Rates of convergence in the functional CLT for martingales

C. R. Acad. Sci. ProbabilitWProbability (Statistique/Sfatist

Paris,

t. 328, SCrie Theory

I, p. 509-513,

Rates of convergence for martingales Bernard

in the functional

CLT

COURBOT

IRMAR, E-mail: (Rqu

1999

LJnivcrsit6 de Rennes I, B.P. bernard.courbotQhumana.univ-nantes.fr le 11 dikembre

Abstract.

1998.

accept6

35042

le 21 janvier

Version

cedex,

France

1999)

We obtain optimal rates of convergence in the Skorokhod embedding framework in the functional CLT for continuous time martingales. 0 Acadtmie des ScicncesElsevier, Paris

Vitesses de convergence pour martingales RCsumC.

Rennes

duns le TCL fonctionnel

Des vitesses de convergence optimales dans le cadre de la m&hode du plongement sont obtenues pour le TCL fonctionnel pour martingales en temps continu. 0 AcadCmie des Sciences/Elsevier, Paris

franqaise

abdgke

I1 est bien connu que la technique du plongement de Skorokhod appliquee 2 l’estimation de la vitesse de convergence d’une martingale vers un mouvement brownien dans le TCL fonctionnel conduit A des vitesses d’ordre au mieux n-4 quand on l’applique au cas d’une somme i.i.d. n-3 XL, & ( > de variables centrkes rCduites. Nous amClioronsici les r&ultats de Coquet, MCmin et Vostrikova dans [I] : en estimant la distance de Prokhorov II(I\II, B), oti A4 est une martingale en temps continu et B un mouvement brownien tous deux unidimensionnels d6finis sur [0, T], nous obtenons des vitesses conduisant ?I une estimation en 71-i logi n dansle cas i.i.d. born& done non amCliorablesau facteur logarithmique p&s par la mCthode du plongement. Si plus gCnCralementM, dont le compensateurp&visible de la mesurede sautsest not6 v, vCrifie L, = lE[lzl” *z+] < 00, nous obtenons : II(M. B) = O(&lognvIt 013~1,’ est la distance de Ky Fan des variations quadratiques pr&isibles (M)

v &ogLI,j~), et (B). Dans le cas

discret indkpendant, la vitesse est un c3(L$ [log L,) ‘)

optimal dans ce cas

Note

pr6sentCe par Paul

au lieu du 0(,5,!$)

DEHEUVELS.

0764~4442/99/03280509 0 AcadCmie des Sciences/Elsevier, Paris

509

B. Courbot

(c$ [lo]), mais ameliore le resultat pour martingales en temps continu de [l] (et en temps discret de [6] pour p > 10). Cette amelioration est obtenue par l’utilisation de l’inegalite exponentielle suivante pour martingales de carre integrable, interessante par elle-meme : pour tous t, a, 6,~ > 0,

ou $ est definie pour z,l~ > 0 par : $(z, y) = (z + y) log (!! + 1) - y. Si dansle cas general desmartingales, la technique du plongement fournit actuellement les meilleures vitesses, une methode de discretisation est applicable si A!f est une martingale a accroissements independants,donnant des estimations dans le cas i.i.d. toujours inferieures a 0 (n- “) mais meilleures que O(n.-i) des que p > 10.

1. Introduction In [I], Coquet, MCmin and Vostrikova used the technique of the martingale embedding into a brownian motion to obtain rates of convergence in the functional CLT for continuous time martingales. Their result, when applied to the elementary case of an i.i.d. sum 5-c; c;=, I!? of ( > centered normalized r.v. &, gives rates in Q(rr,-e) up to a logarithmic factor if the &. admit pth order moments, and in 0(n-i) if they are bounded, whereas it is well known that best rates available with the embedding scheme are in c3(n-“) in this case. We obtain here general estimatesgiving this last rate in the i.i.d. bounded caseby using a convenient exponential inequality instead of Lenglart’s one in [l]. Let us consider a square integrable martingale A4 and a Brownian motion B defined on [0, T] for a fixed T > 0, taking their values in the Skorokhod’s space D endowed with the Jr topology for which it is polish. The measurev is a “good version” of the compensator of the jump measure~1of M, and we write for a function cp: cp* v instead of & J, ip(z, s)v(dlc:, ds); analogous notation is in force for the measure (,u - ZI). (AI) ([Ml) denotes the predictable (optional) quadratic variation of a martingale M (see [7] or [9] for these notations). The Prokhorov’s distance of A4 and B is denoted II(A4, I3) whereas, for two processestaking values in D, the Ky Fan distance (distance in pmhabiliry) is denoted Ic(X, Y); we will use the upper estimate: ICrr(X;Y) = inf {& > 0 : P[]\X - Y/l, > E] 6 E}, where l].]lT is the supremum norm:

IIXIIT = ““Po 0, we consider the martingale with truncated jumps: IMP = M - ‘~n(l~,l>~~ * (~1,- II) and we will control II(M, B) by the following quantities: h;~. = KZ,((M);

(B)), 6: = Ic~((~c*$,~l>p) *v,O)

and L, = iE[I:cl”*~~]

(Lyap~nov’s ratio), when I:r/’ * VT is integrable for a p > 2. The interested reader will find detailed proofs of the following results in (31. 2. An exponential inequality

for local martingale

In [9], Liptser and Shiryaev introduced a cumulant process to get exponential inequalities for semimartingaleswith bounded jumps. For X = A + M , with a predictable process A with bounded

510

Rates of convergence

in the functional

CLT for martingales

variation and a local martingale M, if v is the predictable compensator of the jump measure of X and Xc the continuous martingale part of X, this cumulant is defined by:

as soon as this last integral is defined for A E [0,X”], Xa > 0. exp (XX) is a positive I(GX supermartingale by Ito’s formula, so that a supermartingale inequality for X = k -sn{z,a}*P ((I > 0) gives the following exponential inequality, which generalizes Fuk and Nagaev’s one ([4]) and improves Kubilius-Memin’s one ([S]): Denoting

E(X)

the Doleans exponential

THEOREM

1 (Fuk-Nagaev’s

type inequality).

of a semimartingale

X,

- For all i;, IL,b: 5 > 0, we have:

where II, is dejined for 2, p > 0 by: $(x, y) = (x + TJ)log (I + 1) - TJ.

3. Estimates

of II(M,

B)

Further on, C denotes a positive constant, the value of which can vary from place to place and p is a real greater than 2. The problem is first divided into two estimates:

LEMMA

2. - For sujjiciently small p: (i) II(n/r,MO)

< (a~!)

‘; (ii) II(IcI,lcIO)

< &$*Li.

The first result is established like in [l], by giving estimate of Icu (M, MO) with Lenglart’s inequality, whereas the second one, valid as soon as IzIP * v is integrable, is an easy consequence of Markov’s inequality. LEMMA

3. - Zftc = max

c;(~~.Jlog~z~~[-~ i inf (,6 > 0 : [I]log,0Ib n(bPQ?)

3 CIIY~}) is small enough, then:

< CK+log&

Scheme of the proof. - Like ---in [l],

we embed the martingale 2Mfi in a standard Brownian motion W defined on a space (s2,5, IF, P) and get by this way a If-change of time (T) such that ,&(iMa) = C&VT). By Strassen-Dudley’s theorem we know that II(AJ”! B) = TI(WT, W’lB)) < Icl: (IV,, W(B)) so that it is sufficient to estimate fl[llW7 - W(B) (IT > E] for E > 0. For an o > 0, we divide this estimation into:

~[llwi - YmIIT> c] < qpc

- W(B)IIT > E;117- (WI/, 6 a] + q117- (WIT > 4.

A calculus similar to that in [l] gives the first estimate by considering the modulus of continuity of W. The core of the problem lies on the second term, that we divide once again into: p[IIT - (B)llT

> Q’] 6 11 +

12

+13+

14,

511

B. Courbot

where: I1

=

P[llT

-

[WIIT

>

y],

-@II,2a]. [II{““)- (~)ll,>f]; 14=P[ll(M)

13 = i=’

I3 and I4 do not present any difficulty. For I 2, we apply Theorem 1 to the discontinuous martingale ( [M”] - (MP)) and get:

12 6

2exp{-11,(CF2,CaF2)}

+2P[II(M)-(B)IIT

> t].

The same theorem with now (T - [W7]), w h’ICh .1sa @,)-martingale by the embedding theorem, yields after rather technical calculations for U, 11> 0, vpP2 sufficiently great: I1 < C[exp { --G(vu.-~! Gnu-‘)

} + exp { -li, (COP, Cc@-2) } + 21-2/P + U?ijP

+ww

- (NT > co1 + q”2q,x;,>[f) * vr 2 Cal].

