Immmmmnmwmm ’ W H \lxflllh'lilmWl‘ulmlll O __THS (X) CDN 013—: «$.fiAuaw...‘ I'~‘. . M k”:- ‘. LIBRARY masts This is to certify that the dissertation entitled LIMIT THEOREMS FOR DISCRETE PARAMETER RANDOM EVOLUTIONS presented by Gilles Blum has been accepted towards fulfillment of the requirements for Ph.D. degreein StatiStiCS Major professor Date 1/7/ {V MS U is an Affirmative Action/Equal Opportunity Institution MSU LIBRARIES “ RETURNING MATERIALS: ace 1n boo rap to remove this checkout from your record. FINES will be charged if book is returned after the date stamped below. ——- LIMIT THEOREMS FOR DISCRETE PARAMETER RANDOM EVOLUTIONS By Gilles Blun 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 1982 ABSTRACT LIMIT THEOREMS FOR DISCRETE PARAMETER RANDOM EVOLUTIONS By Gilles Blm Let E. beasubset of R. For y in R and "-13,... let {Xu’y(k): k - 0,1,...} be a Markov chain in E with transition N PN’y. Assume that for every xin EN and .Y 1" R |Ex(xN”(l)-X)-a"p(X.y)-a§r(x.y)I 5 vupnbr). |Ex(X"’y(l)-X)2-d§5(x.¥)l f YNDNU)’ where a" e» R+ . y" e 11+ and p.r,s and pN are functions satisfying some additional conditions. Let {Y(k): k . O,l,...} be an ergodic Markov chain in R with transition P and Z"(k) -{(XN(k), Yik)):k-O.l,..} be the Markov chain. in E x R with transition P" satisfying P"<(x.n.Ax a) - Mummy. Then under some technical conditions it is shown that, as a." v (yN/aN) + O Xu([-/a:]) converges to a diffusion process that we characterize by its_generator in terns of p.r,s and P. He then use this result to obtain a diffusion approximation to the wright-Fisher model in Markovian environments. among other applications. ACKNOWLEDGEMENTS I would like to thank my advisor, Professor S.N. Ethier for his generous help in the preparation of this work. I am also grateful to Professor 0. Blair for assistance in connection with Chapter III. ii TABLE OF CONTENTS INTRODUCTION ...... . ................ Chapter I ~ AN ASYMPTOTIC THEOREM FOR DISCRETE PARAMETER RANOON EVOLUTIONs ............. II APPLICATIONS ....... ' ......... III A LINIT THEOREM FOR GEODESIC RANOON HALKS. . APPENOIx ................... '. . . . BIBLIOGRAPHY- ...................... iii Page 10 26 4O 45 Notation IKE) is the set Of Borel probability measures on E. HE) is the o-algebra Of Borel sets of E. Bor(E) is the set Of Borel measurable functions on E. B(E) is the set Of bounded Borel functions on E. . If fe Bor(E) and the integral of f with respect to the positive measure m on E exists we shall write it I f dm or m(f). E C(E) is the set Of continuous functions on E. E(E) is the set Of continuous functions on E vanishing at infinity. CK(E) is the set Of continuous functions on E with compact support. C§(E) is the set Of r tines continuously differentiable functions on E. with compact support. If f is a function on E and EN is a subset Of E we will also denote by f the restriction of f to EN' iv INTRODUCTION The primary purpose Of this work is to prove limit theorems for discrete parameter random evolutions. BefOre describing the kind of theorems we have in mind we first recall a wa definitions and results pertaining to the continuous case. Intuitively a continuous parameter random evolution describes a situation in which a process controls the development Of another process. when the controlling (or driving) process is continuous parameter Markov the main questions concerning representation and asywtotic theorems have been answered. ' To describe the type of results which have been obtained we consider one Of the simplest examples of a continuous parameter random evolution.' The model is that of a particle moving on the real line at one Of n possible velocities v1.....v . It changes velocity at n random according to a pure Jump Markov process Y(-) in {1.2,...,n} with generator O. Let x(t) and be respectively the position (t) v V and velocity of the particle at time t. The Markov process Z(-) - (X(-), Y(-)) has generator d n Af(x,i) - vi a; f(x.i) + 321 quflxJ) for f 6 C"°(Rx{l.2,....n}) and if for such an f we define u by u(t.x.i) ' Ex,if(l(t)) . then u is the solution Of the hyperbolic system 3—: (t.x.i) = v1 5-; (t,x,i) n + Z qu “(t’x93)9 1 f l f n. 3'] It is a well known fact that for certain elliptic or parabolic equations the solution can be expressed as the expectation of a function Of a Markov process. Random evolutions provide an example where this is also true for hyperbolic equations. It is this type of consideration ' that led Griego and Hersh (l969) to their definition Of random evolutions and, as in the elliptic case. a purely probabilistic analysis Of random evolutions can lead to new methods Of proof for problems arising outside probability. For continuous parameter random evolutions, asymptotic theorems involve a balance between two limits:l one takes the limit Of small ' stochastic disturbances over long periods Of time. Depending on the way this scaling is done at least two types Of limit theorems have been Obtained: one corresponds to the weak law of large numbers (a first order limflt theorem). another to the central limit theorem (a second order limflt theorem). For instance for the model of the particle moving on the real line considered above. Of interest is the limiting behavior, as e +'O. Of the process I in the first order case