RATES OF APPROXIMATION IN THE MULTIDIMENSIONAL INVARIANCE PRINCIPLE FOR SUMS OF I.I.D. RANDOM VECTORS WITH FINITE MOMENTS

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1 RATES OF APPROXIMATION IN THE MULTIDIMENSIONAL INVARIANCE PRINCIPLE FOR SUMS OF I.I.D. RANDOM VECTORS WITH FINITE MOMENTS F. Götze 1 ad A. Yu. Zaitsev 1,2 Uiversity of Bielefeld 1 St. Petersburg Departmet of Steklov Mathematical Istitute 2 Abstract. The aim of this paper is to derive some cosequeces of a result of Götze ad Zaitsev [5]. We shall show that the i.i.d. case of this result implies a multidimesioal versio of some results of Sakhaeko [12]. We establish bouds for the rate of strog Gaussia approximatio of sums of i.i.d. R d -valued radom vectors ξ j havig fiite momets E ξ j γ, γ > Itroductio The aim of this paper is to derive some cosequeces of the mai result of Götze ad Zaitsev [5] (see Theorem 2 below). We shall show that the i.i.d. case of this result implies the multidimesioal versio of a result of Sakhaeko [12]. We shall obtai bouds for the rate of strog Gaussia approximatio of sums of i.i.d. R d -valued radom vectors ξ j havig fiite momets E ξ j γ, γ > 2. We cosider the followig well-kow problem. Oe has to costruct o a probability space a sequece of idepedet radom vectors X 1,...,X (with give distributios) ad a correspodig sequece of idepedet Gaussia radom vectors Y 1,...,Y so that the quatity k k (X,Y ) = max X j Y j (1.1) 1 k Key words ad phrases. multidimesioal ivariace priciple, strog approximatio, sums of idepedet radom vectors. 1 Research supported by the SFB 701, Uiversity of Bielefeld 2 Research supported by Russia Foudatio of Basic Research (RFBR) Grat , by RFBR-DFG Grat , by the Grat of leadig scietific scools NSh ad by a program of fudametal researches of the RAS Moder problems of fudametal mathematics. 1 Typeset by AMS-TEX

2 2 F. GÖTZE AND A.YU. ZAITSEV would be as small as possible with sufficietly large probability. The estimatio of the rate of strog approximatio i the ivariace priciple may be reduced to this problem. We omit the detailed history of the problem referrig the reader to Götze ad Zaitsev [5] ad Zaitsev [13]. Below we eed some otatio. The distributio of a radom vector ξ will be deoted by L(ξ). The correspodig covariace operator will be deoted by cov ξ. We write log b = max {1,log b}, for b > 0. By x we shall deote the iteger part of a umber x. The aim of the preset paper is to obtai multidimesioal aalogues of the followig result of Sakhaeko i the case of i.i.d. summads. Theorem 1 (Sakhaeko [12]). Let ξ 1,...,ξ be idepedet radom variables with Eξ j = 0, j = 1,...,. Let γ 2 ad L γ = E ξ j γ <. The oe ca costruct o a probability space a sequece of idepedet radom variables X 1,...,X ad a correspodig sequece of idepedet Gaussia radom variables Y 1,...,Y so that L(X j ) = L(ξ j ), EY j = 0, Var Y j = VarX j, j = 1,...,, ad cγ 2γ L γ, (1.2) where c is a absolute costat. It should be metioed that, i Sakhaeko [12], somewhat more geeral results are proved. Sakhaeko [12] oted that the iequality (1.2) implies the well-kow Rosethal [10, 11] iequality (see Lemma 1 below). After the atural ormalizatio, we see that (1.2) is equivalet to E ( (X,Y )/σ ) γ cγ 2γ L γ /σ γ, where σ 2 = Var ( ξ j). It is clear that Lγ /σ γ, 2 < γ 3, is the well-kow Lyapuov fractio ivolved i the Lyapuov ad Essée bouds for the Kolmogorov distace i the CLT. I this paper we shall derive Theorems 3 ad 4 which are, i fact, rather immediate cosequeces of the followig Theorem 2 which was proved i Götze ad Zaitsev [5]. I Theorem 2, we cosidered the case of (geerally speakig) o-i.i.d. summads. Theorem 3 is derived from Theorem 2 i the particular case, where the summads are idetically distributed. Theorem 3 is a multidimesioal versio of Theorem 1 for i.i.d. summads.

