An alternative proof of a theorem of Aldous. concerning convergence in distribution for martingales.

Transcription

1 A alterative proof of a theorem of Aldous cocerig covergece i distributio for martigales. Maurizio Pratelli Dipartimeto di Matematica, Uiversità di Pisa. Via Buoarroti 2. I Pisa, Italy pratelli@dm.uipi.it We cosider regular right cotiuous stochastic processes X = (X t 0 t 1 defied o the fiite time iterval [0, 1: let P X be the distributio of X o the caoical Skorokhod space D = D([0, 1; R of càdlàg paths. We cosider o D, besides the usual Skorokhod topology referred as S topology (Jacod Shiryaev is perhaps the best referece for our purposes, see [4, the pseudopath or MZ topology: we refer to the paper of Meyer Zheg ([6 for a complete accout of this rather eglected topology (see also Kurtz [5. We will use the otatio X = S X (respectively X = MZ X to idicate that the probabilities P X coverge strictly to P X whe the space D is edowed respectively with the S or the MZ topology. We will write also X = f.d.d. X to idicate that all fiite dimesioal distributios of (Xt 0 t 1 coverge to those of (X t 0 t 1. The followig theorem holds true: Theorem. Let (M be a sequece of martigales, ad M a cotiuous martigale, ad suppose that the followig itegrability coditio is satisfied: (1 all radom variables ( sup 0 t 1 M t, = 1, 2,... are uiformly itegrable. The the followig statemets are equivalet: (a M = S M, (b M = f.d.d. M, (c M = MZ M. The implicatio (a= (b is quite obvious, sice Skorokhod covergece implies covergece of fiite dimesioal distributios for all cotiuity poits of M (see [4.

2 The implicatios (b= (c is a easy cosequece of the results of Meyer Zheg: i fact the sequece (M is tight for the MZ topology ([6 p. 368 ad, if X is a limit process, there exists ([6 p. 365 a subsequece (M k ad a set I [0, 1 of full Lebesgue measure such that all fiite dimesioal distributios (M k t t I coverge to those of (X t t I : ecessarily P X = P M. Aldous (i [2 gives a proof of the implicatio (b = (a, but (although he does ot metio the MZ topology the implicatio (c = (a is more or less implicit i his paper (see [2 p The purpose of this paper is to give a proof of the implicatio (c = (a, completely differet form the Aldous origial oe ad strictly i the spirit of the paper of Meyer Zheg; I hope that this cotributes also to a better kowledge of the result of Stoppig times ad tightess II ([2, which is i my opiio very importat ad seems to be almost ukow. The proof will be postpoed after some remarks. Remark 1. I wat to poit out that Aldous proof of the implicatio (b = (a requires the followig weaker itegrability coditio: (2 all radom variables M1, = 1, 2,... are uiformly itegrable. Coditio (2 implies that all r.v. of the form MT, = 1, 2,..., with T a atural stoppig time for M, are uiformly itegrable; istead our proof eeds a more striget coditio, i.e. that all r.v. of the form MS, = 1, 2,..., with S a radom variable i [0, 1, are uiformly itegrable. Remark 2. The extesio of the Theorem to processes whose time iterval is [0, + is straightforward: i that case the correct hypothesis is that, for every fixed t, the r.v. sup 0 s t Ms, = 1, 2,... are uiformly itegrable. I fact, if the limit fuctio f is cotiuous, f f for the S topology (respectively the MZ topology o D(R + ; R if ad oly if the restrictios of f to every fiite time iterval coverge to those of f (for the S or the MZ topology. Remark 3. The Theorem fails to be true if the limit martigale M is ot cotiuous ([2 p. 588, ad fails for more geeral processes, e.g. for supermartigales. Let ideed T be a Poisso r.v. ad put, for every : X t = ( I {t T } t T ( (t T I {t T } 1. The processes X are supermartigales whose paths coverge i measure (but ot uiformly to the paths of the cotiuous supermartigale X t = (t T. Remark 4. Suppose that the processes X are supermartigales, ad cosider their Doob Meyer decompositios X = M A. If separately M = MZ M ad the martigale M is cotiuous, ad if A = MZ A ad the icreasig process A is cotiuous, the X = S X = M A (remark that, for mootoe processes, covergece i the MZ sese to a cotiuous limit implies covergece for the S topology. A applicatio of the latter result ca be foud i [7, theorem 5.5.