and of the process 3 t X€(t) a x + g (I/€)VY(s/€2)ds in the second order case. The connection between the first order limittheorem (resp. the second order limit theorem) and the weak law Of large numbers (resp. the CLT) can easily be seen from this example. Consider fOr instance the second order case. Let 00.91.... be the successive states occupied by Y(-), 10.11,... the time spent there, M(t) the number Of jumps in the time interval [0.t] and for j 8 1,2,... let 30, SO -l 55 ' 12w 11. _ .Assun e 8 N'5 and write XN‘(t) for Xe(t). Making a change Of variables ' Nt , xfl(t) ' X + N‘sig VY(s)d3~ M(Nt)-l -k i ' ' . + N + "t- e 'i. Suppose now that instead of being exponentially distributed, for J ' 0.1.... Tj satisfies P(TJ ' I) e l and let t 8 l and x 8 0. He Obtain then ' l "‘ "Nm " '3; k2. Vuk-T)’ and the second order limit theorem, which is now a result about discrete parameter randomTevolutions, reduces to the CLT for a Markov chain in {v}....,vn}. Asymptotic theorems for discrete parameter random evolutions (both driving and driven processes are discrete parameter) have been Obtained by several authors. Here are two examples. For N - i,2..... let. {(x"(k), v"(k)): k .. O.i....} be a homogeneous Markov chain in It” all" . Supposing that asylptotically as N‘+-~ the "infinitesimal“ covariances and means Of’ X”([-/eN]) are a1j(x.y) and bi(x,y) and those Of YN([-/6NJ) are 0 and ct(x.y) and assuming "LG" 8 liar” GN/GN 8 O and the zero solution Of 9 8 c(x.y) is globally asymptotically stable, Ethier and Nagylaki (l980) show that X"([-/€N]) converges weakly to a diffusion process with coefficients “15(x’0) and b1(x,0). This could be regarded as .an example Of a limit theorem for a random evolution with feedback: x"(-) is driven by v"(-) which in turn depends on x"(-). _In a recent article Kushner and Huang (1981) developed a general method fer proving weak convergence to a diffusion process of the sequence of appropriately scaled and interpolated solutions to the equation XN(k+l) - xflm - a: kuixuik). ruin) + «Noflufliki. vain) + oiofii. where YN(k) are random variables-satisfying certain mixing conditions. In this example XN(-) can be considered as the driven part of the random evolution (xN(.), YN(o)). (He were unaware of Kushner and Hueng's article while doing this work and there is some overlap between their results and ours.) The main result of this work is motivated by a problem Of Karlin and Levikson (1974). The problem is to derive a difquion approximation to the wright-Fisher genetic model with selection coefficients in a random environment. The mathematical formulation is as follows. Let 0 < BN < a" fOr each N 3 1. and suppose a" +TO. Let den and EN'{O.%....,1}. For N8l,2,... Tet {sz- (Xu(k), Y(k)): k 8 0,1,...) be a homogeneous Markov chain in EN x I! with transition P" ' satisfying (0.1) P'I(X.y).A x a) = P""(x.A) P(y.B). 'where P is the kernel of a stationary Markov chain in I! and fOr every . N y y in R P ’, (x -) is binomial (N.P ) where ’ anNU) - ' (0‘2) . Px,o ' 1 I :xx and (0.3) cum - “r + anym- I). The problem is to Obtain a diffusion approximation for the sequence {NN(k): k 8 O,l,€..} when it has been suitably scaled. In their article Marlin and LeviksOn study the case oN 8 $15.8N 8 N'I, {Y(k): k 8 0,1,...) i.i.d. with and) - O. Our main result can be roughly stated as fellows (see 52 fer the exact fOrmulation). Let E be a closed interval of It EN a subset of E. m 6 PM ), "N e P(EN),u e P(E) and assume u" a in. For N = 1,2,..., let {ZN(k) = (XN(k), Y(k)): k = 0,1,...} be a homogeneous Markov chain in E x R with initial distribution ON x m and N transition PN satisfying (0.1). Assimie that for each yin R, PI"-y is the transition Of a homogeneous Markov chain {XN’y(k): k 8 0,1...} in EN and P is the kernel Of an ergodic Markov chain in I! with invariant measure m. Suppose there exist functions p, r1. r2. s on E x R and ”N in L‘/3(dn). asst-ed to satisfy some additional conditions, such that fOr every x in EN. (004) IEx(xN’y(1)'x)'.'aN P(XsY)‘G§ r1(xs.Y)’B" r2(xe.Y)| f YNDN(Y)9 (0.5) IEx(X"”(1)-x)2-a§ shall 5 YNONU). (0.6) EXIXN”(1)-XI3 5 inNLv). where yum" v as)“ + O. Then as N + - the finite dimension“ distri- butions of ”(II-[(0% v Bull) converge to those Of a diffusion process A x(-), with initial distribution (“that we characterize by its generator in terms Of p, r1, r2, s, m and P. Depending on the relative values of .a and a" in (0.3) and. N on the scaling there are several possible limiting diffusions. The case a: 8;] + 0 corresponds to a first order-limit theorem and leads to a deterministic process. If afi - a" (resp. a: a: + -) condition (0.4) takes the form (M)- IEx(X"”(1)'-x) - a" P(X..v)-a,fr(xcv)| 5 in rum where r 8 r1 + r2 (resp. r 8 r1). Replacing (0.4) by (0.4)' the cases ' a: 8 5N and is: oil. 4 8 can, without loss Of generality.be treated simultaneously . Using our theorem we can generalize the result Of Karlin and Levikson to Markovian environments. More precisely let {Y(k): k 8 0,1,...} be a stationary Markov chain with Markov kernel P, Q the linear con- traction on L1(dm) defined by QfCV) ' I f(z) P(y.d2). and assume there exists n in L4(dm) and A in L2(dm) such that (0.7) (1-0)“ ' IdFs , (0.8) (1-0» - lyzdm - yz. Then X N([NtJ) converges to the diffusion process whose generator A. with domain CZEO. l]. is given by (0.9,) . . A - x(l-x)[t(l-2x) + (O- xv)] g; + % x(l-x)[l + x(l-x.)