We finish the proof by gathering the previous estimates and optimizing the various parameters. 0 Considering both Lemmas 2 (i) et 3, we find easily that: rI(M,B)

= +q1ogm/b

+KQOgPq:

result which does not improve that in [l] without supplementary integrability hypothesis, since the dominating term is II(A!f, A4i’). However, if M is an i.i.d. sum of bounded T.v., we find in [I] the >

whereas a straight application of Lemma 3 yields: rI(M, B) = O(d

log+ 7%)

i.e. an unimprovable result up to a logarithmic factor in the embedding framework. When IxI’ *VT is integrable for p > 2, we use Lemma 2 (ii) instead of (i) and find the following estimate: THEOREM

4. - II(M,B)

= 6(h-~~logn1~Ii V L$‘llogL,l~).

In the i.i.d. case, this result writes: II(M,B)

= O(&ogI,,I~)

= +-qlognlq

optimal for sum of independent r.v. ([IO] for example),

4. Remarks 1) For discrete time martingales, this result improves for p > 10 that of Hall and Heyde ([6], Lemma 4.9, where the rate in the i.i.d. case is always less than n-i. Haeusler found in [5] a better estimate (J n-G%+ V n,-i log3 7~ but under restrictive conditions: in particular, 1~1~* V~ has to be ( > a.s. bounded instead of integrable.

512

Rates of convergence

in the functional

CLT for martingales

2) Theorem I allows an improvement of the estimate in [2] where an only optional characteristic is used: we obtain:

where El.7 = Icu( [M]? [B]) is the distance of the quadratic variations [M] and [B]. As for the predictable estimates, the rates are O(K~) up to a logarithmic factor in the i.i.d. bounded case, but for r.v. such that [nil] is deterministic (Rademacher r.‘.v. for example). 3) Since these estimates are optimal in the embedding framework, one can naturally ask whether they can be improved by another technique. The answer is unknown in the general case but, if h/l is a martingale with independent increments, we can use optimal results in the discrete time case for a discretization of the processesn/r and B. More precisely, we write:

-where B” is a Brownian motion with variance (M”) and where AJL’ (resp. B, B”) are the processes obtained from IM!’ (resp. B: B”) after discretization on a subdivision of [0, T]. II(B, B) and -II(M”, JJ”) are estimated with exponential inequalities, II(BP, B) by the embedding scheme; moreover, Sakhanenko’s theorem ([ 1I], p. 4) gives us an optimal upper estimate of II(A@, W). After optimization, it follows that in this particular case of P.I.I. we have:

which yields in the i.i.d. case rates always less than O(?r,-i),

but better than O(r),-f)

for p > 10.

References [I] Coquet F., Mtmin J., Vostrikova L., Rate of convergence in the functional central limit theorem for likelihood ratio processes, Math. Meth. Statis. 3 (2) (1994) 89-I 13. [2] Coquet F., MCmin J., Vostrikova L., Majoration de la distance de Levy-Prokhorov entre une martingale et le mouvement brownien, distance entre les processus associts, Seminaires de Probabilite de Rennes, IRMAR Rennes, 1993. [3] Courbot B., Vitesses de convergence pour les martingales dana le theoreme central limite fonctionnel : comparaison de methodes, cadres uni et multidimensionnel, These de doctorat de I’UniversitC de Rennes I, 1998. [4] Fuk D.K., Nagaev S.V., Probability inequalities for sums of independent random variables, Th. Probab. Appl. 16 (4) (197 I ) 643-660. 151 Haeusler E., An Exact Rate of Convergence in the Functional Central Limit Theorem for Special Martingale Difference Arrays, Z. Wahrsch. verw. Gebiete 65 (1984) 523-534. (61 Hall P., Heyde CC., Martingale Limit Theory and its Application, Academic Press, New York, 1980. [7] Jacod J.. Shiryaev A.N., Limit Theorems for Stochastic Processes, Springer, Berlin-Heidelberg-New York. 1987. [8] Kubilius K., MCmin J.. InegalitC exponentielle pour les martingales locales, C. R. Acad. Sci. Paris 319 (1994) 733-737. [9] Liptser R.S., Shiryaev A.N., Theory of Martingales. Kluwer Academic Publishers, Dordrecht, 1986. [ 101 Sakhanenko AI., On unimprovable estimates of the rate of convergence in invariance principle, in: Nonparametric Statistical Inference, Colloquia Math. Sot. Janos Bolyai 32, Budapest, 1980, pp. 779-784. [I I j Sakhanenko A.I., Convergence rate in invariance principle for non-identically distributed variables with exponential moments, in: Borovkov A.A. (Ed.), Limit Theorems for Sums of Random Variables. Adv. Probab. Th., Springer, New York. 1985, pp. 2-73.

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