - 01 E 2 Xj-l + 40 E 2 XJ_1 DJ—l J=l _ j=l ‘ 3 + 4u3 Ejil XJ_1 DJ-l’ 2 2 2 2 3 where 01 = 15(8l - a ) and "3 = E(81). t t 2 2 2 2 (v) E(Dt- a It)( 2 (1 - to ) = 2u3 E 2 Xi_1D1_1 1:1 1:1 2 + 01 BIt (vi) E(D2 - 02I )D - E 2 x3 + 202 E 2 x2 D t t t ' “3 1-1 1-1 1-1' 1:1 1:1 finnni. Recall that {Dn' in} is a mean zero martingale. Moreover. by the Schwarz inequality and the Markov inequality 1/2 1-:|11n|1{t > n} g "Dun2 P (t > n) -1/2 1/2 S Ianll2 n E t l{t ) n} (2.4.12) 4 0 as n 4 w. by Lemma 2.1. and the Dominated Convergence Theorem. Therefore. by the optional stopping theorem (2.4.13) E(Dt|$n} s Dn 11 c 2 n and hence E Dt = 0. As for (ii). it follows from (2.4.12). Theorem 1 and Lemma 6 of CRT (1965) and the independence of Xi-l and 6i 62 t t 2 2 2 2 2 that EDt = E 2 x1_1:1 = E 2 E(X1_1£1I31_1) 1:1 1:1 2 - a E It' As for (iii). let M = D2 - 021 . From (2.1.1) we n n n have that (2.4.14) (Mn. 9“: n 2 1} is a mean zero martingale. with X2 82 + 2X 8 D — 02 X2 as the martingale n-l n n-l n n-1 n-1 difference sequence. As for (iv). it follows from (2.4.8). (1). and the Wald's lemma that (1-62)EIt 3 5x3 + 02 Et < m. Therefore from (ii) (2.4.15) E 0% < m. Furthermore. EIM |1 g 3021 + 02 E I 1 n {t > n} n (t > n) n {t > n} 2 2 2 ' E DtAn - EDt 1{t S n) + a E In1{t > n} (2.4.16) 4 O as n 4 m. because 2 2 2 2 2 E DtAn = a E ItAn 4 a E It' E Dt1(t S n} 4 E Dt and E In1{t > n) S E Itl(t > n) 4 O as n 4 m. Assertion (iv) now follows from (iii) and Theorem 1 and Lemma 6 of CRT (1965). As for (v). use - 2xy = (x-y)2 - x2 - y2 to write t t 2 2 1 2 2 2 (2.4.17) - Mt( 2 (i-ta ) = 5 (Mt - 2 8i + to ) 1:1 i=1 1'. - ME - ( 2 8 — taz)2 63 By the Wald's lemma t 2 (2.4.18) E[ 2 (cf - 02)] = 6(af - 02)2 E t. i=1 Moreover. using arguments in the proof of the Wald's lemma (see Woodroofe (1982). p. 9) it can be shown that n 2 (2.4.19) El 2 - no I l{t ) n) 4 O. as n 4 m. i=1 2 i n Therefore. the fact that Mn and 2(8? - 02) are 1 martingales. (2.4.16). (2.4.19) and another application of Theorem 1 of CRT (1965) yields. t E(M - 2 :2 + t02)2 t 1 i=1 ‘ 2 2 2 2 2 2 e E 1:1 E[(X1_1(8i-a ) + 2x1-1‘1D1—1 - (ti-a )) [31-1] 2 2 2 t 2 2 2 t 2 2 (2.4.20) = E(81 - a ) E E (X1_1-l) + 4 E31 E 2 X1_1D1_1 1:] i=1 1'. 2 2 2 Now take expectation on both sides of (2.4.17) and use (iv). (2.4.18) and (2.4.20) to get the required result. The proof of (vi) follows from (2.4.12). (2.4.16). (ii). (iv) and arguments similar to the ones in the proof of (v). Hence the lemma. 0 Using the above Lemmas and the results in Sections 2.2 and 2.3 we will now carry out the analysis of terms (a). (b) 64 and (c) in (2.4.2). In what follows we will assume as many moments as we need. Assume w.l.o.g that a2 1. Now. by Lemma 2.17 (ii) it readily follows that (2.4.21) (a) ll 0 As for (c). write -1 -2 2 t2 (2.4.22) A t It- a = a t(1t - 8t) + 6(a' - 1) = (a’lt) a'1[It - at] + 0(a'1t - 1)2 + 20(a‘1t - 1). Since by definition A;3 S 0-3 + (A t"1 121)3 we have 2 -3 -1 2 E|(Dt - It) At (A t It - 0) | g 9‘3 5|(0f- It)(A-1t 1 - e)2| + E|(Df — It)‘ 1’. -(A t'1 1:1)3(A'1t It - 0)2| = (c1) + (c2). say.. We will show that both (cl) and (c2) are bounded as A 4’". by using the decomposition in (2.4.22). Consider (cl). Now. 2 a-2 EI