(v + 21)] i?- where v 8 EEY2(k)J and t, a constant, can be computed in terms Of’ P. Mhen {Y(k): k 8 0.1....) are i.i.d., which is the case studied by Karlin and Levikson. t 8 0. He can also easily prove a central limit theorem fer ergodic Markov chains. another one fOr chain dependent random variables and Obtain diffusion approximations to a certain type of stochastic difference equation. The CLT fer Markov chains we have in mind can essentially be stated as follows. Let {Y(k): k 8 0.1,...) be an ergodic Markov chain satisfying (0.7) and (0.8). Let 8 (0.10) o2 . “an - (0n)2)dm- Then Y(l) +"'+Y(N2- g N(0,l). Conditions (0.7) and (0.8) are clearly very restrictive and limit the applicability of such a result. Nevertheless it must be noticed that we do not impose any mixing condition on {Y(k): k 8 0,1,..3}; conditions (0.7) and (0.8) are assumptions on the one step transitions of the Markov. chain. In a certain sense this result is a very natural generalization to the Markov case Of the CLT for i.i.d. random variables. Generalizing this CLT we Obtain a diffusion approximation for suitably scaled solutions to the equation xflik+ii - Xu(k) - 6(XNIK). Su(k)) where fer k 8 0,1,... S"(k) 8 “M9 + a" Y(k) and 6 satisfies some . differentiability assumptions. An application of this can be found in Guess and Gillespie (1978). He do not pursue such applications as they have been recently treated by Huang and Kushner (1981). In O'Brien (1974) a CLT for random variables defined on a count- able Markov chain (or chain dependent random variables) is obtained. He prove a similar result without any'countability asswtions. The last result of this work is a CLT for random variables taking their values in a Riemann manifold. This is essentially a discrete parameter analogue of a theorem of Pinsky (1976, 1978). Two approaches have been particularly successful to prove asymptotic theorems fOr random evolutions in the continuous case: the semigroup approximation theorems of Kurtz (I969, _1973, 1975) and the martingale problem of Stroock and Varadhan (Papanicalaou, Stroock, Varadhan, 1977). In the discrete parameter case we can adapt either of 'these methods. He have chosen the semigroup approach; the approximation theorem we use is a discrete parameter analogue Of a theorem of .Kurtz (1975 Theorem 3.15). .Remarking that ZN(k) = -(XN(k), Y(k)) is a Markov chain in EN x R and defining L" {if e Bor(EN x R): 5:9 E|i(zN(k))l < .} we show that fbr every f in a core for the generator A Of a Feller semigroup there exists a sequence fN in i.N such that Sign ElfN(ZN(k)) - f(XN(k))l + 0 and . ski-P E'IANfflizuikn- Af(XN(k))| + 0 where AN f(x.y) = «.12 (i:My f(ZN(1))- f(x..v)). Assuming convergence of the initial distributions this implies that the finite dimensional distributions of “(D/ail) converge to those Of a Markov process with generator A. . ‘ CHAPTER I AN ASYMPTOTIC THEOREM FOR DISCRETE PARAMETER RANDOM EVOLUTIONS Throughout this section E is a closéd interval Of 1!. possibly unbounded, EN a Borel subset Of E. F a' Borel subset Of R , {Y(k): k 8 0.1,...) an ergodic Markov chain in F with transition :P and invariant measure m. If f’e L](dm) we define the function Qf in L1(dm) by setting (1.1) MM = I f(z)P(y.dz). He note that Q ,is a contraction in Lp(dm), l 5 p 5 ., Theorem l.l: Let k].k2 be two integers. k2 > k1 > 0. Let p,r and s be in Bor(E 8 F). and Of the fOrm (1.2) P(xo') . ao(X)bo(.v). "T (1.3) r(x.y) . I a1(X)b1(y). . k2 (1.4) s(x.y)’8 kX+1 a1(x)bi(1). l where, for i - 0,1,....k2, a1 is bounded twice continuously differenti- able, bole L4(dm) and for i 8 l....,k2. b1 5 L2(dm). Assume that for i 8 0,1,...,k2 there are functiOns n €L4(dm) and 11 e L2(dm) satisfying (1.5) ' (I-Q)n - b0. (1.6) (I-o)x0 = (bon - b3)-:. 10 ll where (1.7) i = 15{I(Qn2 Jinnizidm - [b3 din}, and for ‘i =1,...,k2, (1.3) (no);1 - ti - [bimh Let u€ P(E). For N -1,2,...- and ye F Tet (x"'5’(k): k - 0.1....) be a homogeneous Markov chain in EN with transition denoted by Pu” and initial distribution u". Suppose that for every A in 8(EN) the function (x,y) + Pfl’y(x,A) is Borel measurable and u" u u as N -> a. Let a" be numbers satisfying 0 < a" + 0 as N ~> a and assume there exist functions on in L‘/3(dll), ntmibers y" > 0 and M > 0 such that. for every x in E" and y in F. "(1.9) (EXM'Y-(ii - xi: - a" rim-a: P(mll 5 this). (1.10) IEx(X"”(1) - xi2 «1% 5(x.y)I 5 Wm. (1.11) ' . Ex|x"”(i) - x|3 5 menu). (1.12) [pa/3C1 < N and me: + O as N + - . Let {zflm - (XN(k), Y(k)): k - 0.1....) be the Markov chain in E" x F with initial distribution 11".". and transition PN satisfying, for A e 8(EN) and B e 8(F), (1.13) P"((x.y),A x a) - PN’y(x,A)P(y,B). Let A be the linear operator on C(E), with domain c§(E). defined by 12 2 (1.14) A = [11036 + Ir(-,y)m(dy)] g; + [1'33 + lefs(-.y)m(dy)] if. a) Assume ‘A' generates a strongly continuous,positive, conservative, contraction semigroup {T(t)} on C(E). There exists then a Markov process X(-) with sample paths in CE[0.8), semigroup {T(t)} and initial distribution u such that the finite dimensional distributions Of xN(['/“NJ)' converge to those of X(-). b) Furthermore if sup |b1(y)| s - and sup IpN(y)| < a (this being i.y NJ essentially