3 The proof of the implicatio (c = (a of the Theorem is rather techical, ad will be divided i several steps. Step 1. Give ɛ > 0, there exists δ > 0 such that, if S is a r.v. with values i [0, 1 ad 0 d δ : (3 E [ M S+d M S ɛ. This is a easy cosequece of the path-cotiuity of the limit process M, ad of the itegrability of M = sup 0 t 1 M t. Remark that the fuctio f sup 0 t 1 f(t is lower semi-cotiuous o D edowed with the topology of covergece i measure (i.e. the MZ topology; therefore the itegrability of M is a cosequece of coditio (1 of the theorem. Step 2. Suppose that (a is false; the the sequece does ot verify Aldous tightess coditio ([1 p. 335, see also [4; therefore there exists ɛ > 0 such that for every δ > 0 it is possible to determie a subsequece k ad, for every k, a atural stoppig time T k (i.e. a stoppig time for the filtratio geerated by M k ad 0 < d k δ such that (4 E [ k M k T k +d k M k T k ɛ. (I the sequel, for the sake of simplicity of otatios, we will assume that idices have bee reamed so that the whole sequece verifies (4. We choose δ such that, for ay r.v. S whatsoever, we also have (step 1 E [ M S+2δ M S ɛ 4. Step 3. There exists a radom variable T with values i [0,1 such that (M, T coverge i distributio to (M, T o the space D([0, 1, R + [0, 1 equipped with the product topology (D beig equipped with the MZ topology. I fact the laws of (M, T are evidetly tight sice the laws of M are tight o D ([6 p. 368; we poit out that the limit r.v. T is ot a atural stoppig time for the stochastic process M (but it ca be proved that M is a martigale for the caoical filtratio o D [0, 1, i.e. the smallest filtratio that makes M adapted ad T a stoppig time. Step 4. For c ad d i [0, 1, we have the iequality (5 E [ M T +δ+c MT d ε 2. (It is techically coveiet to regard each process M as exteded to [ 1, 2 by puttig M t = M 0 for t < 0 ad M t = M 1 for t > 1 : this eables us to write M T +δ istead of M (T +δ 1. Cocerig the iequality (5, firstly we ote that ( M T +d M T = E [ M T +δ+c M T FT +d ad therefore E [ M T +δ+c M T E [ M T +d M T ɛ.

4 The we remark that (T c is ot a stoppig time, but the r.v. MT c is F T measurable: i fact MT c.i{t t} = M(T t c.i {T t} ad (T t c is F t measurable. Let X = ( MT +δ+c M ( T, Y = M T MT c ad G = FT : Y is G- adapted ad E[X G = 0. We remark that E [X + G = E [X G = 1 2 E [ X G, ad that X + Y X +.I {Y 0} + X.I {Y <0}. Oe gets E [ X + Y G 1 2 E[ X G ; ad, takig the expectatios, the iequality (5. Step 5. There exists a subsequece ad a set I [ 1, 1 of full Lebesgue measure such that the fiite dimesioal distributios of ( MT +t coverge to those of t I (M T +t t I. The proof of this step is a slight modificatio of the argumet give i [6 (p. 364: Dudley s extesio of the Skorokhod represetatio theorem implies that oe ca fid o some probability space (Ω, F, P some radom variables (X, S ad (X, S with values i D [0, 1 such that the laws of (X, S (resp. (X, S are equal to those of (M, T (resp. (M, T ad that, for almost all ω, (X (ω, S (ω coverge to (X(ω, S(ω : to be accurate, the paths t (Xt (ω coverge i measure to the path t (X t (ω ad S (ω coverge to S(ω. We remark that the Skorokhod theorem caot be applied directly sice D is ot a Polish space ([6 p. 372, but Dudley s extesio works well (see [3. By substitutig X with arctg(x, we ca suppose that X ad X are uiformly bouded: therefore we have ( +1 (6 lim dt X T (ω+t (ω X T (ω+t(ω dp(ω = 0. 1 Ω By takig a subsequece, we fid that for every t i a set I [ 1, 1 of full Lebesgue measure, (7 lim X T (ω+t (ω X T (ω+t(ω dp(ω = 0. Ω Hece oe gets easily the covergece of fiite dimesioal distributios of ( M T +t t I. Step 6. We choose 0 d, c 1 such that d + c < δ ad that ( MT +δ+c, M T d coverge i distributio to (M T +δ+c, M T d ; sice the r.v. ivolved are uiformly itegrable, the iequality (5 gives i the limit ad fially we have a cotradictio. E [ M T +δ+c M T d ε 2

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