the case when F is compact), convergence is in distribution in DEEOuP). m: Since (1.140) I (I-tlln dn 8 In(z)m(dz) - Iii-(dill n(z)P(y.dz) 8 0. ' a necessary condition for (1.5)to be satisfied is . (1.15) . [boon = O. For the'same reason (1.6) can hold only if (1.16) ‘ T - .1 (bon - 53m. 8y (1.14a) with n replaced by n2, [Onzdn - 1 nzdn. Applying then (1.5) we easily see that ] (Qn)2dm - j (n2 - zoon + t§)dn. SO (1.16) follows from (1.7) and (1.16) is thus always true. 13 In general to find sufficient conditions for (1.5), (1.6) and (1.8) is more difficult. One case where they exist is the following. Assume that for i 8 O,l,...,k2 b1 is a polynomial Of degree less than or equal to two. (This is typically what happens in the applications we consider.) Let o,c1 and 81 be real numbers with o and a1 not equal to one. u 8 I ym(dy), v 8 j y2m(dy) and suppose {Y(k): k 8 0,1,... is such that if i1 and f2 are defined by f1(y) - y and f2(y) - y2 then ' (1.17) Qf1(.v) 8 av + P(l-u) and. (1.18) . QfZU) 8 a1y2 + 81.1! + v(l-a1)-e]u. Let "0 and A be defined by 1100') . 1%; and 2 w W" «a, i m Then (I-Q)no8y-u and (I-Q)A8y2-v.- A case of particular interest is when bo(y) ' 90“-”) and for 181,...k2, .b1(y) 8fb1dll'l ' 910“") + 71024). 14 I where 81 and vi are constants. Since n is then a polynomial Of degree one satisfying (1.16) there exists a and y in I! such that 2 2 T-(bOn-bo) ' 6(y-u) + Y(y 8V). The fOrm of n and 1i in (1.5), (1.6) and (1.8) follows then easily. He give new two examples Of Markov chains satisfying (1.17) and (1.18); Example 1. Let {Y(k): k 8 0,1,...} be the Markov chain in Z+ with transition P(-,-) given by p(y,.) a bin(y,p)8POisson(e), where O < p < l and e > 0- If A 8 e/(lepxPoissonh) can serve as invariant measure for {Y(k): k 8 0,1,...}. (To prove this one can use generating functions: 2 (z s’POammi y z . I (q + SP)’ 2““) —-1—‘y°-A y y' . e(”PMs-1) a e(e/(]-p))(s'1).) Using generating functions again it is then easy to show that 0f1IY) 8 Pyr+ XII-P) ,.]5 and 2 2 szb') 8 P y + P(l-P)(2A + l)y ’8 A(l-P)[A(l-P) + 1] Conditions (1.17) and (1.18) are thus satisfied here. Exflle 2..Assimie Y(D) is N(O.1) and mm. x(1),... arei.i.d. N(0,l) and independent Of Y(O). Let Y(n+1) - .nn +/1-o2 x(n). where -1 < p < 1. {Y(n): n 8 0,1,...) is a stationary Markov chain with invariant measure the standard normal. Conditions (1.17) and (1.18) are easily seen to be satisfied with o 8 p, o] 8 pz. 81 8 0, 'v - 1. He also note that here n(y) - y/(i-p) .and - t 8 hi] (anz - (Qn)2)dl - Iyzdll‘ 8511((2H-L17m-11 ‘0).(1'0) 8 P/(18P)- If we do not want to assume (1.17) and (1.18) the fOllowing result (Revuz 1975 Theorem 6.3.10) gives sufficient conditions for (1.5), (1.6) and(l.8). Let B°(F) - {f e an): m(f) -' 0}. If {Y(k): k 8 0,1,...) is a quasi compact Harris chain then (I-QIBIF) . 3°(F). He can use this result if for i = 0,l,...k2 bi is bounded. 16 Proof Of Theorem 1.1: He will prove part a) of this theorem using Theorem A.l.a). To prove part b) one can use Theorem A.l.b). ' Let K 8 {g c B(E): g 8 constant + f with f 6 C(E)}. Clearly f E C(E) and g e K implies f g G K. The existence of a Markov process with sample paths in CE[°’°) and initial distribution u corresponding to {T(t)} follows from Theorem A.2. Let ~ be the equivalence relation on Bor(EN x F) defined by f 8 9 iff Elf(ZN(k)) - 9(Z"(k))| 8 O for every k 3 O, and let (1.19) Lu- 8 {f e Bor(E" x F)/~: sup E|f(z"(k))l < -i. For. f'E LN’ x g E". y e F let (1.20) TN f(x,y) 8 ff(u.v) PN (x,y,du,dv) (1.21) AN f 8 cf (TN - I). For it B(E) define a" fe LN by (1.22) a" fixer) 8 fix). and let 0 8 C;(E) -be the space of infinitely differentiable functions with compact support.* (0 is a core fOr A.) Condition (A.2) of in L Theorem (A.l) is assumed; He construct a sequence f N satisfying (A.7), (A.8), (A.9) and (A.lO). N 17 Let x 6 EN. y E F and B be the linear operator on L defined by i (1 23) _ 39(XaY) 8 19(X.V)P(y.dv) - 9(X.y). Let f be in D, f]. f , h h k in L . 1. and let N f 8 n f + a h + a 2k -N N N N ‘ Using the triangle inequality and denoting f by f, n N (1.24) "aN2(TNfN-fN) - nN Affl IA -2 2 - aN (lTNf - f - nfli1 - 1.." f2“ + an‘ui1 + Bhll + .1' - aN uThh - h - Bh - ouhlu 4. uh1.+ f2 + Bk -' a" Affl + “TNk - k - Bk“, where for f'e L".flffl 8 srp E|f(2"(k))|. To finish the proof we find f1. f2. h. h1 and k such that (1.25) eathNf -, f - aNf] - afifzu .. 0., (LE) q+eh8m (1.27) oilflTflh - h - 8h - auh1u + 0. (1.28) h + f + Bk - n Af 8 O VN, 1 2 N 18 (1.29) (ITNk - .k .. Bk" + 0. Ear XE EN’ 3! 6 F, let (1.30) f1(X.y) 8 P(X.y)f'(X). and (1.31?) f2(X.y) 8 r(X.y)f'(X) + %S(X.y)f"(X). Clearly f1 and f2 belong to LN' Using a Taylor expansion with remainder term Of order 3 and (1.20), ITNf(x.y) - f(x) 8 f'(x)f(u-x)PN’y(x.du) 8 15 f"(x)f(u8x)2PN’-y(x,du)| 5 K1 Ilu-xl3 Pu’y(x.du), . where it1 8 supili'(x)|1. By (1.9). (1.10). (1.11) 'and the triangle X ' . inequality, ITNf(x.y) 8 f(x) - auf](x.y) - «affixed! (Tuf(x.y) - f(x) - f-(xijiu-x)P“"(x.du) - ’5 f“.(x)l(u-x)2P""(x.du)| - + |f'(x)I(u8x)P""(x.du) + ’s f“(x)I(u-X)2P"”(x.du) - 1‘:qu (11.x) - u§f2(x.y)l 5 3K2 1rii"N‘3')8 where K2 8 supikp |f'(x)|. |f"(x)|}. (1.25) follows then from (1.12). For x6 Ewye F , let (1.32) h(X.y) 8 n(y)ao(X)f'(X). 19 (Note that “h“ 5 sup|a0(x)f'(x)| “n“ implies h E LN)' 8y (1.23) and (1.5). Bhlxo') 8 Ih(X.2)P(y.d2)- h(X.y) 8 aolx)f'(x)(In(z)P(y.dz) - n(y)) ' ‘f'l (xey). (1.26) is thus proved. For XE EN. 9 € F.19t (1.33) h1(X.y) 8 p(x..v)1§-§‘; (X.v)P(.Y.dv)- By (1.32). (1.2) and_(l.5). 'h’,(x...v) 8 80(X)bo(y)(ao(X)f'(x))' Inizle'AZ) 8 aoixxhoixii'ixir [bo(y)n(y) - bfibii. Since b0 and n are in L4(dm) and ab(x)(ao(x)f'(x))' is bounded. h1 is in '1' Using a Taylor expansion in the first variable with remainder term of order two and (1.32), ITNNXJ) - h(X.y) - BMXJ) - I(u-X) £3} (x.v)-P"(x.¥.du.dv)| 5 x3 iiu-xiziniv)(‘P"(x.v.ou,av) where K3 - sup{|(ao(x)f'(x))'|, |(ao(x)f'(x))"|}. By the triangle x . inequality, (1.13) and (1.33), 20 (1.34) ITNh(xey) "’ 11(ng) " Bh(X,.Y) " aNhI (xeyH 5 ITNh(X.y) -rh(X.y) - Bh(X.y) -I(U-x) %%(x.V)PN(x.y du.dV)l + |I(u-x) %g(x.v)P"(X.y.du.dV) - a" h1(X.y)| 5 K3 I(U-X)2ln(V)IP"(x.y.du.dV) + ( [(u-x) :2 x.v)P"(x.v.du.dv) - a" h1 (X.y)| 5 K3 f(u-x)2P"’y(x.du) 1 |n(v)|IP(y.dv) + .II(u-x)P"”(x.du) iggixnmmv) '- oNP(X.y) igfaxnipwvii. e L2(dm) such By (1.9) and (1.10) we can find s e L2(dm) and r [that 1 1 (1.35) . I(u8x)2P"”(x.du) 5 a: ‘1‘” + mm. (1.36) |I(u-x)P"”(x.du) - opium 5 83 r,(v) + mono). Using then the inequality 1f%£(x,v)P(y,dv)| 5 K3 ] |n(v)|P(y,dv), (1.35) and (1.36). ' ITNMXJ) -lh(X..v) - 81101.1!) - aflhfixw)! 5 k3(2v"o"(.v) + ofiiqiv) 8 s1 (vii) I ln(v)lP(y.dV)- Let then a e L‘(dw) be defined by O(y) 8 f|n(v)|P(y,dv). Since 9" E L4/3(dm). the H31der inequality implies 21 (ITNh - h - Bh - 0."th 5 K3121Nllonell + afi "(r1 8 s1)ell) f K3(YN(fog/3d!!!)3/4U04dll)ll‘ i’ 03(“1‘1‘” 5] )ZMITUOdefés and (1.27) follows from (1.12). He now have to find k in -LN such that (1.28) holds. ‘ He first rewrite A in a more condensed form. Let x 6 EN’ c0(x) 8 a0(X)aé(x)f'(x) + afi(x)f“(x). ai(x)f'(x) for 1 5 i 5 k1 c101) 8 ‘5 a1(x)f"(x) for k.l < i < k2, Using the definition of A. fé, h1 and h we get TNAf(x.y) 8 tco(x) .+ £1 c1(x) I- b1(y)8(dy). . f2(x.y) 8 r(X.y)f'(x) + ‘5 5(X.y)f”(x) "1 "2 ' I; ‘i(x)bi(y))f'(x) + 1. (E1+181(x)b1(y))f“(x) k2 '§ c1(x)b1(y)e 11101.1() - P(xo') 1%} (x.z)P(.v.dz) 8 ao(X)bo(y)l:a5(X)f'(x) + aolwad] In(2)P(.v.dZ) 22 = co(X)[bo(y)n(y) - bfiiyii. Let k2 k(X.y) 8 Z C1-(X)Ai(y). 0 k2 (Note that "k“ 5 (sup |c1(x)|) ( 2 flkiu) < 8 implies k E LN') i.x - 180 By (1.20)., Bk(x.y) 8 Ik(X.2)P(y.d2) - k(X.y) k 2 . 8 Z c1(X) (Ili(2)P(y.d2) - 11(y))- 0 By (1.5), (1.6), (1.7) and (1.8). k 2 Bk . co[t 8'(b0n8‘bg)l + 1;] ci[fb1(z)m(dz) - bi] 8 nNAf - h] - f2. . (1.28) follows. Let k2 A(Y) ' 2 lAf(y)|e 1'0 any) 8 [A(z)P(y.dz). K4 8 sup Ic;(x)|. i k x: 2 Since k(x,y) 8 120 c1(x)ki(y)a using the mean value theorem, (1.20) and (1.10), ITNk(xey) ‘ k(xay) " Bk(xsy)l f K461(Y) I (u-X)PN’y(X,dU) 23 K4610) (1(u-x)2P"’y(x.du))” IA K41vaN1y) + a§)s(x.v)()* 61(y) IA IA K4 .NivNagipN1y) + 51(y))8,1(y), Using the fact that 81 e L2(dm), pg 6 L4/3(dm), s1 5 L2(dm) imply that 61(pN + 51)}5 e L1(dm) and assuming N big enough for yNaiz to be smaller than one, we get llTNk - k - Bk" 5 a" R4 “(”11” s1)“eu. By the Holder inequality mw+snRP=1MW+shiemF 3 [(ON 4' s])cm1f ezdm. By (1.12) the term on the right side of the inequality is “bounded by a nuiiber independent Of N and (1.29) follows. Since {fN} clearly satisfies (A.7), (A.8) and (A.9) Theorem 2.1 is proved. Remark. It is possible to give an abstract semigroup version Of the previous theorem. For N 8 1,2,..., let LN be a Banach space, TN a linear contraction on LN’ EN a positive number and put AN 8 (Tn-1V6”- Let BN’ C", DN be linear Operators on LN. Assume that for every f in NB") n NC") n'D(DN) llTNf - f - (BNf + JEN' cNf + EN but)" 8 om"). Let L be a Banach space, TN: L -> LN a bounded linear transformation 24 with sup “an1 < 8. Let {T(t)} be a strongly continuous contraction N semigroup on L with generator A. Let A be a core for A and assume that for every f in A there exists hN and kN in D(BN) n D(CN) n 0(0N) such that 11 f e D(BN)n 9(CN) ii D(DN) and N (1.37) BNhN 8 - BN(an), CNan 8 O (1.38) 8 INAf 8 (CN hN + DNn W) (1.39) ‘3” (noun. ikflu. noun"). NCNkNfl.fl0NhNu. uoNkNu) < . . Then, as 6N + 0, for each f E L, TIENt/GN:l n f + T(t)f for all t i 0, uniformly on bounded intervals. N _ The proof of this result goes as follows. Let fN 8 an i- fihN IENKH' Since flfN 8 iiN fl) .. 0, by Kurtz's approximation theorem for discrete paranter contraction semigroups (W 1969, Theorem 2.13) it is enough to show that (1.40) ' ((ANfN - .NAfu .. O as 6N + 0. Denote for simplicity wa by f. By (1.37) uTNfN - fN - (ENgN.+.E’ NNc fN + E" DNfNHl . - . . an “N fN fN (ficNf * ENDNf ” “q BN’N * eN"N " eN 2°N"N + eNBNkN + 63,211 C kN +eN nNkN)|l. 3/2 ' ”TNfN ' fN ' eN‘BnkN I cN"N I l3N") I 6N Using then (1.38) and (1.39), (ONNh +cNkN)+eNDN kNll. 25 ”7qu ‘ fN ’ €N“N”“ ' O(EN) and (1.40) follows. If we go back to the proof of Theorem 2.1, and for g, the restriction to EN 8 F of a function on E x F twice differentiable in x, A CN9(x.y) 8 P(X.y) I§%X.2)P(y.dz). 2 DN9(X.y) -- r(x.y) i-gamwwz) + s. 5(X.y) I immune), and . A 8 CN(E). (In‘thissetting (l.5),(l.6) and (1.8) imply (1.37), (1.38) and (1.39). CHAPTER II APPLICATIONS In this chapter we give an application Of Theorem 1.1 to the problem of Karlin and Levikson lentioned in the introduction (Theorem 2.1). another one to sum of chain dependent processes and to the central limit theorem for ergodic Markov chains (Theorem 2.2). He discuss then briefly a method for Obtaining diffusion approximations to sequences Of suitably scaled stochastic difference equations. Throughout this chapter {Y(k): k 8 0,1,...} is an ergodic Markov chain in F, a Borel subset Of R , with invariant measure in and transition P. He assimie I ym(dv) 8 0. WWI“ 2.1: Lat E ' [09]]. 11 E P(E), EN ' {0,%-,...,1}aHN E P(EN). Assimie "N .. p, jy8m(dy) < .. and for i 8 l,...,4 there exist functions 11 and A1 satisfying (1.5), (1.6) and (1.8) with bow) ' ye b1“) ' b3“) .19 bziy) . my) . v2. Let aN 8 11.5, aN 8 N"1 and for N8 0,1,..., iz"(k)'8 (x"(k).v(k)): k 8 0,1...}be the Markov chain with transition P" given by (0.1), 26 27 (0.2) and (0.3). Then as N +.. the finite dimensional distributions of XN([NtJ) converge to those Of X(-). the diffusion with initial distribution u and generator A given by (0.9) where T is defined by (1.7). Proof: For y in R and N80,l,... let {x""(k): k80,1,...} be the Markov chain in EN with initial distribution “N and transition Pu’y. He show first that conditions (1.9) to (1.12) are satisfied with WM) 8 X(l-X)y. r(x..v) . (e-xvz)X(l-X). 5(x..v) . xii-xm + xii-x1112). Using the'fact that Xu’yw) 8 x imlies NX"”(1) is binomial WNW) with px,oN(.Y) ' <1 + «Nomi/(1 + ought). (8” ENPJUI-V'PRWO)'* . 8 aNiy)X(l-XI-a§(y)x2(l-x) + afiulxad-‘éflmh He have the following two relations (2.2) cum . (11". + N"‘v)v(-is) 8 (ii-18 + ".1,” - [(N-IO + N'I‘y) 1* 811(“’.,fl(oN(y)) and 28 (2.3) «NM? [(1141) + (Vivid-1.))? (11". + 11-8,)? - [W‘s + n'tviz - 8.1 1,0,“,5] (anon. From now on we denote 1(_m’_8](oN(y)) by 6N(y). By (2.1), (2.2) and (2.3). Ex(XN’y(l)8x) 8 N"‘yx(1-x) + h"(e-y2x)x(1-x) '3/29y)x2(18x) - [(N'19 + N-gy) 4' 3531((18’Q5NLY) 3/26y 81 2 2 2 3 18x 8 (N 22. + 211-11 .v + ox (l-x)6N(y) + oN(y)x W. and we get - (N 29 2+ 2N IEx(xN’y(l)-X) - N'ByMl-n-N'.‘(e-yzx)X(l-x)I 5 N'B’zoN’fiy). 8 N ' ‘ wh 8 ith 9" pN,1 k2] “’k 1" im 8 N'I’ez. 121v) 8 .20). 11%) 8 We +N3’2)6N(y). :) . mun). i5iv) . (N '28. )GNO). .5111) 8 26y6N(y). My) 8 v’fiyz GNM. v20) 8 N3/2laNb/HZ- is a bounded sequence. By Minkowski's inequality it is enough to show that for k 8 1,2.....8.I111:|4/3dm are bounded sequences. This is clear for k 8 1,2,6 and 8. He check that the condition is satisfied for To complete the proof Of (1.9) we have to show that {IION 1(y)l k 8 7. The other cases follow in the same way. 29 If e > 0, on(y) _<_ harm/2)“) and -Jfiyz 4 -/N72 4 flu';(y)|4/3m(dfl : 24/3 I [(N/2)y214/3m(dy) s 2 ”I y m(dy)+ 0. The case a 5 0 follows by a similar argument,- This completes the proof Of (1.9). He show now that (1.10) is satisfied. Note that N,y _ 2 3 N,y _- 2 _ 2 (2.4) EXIX (1) X) Ex(x (1) anONLY)) + (px,oN(y) X) . Using a property of the binomial distribution we get N.) _ 2 a l - Ex(x (II) pX,ON(Y)) N pX,ON(,Y)(] perN(y)) 8 1{X + 0 i )X Ll:§%—y-J[18X8O I )X ( 1-x )J _ N ' N y 1+0" y x . N y , 1+ONIy)x 1 1 oN(y)x(i-x) ,oN(y)X(l-X) = fi’ X(1-X) '1' N1+ON(,Y)X [I'ZX- T—WN .Y X J This implies N.y 1 1 (2.5) IEx(x (1) - px.oN(y)) - Nx(1-x)| 5 N|0N(y)|(1 + (oN(y)|). Hefindnextabo df - 2 ° un or (px’ONU) x) . Since (PM,NM - x) 8- aN(v)x(1-x)-o§(y)x2(1e;fim . (2.6) (i>,‘,‘,NN,,-x)2 8 o,‘°;(v)x"‘(1-x)2 + 11(aN(y)x). 30 where w] satisfies Iv](oN(y)X)l 5 o:(y) + zloN(y)|3. Using then (2.3), “183/2 0:0)" _._ y + N w2(6’y) 'l' [¢§(B.y)JGN(.Y) a where $2 and pg are polynomials Of degree 2 in y. He then get |(PX,ON(y)-X) - N'1y2x2(l-X)2| JB/lez(e,y)| + |u3(6.YI|6N(y) + Iv1(aN(Y)X)|- Using the triangle inequality. (2.4), (2.5) and (2.6), lEx(xN’y(ll-X)2 - N"X(l-X)(1 + X(l-X)y2)l 5 N'BIZPN’Ziy). 3/2 83/2 where pn,2(y) 8 N [N'1|0N(y)|(l + IoN(y)|) + N IP2(e.y)| + |i§(e.y)|6N(y) + 11(oN(y)X) J. The fact that {IP:/3 dm} is a bounded sequence can be shown easily and (1.10) is proved. In the same way it is possible to show that Ex|x"*’(1)-x|3 5 N'3/2(ll + Zy4)3/2. If we define pN 3(y) 8 (11 + 2f)”2 then ja4/3 dm is finite, does not depend on N and (1.11) is satisfied. Taking then pN1 + pN2 + pN3 = (1.12) follows. pN 31 We find now the form of A. Using the notation of Theorem l.l, a0(x) = x(l-x), lr(x.y)m(dy) = (B-XV)X(l-X). fs(x,y)m(dy) = x(l-x)(l + x(l-x)v).. By (l.l4) A is given by (0.9) and by Theorem A.3 I' is the generator of a strongly continuous positive conservative contraction semigroup on. C[0,l]. This proves Theorem 2.l. Definition: Let {Y(k): k = 0,1,...) be as in Chapter 1. Random variables {X(k): k = 1.2,...) defined on the same space as {Y(k): k'- 0.1,...} are said to be chain dependant iff for every x in R and ksl,2,..., P[X(k) IA x|Y(O),X(l),...Y(k-2),X(k-l),Y(k-l)] P[X(k) 5 x|Y(k-1)J. Theorem.2.2: For y in R let Py€P(R) be defined by Py(-.XJ = P[X(k) 5 x|Y(k-l) = y] and let boo) = IxP’(dx). b2(y) - lszy(dX). em = unified. ’ 32 Assume b0 6 L4(dm), b2 6 Lz(dm) and 6 e L4/3(dm). Suppose (1.5), (1.6) and (l.8) hold with k1 = l, k2 = 2, b0, b2 as above and b1 = 0. Let 02 = I(Qn2 - (Qn)2)dm- Then X(l)+ ...+xm £u(o,1). 0 Proof: For y in R let {Xy(k): k=l,2,...} bei.i.d. with distribution P’ and {XN’y(k): k = 0,l,...} be the Markov chain defined by XN’y(0) has a given distribution u and - - Y y 0 'Let PN"y bethetransitionof XN’y and let ZN(k) = (XN(k),Y(k)) be the Markov chain in R x R with transition P satisfying PN((x,y),Ax B) ' PN’y(x9A)P(.YsB)o (Note that {XN(k): k = 0,l,...} is given by xN(k) = XN’y(0) + XU )+...+X(k) .) 0 Conditions (l.9) to (l.ll) of Theorem l.l take the form st(x"”m-x) - (amflboon = o. st(x"" - XN(k) = G(XN(k).oN(k)) Let x(-) be the Markov process in E whose generator A, with domain ' C2[0,l], is given by A = [Tff' + ef + 7‘"; ng/dx + m2 + $1 fZJdZ/dxz Then. as ,N d.., the finite dimensional distributions of XN([-/a:]) converge to those of x(-). Proof: He just give an outline of the proof. Let a" - die + any and define {x""(k): k = 0,l,...} by: N,y X (0) has distribution v. x""(k+i)-x""(k) = e(xN(k).aN). (Note that we define like this a deterministic difference equation.) If we call PN’y the transition of XN’y the chain {ZN(k) = (XN(k), Y(k)): k = 0,l,...} has transition PN’y(x,-)xP(y,-). He can use Theorem l.l and here (1.9) and (1.10) take the form 35 Ext(xN”(1)-x)-G(x.a 3 = o. N) EXE(XN’y(1)-X)2*G(x,oN)2]= 0. Using (2.10) we obtain p(x.y) = yf(x). r(x.y) = ef(x) + y29(X), 5(x.y) = yzfm. Theorem 2 . 3 fol lows . CHAPTER 111 A LIMIT THEOREM FOR GEODESIC RANDOM wALks Let M be a Riemann manifold of dimension n. The Brownian motion in M is a stochastic process {X§: t 3 0} with continuous sample paths such that X3 = x and x t x f(xt) - f(x) + 6 (Af)(xs)ds is a martingale. (For the existence of such a process see Pinsky(1978)). The aim of this section is to approximate the Brownian motion by geodesic random walks. The result we prove is a discrete parameter version of a result of Pinsky (l978).. Let x.€ M, Tx(M) be the tangent space of M at x, Sx(M) the unit sphere of Tx(M), S(M) the bundle of tangent unit spheres. Let Ex 6 Sx(M) and x(é,€x) be the unique unit speed geodesic starting from x in the direction Ex. Let {1(k): k 8 0,l,...} be i.i.d. in 12" with distribution P and assume E[T(k)] s‘E[1“Z(k)] - l8 l. Let {Z(k): k - 0,l,...} be the random.variables in S(M) such that 2(0) - (xo.Eo),....Z(k) 8 (xk’Ek) where xk = xk-1(t(k-1)’€k-1)’ Ek‘fi *k and the conditional distribution of 5k given {(x0.€0)....,(xk_].Ek_1)} is the uniform distribution on Sx denoted by u. The sequence k 36 37 {x0,x],...} is called a geodesic random walk. Theorem 3.1. For n . 1,2,... let {ZN(k) = (x:,Ek): k = 0,1,...) be the random variables such that x3 = x0 and N t k-l xk ' x2-1 (‘§§7"l"' Eiii-1)- Assume the Ricci curvature of M bounded from below. Then for every . N a x f in C(M), m E[f(X[M])J E[f(X£)]. n 2599:, To prove this theorem we need the following Lemma: Lemma: Denote by P.(x,g) the parallel transport along the geodesic X(-.g). Let f e ciao») and let "(tsxaisn).= f(x(ts£)s Pt(xo€)(N)L where x E M, 1; e SX(M), n e Sx(M). Then a1. .ee_iia_u. 3t (Oexogen) 51 3X1 Pj’ke "k 351. 2! n 5 He will denote at by DE(u) and DE by DE . We also remark that 2 . af ' if f E CK(M)’Dgf E 51 5;} . He use Theorem A.lb. In the notation of this theorem EN = S(M), E . M, o . c§(n). For f in 6(S(M)) we define inane) =1 axe/gt). meXuEHanP u(dn). JR" AN = N(TN-I), 38 l. Af=-nAf for 1’60. 1 Let fe D,h=D£f,k=DDf and f =f+N';’h+N' g; N k. We have to show that “ANfN - Afu + O as N + + on. Using the lenma we have the following Taylor expansions for f and h (we replace 1(1) by 1]): (2.1) |f(x(r] mm) - f(x) «(r1 Mi) DEf(x) + (r§/N)D£Dgf(x»| 5 K 11'3/213. (3'2) lh(x(T]/'fis£)s PT-I/N(x’g)(n)) "' h(xsn) ' (Tl/N)02h(xsn)l f KN-1T§ . For g e C(S(M)) let 8 be defined by Bg(x,§) 3 I 9(Xon)u(dn) ' 9(xs€)° sxon ' By (2.l), (2.2) and the definition of TN , (2.3) NITNf(x,£) - f(x,£) - (MJDEflx)-15N'-]DEDEf(x)| 5 Nikk Elii’). where K is a constant depending on M only . (2.4) Ii)i,,h(x.e) - h(x.£) - Bh(X.£) - MT)" Ing'Mxminwnil . 3 W)" K ELI?) " MT] K. 1K. (2.5) ITNk(X.£) - k(X.£)'- Bk(X.£)| < MW1 K Ehl = N ' Using then a triangle inequality , IANfN(x.a) - Af(x)| 5 Mi)“ K(E|11|3 + 1 + 1) + |Bh(x.€) + 015le + |Bk(x,g) + D£D€f(x) + I 02h(x.n)du(n)-Af(x)l. 39 To conclude the proof we show the following equalities: (2.5) Bh(x,£) + 05f = o, (2.7) I02h(X.n)u(dn) = 0. (2.3) Bk(x.§) = Af(x) — DEDEf(x). Using the definition of B and the fact that In1u(dni) = 0, af . ‘ a: . " E1 '37" “£1 -DEf(x)i . Bh(x9€)1 8 In.‘ 3%Mdni) 3X1 This proves (2.6). For (2.7), ah i 3h ' £1 571(on) "' Fj’kijnk 5:1(xm) n 05h(xsn) 32f i af 6 n ----- P E n --. i k axiaxk J,k j k‘axk _ (A.7) follows then from the fact that W )(d)-< ———°2f () " 3‘) (dnl'O I 5 xi“ " “ 51 ax‘axk * ' ri,k‘j 32k ("kP k ‘ ' (A.8) follows essentially as in Pinsky's article. He have to show Bk(x.£) = ID€D£f(x)u(dE) - DEEf = Af - DEDEf' This reduces to [0505f(x)u(d§) = Af (see Pinsky l978 p. 209). To conclude we use the following result of Yau. Lemma: Let M be a complete Riemannian manifold with Ricci curvature bounded from below by a constant. Then the Brownian motion semigroup preserves the class of continuous functions which vanish at infinity. 40 Remark: In his article Pinsky assumes that for k = 0,l,2,.. Tk is exponential with parameter one. He then introduces the geodesic transport process X(-) defined by x(t) = xk_](t-(11+...+tk_1), £k_1) where 11+...+rk_1 5 t < 11+...+rk. He then shows convergence of a scaled geodesic transport process to Brownian motion. APPENDIX In this appendix we give conditions for sequences of discrete parameter processes to converge to Markov processes. Theorem A.l is a discrete parameter version of Theorem 3.l5 in Kurtz (1975). Throughout this section (E,r) is a complete locally compact separable metric space. Theorem A.l: a) Let K be a Banach subspace of B(E) which contains 0(E) and such that -f e 0(E) and g e K imply f g e K. Suppose {T(s)} is a strongly continuous contraction semigroup on K with generator A corresponding to a Markov process x(-) with initial distribution u and sample paths in DE[0,~).V For N - 1,2,... let {ZN(k): k - 0,l,...} be a sequence of Markov chains with transition P and measurable state space 0". Let 6N be a positive constant, EN +v0 as N +-a, and nN: GN +~E a measurable mapping.) Let (M) XN(t) = nN(zN(te,]‘tJ)) and assume (A.2) ' lim E(f(XN(0))) = E(f(X(0))) N-ic for every f in K. Let ~ be the equivalence relation on Bor(GN) defined by f ~ 9 iff E|f(ZN(k)) - 9(ZN(k))| = 0 for every k 3 0 4) 42 and let (A.3) LN = {f e Bor(GN)/~: ufn = s:p E|f(ZN(k)N < a}. (A.4) TNf(x) = ff(z)PN(x,dz), f e LN, (A.5) AN- €i“T~-I>» and (A-6) - an(2) = f(nN(2)). f e B(E)- Let D be a core for A and assume that for every f in 0 there is a sequence {fN} in LN such that (A.7) sap "fun < ° 9‘ (11.8) 5:" uANfNu < . ., (A.9) limN+¢ qu-anu = o , (A.l0) limN+¢ “ANfN-nNAfu = 0. Then the finite dimensional distributions of' XN(-) converge to those of X(-). b) If in (A.7) - (A.l0) L is replaced by B(GN) with the sup norm, N then convergence is in distribution in DE[0,o). Proof: a) Let V be an independent Poissonprocess with E[V(t)] = t. Remarking that LN is a Banach space and TN is a linear contraction on LN it follows that A is the generator of ZN(V[€i1t]). Let 43 YN(t) =ZN(V[Ei1tJ). Conditions (A.7) to (A.l0) imply conditions (3.l7) to (3.20) of Theorem 3.l5 in Kurtz(l975), and the finite dimensional distributions of nN(YN(t)) converge to those of x(t). The convergence of the finite dimensional distributions of XN(t) to those of x(t) fbllow then by a standard argument. The next theorem,due to Blumenthal and Getoor(1968, Theorem 1.9.4), gives conditions for a semigroup to be the semigroup associated with a Markov process. Theorem A.2: Let E be as in Theorem 2.1 and {T(t)} be a strongly continuous positive contraction semigroup on 0(E) whose infinitesimal generator A is conservative in the sense that there exists a sequence {ffllc D(A) such that b. p. lim f = l and b.p.lim Af - 0, where N N b.p.lim fN a f means fN(x) +-f(x) for every x in E and sup quM < n. Then, for each u e P(E) there exists a Markov process N X corresponding to T(t) with initial distribution u and sample paths in DE[0’°)' Suppose also that the generator A of {T(t)} satisfies the following condition: for every x e E and neighborhood V of x there exists f'e‘D(A) and a neighborhood U of x such that 1" 5 f 5 lv and Af a 0 on U. Then almost all sample paths of X(-) belong to CE[0,~) ‘ The next theorem gives conditions for a one-dimensional diffusion operator to be the generator of a contractionsemigroup-. (Ethier 1978). 44 Theorem A.3: Let E be a closed interval of It, possibly infinite, with end-points r0 < r1. Let a and b be continuous functions on E with a 3 0, having bounded deviratives a", b', b". Suppose a(rl) g 0 f (-l)1b(r1) 1f lrjl < o and fbr f'e CE(E) let Gf = af" + bf'. A = 0' generates a semigroup {T(t)} satisfying the conditions of Theorem A.2. 10. BIBLIOGRAPHY R.M. Blumenthal and R.K. Getoor (1968). Markov Processes and Potential Theory. Academic Press, New York. S.N. Ethier (1978). Differentiability preserving properties of Markov semigroups associated with one-dimensional diffusions. Z. Hahrscheinlichkeitstheorie verw. Gebiete 45, 225-238. (I) .N. Ethier and T. Nagylaki (1980). Diffusion approximations of Markov chains with two time scales and applications to population genetics. Advances in Applied Probability 12, 14-49. :0 . Griego and R. Hersh (1969). Random evolutions, Markov chains ~and systems of partial differential equations. Proc. Nat. Acad. Sci. U.S.A. 62, 305-308. J. Gillespie and H. Guess (1978). The effects of environmental autocorrelations on the progress of selection in a random en- vironment. The American Naturalist 112, 897-909- S. Karlin and B. Levikson (1974). Temporal fluctuations in selection intensities: case of small population size. Theoretical Population Biology 6, 383-412. T.G. Kurtz (1975). Semigroups of conditioned shifts and approximation of Markov processes. Annals of Probability 4, 618-642. . AT.G. Kurtz (1973). A limit theorem for perturbed Operator semigroups with applications to random evolutions. Journal of Functional Analysis 12, 55-67. H.J. Kushner and H. Huang (1981). On the weak convergence of a sequence of general stochastic difference equations to a diffusion. SIAM Journal on Applied Mathematics 3, 528-536. G.L. O'Brien (1974). Limit theorems for sums of chain dependent processes. Journal of Applied Probability 11, 582-587. 45 46 11. G.C. Papanicolaou, D.Stroock and S.R.S. Varadhan (1977). Martingale approach to some limit theorems. In Statistical Mechanics and Dynamical Systems (by D. Ruelle)‘and;papers from the—1976 Duke TUrbulence Conference. Duke University, Durham, N.C. 12. M.A. Pinsky (1978). Stochastic riemannian geometry in Probabilistic Analysis and Related Topics (Bharucha-Reid, ed.) Academic Press, New York. 13. D. Revuz (1975). Markov Chains. North Holland Publishing Company, Amsterdam. "'TITI'ITIEIIWLEMLHI[ifllilflffl'l‘flflfliflijflflllflm