3 MULTIDIMENSIONAL INVARIANCE PRINCIPLE 3 Theorem 2. Suppose that α > 0, ad ξ 1,...,ξ are idepedet R d -valued radom vectors with Eξ j = 0, j = 1,...,. Let γ 2 ad let the quatity L γ be defied by L γ = E ξ j γ <. (1.3) Assume that there exist a positive iteger s ad a strictly icreasig sequece of o-egative itegers m 0 = 0, m 1,...,m s = satisfyig the followig coditios. Let ζ k = ξ mk ξ mk, cov ζ k = B k, k = 1,...,s, (1.4) ad assume that, for all v R d ad k = 1,...,s, where w 2 v 2 B k v,v C 1 w 2 v 2, (1.5) w = C 2 L 1/γ γ /log s, (1.6) with some positive quatities C 1 ad C 2. Suppose that the quatities λ k,γ = satisfy, for some 0 < ε < 1, m k j=m k 1 +1 E ξ j γ, k = 1,...,s, (1.7) C 3 d γ/2 s ε (log s) γ+3 max 1 k s λ k,γ L γ, (1.8) with a positive quatity C 3. The oe ca costruct o a probability space a sequece of idepedet radom vectors X 1,...,X ad a correspodig sequece of idepedet Gaussia radom vectors Y 1,...,Y so that L(X j ) = L(ξ j ), EY j = 0, cov Y j = cov X j, j = 1,...,, ad a1 ( ε 1 d 21/2+α log d ) γ Lγ, (1.9) where a 1 is a positive quatity depedig o α, γ, C 1, C 2 ad C 3 oly. Coditio (1.8) is rather cumbersome, but we shall show that it is trivially satisfied if ξ 1,...,ξ are i.i.d. radom vectors, ad is sufficietly large. The mai results of this paper are the followig Theorems 3 ad 4.

4 4 F. GÖTZE AND A.YU. ZAITSEV Theorem 3. Suppose that α > 0, ad ξ is a R d -valued radom vector with Eξ = 0 ad E ξ γ <, for some γ 2. Let σmax 2 ad σmi 2 be the maximal ad miimal strictly positive eigevalues of the covariace operator cov ξ respectively. Let be a arbitrary positive iteger. The oe ca costruct o a probability space a sequece of idepedet radom vectors X 1,...,X ad a correspodig sequece of idepedet Gaussia radom vectors Y 1,...,Y such that L(X j ) = L(ξ), EY j = 0, cov Y j = cov X j, j = 1,...,, ad the followig iequality is valid: where A = A(γ,α,d) = max a2 A ( σ max /σ mi ) γ E ξ γ, (1.10) { (d 21/2+α (log d) 2) } γ γ(γ+2), d 4 (log d) γ(γ+1) 2, (1.11) ad a 2 is a positive quatity depedig o γ ad α oly. Theorem 4. Let the coditios of Theorem 3 be satisfied. The oe ca costruct o a probability space a sequece of idepedet radom vectors X 1,X 2,... ad a correspodig sequece of idepedet Gaussia radom vectors Y 1,Y 2,... such that L(X j ) = L(ξ), EY j = 0, cov Y j = cov X j, j = 1,2..., ad the followig iequality is valid: a3 A ( σ max /σ mi ) γ E ξ γ, for all, (1.12) where A is defied i (1.11) ad a 3 is a positive quatity depedig o γ ad α oly. Applyig the Chebyshev iequality, we see that, i the coditios of Theorem 4, P { (X,Y ) x } a 3 A ( σ max /σ mi ) γ E ξ γ /x γ, (1.13) for all x > 0 ad all = 1,2,... Clearly, the statemet of Theorem 4 is stroger tha (1.13). A costructio for which (1.13) is valid for d = 1 for fixed ad x = O ( log ) with a 3 depedig o γ ad L(ξ) was proposed by Komlós, Major, ad Tusády (KMT) [7], see also Borovkov [1] ad Major [9] i the case where 2 < γ 3. The Sakhaeko [12] proved Theorem 1 which provides the validity of oe-dimesioal versio of iequality (1.13) for all x o the same probability space. Eimahl [4] obtaied a multidimesioal versio of the KMT result without restrictios o the values of x. Theorem 3 is formulated for fixed. This meas that the probability space depeds o this. However, a applicatio of the result for fixed allows to modify the costructio i such a way that (1.12) is satisfied for all simultaeously o the same probability space (Theorem 4). It suffices to use idepedet costructios

5 MULTIDIMENSIONAL INVARIANCE PRINCIPLE 5 which exist by Theorem 3 with fixed 2 m, m = 1,2,... A versio of Theorem 1 which is valid for all simultaeously o the same probability space was obtaied by Lifshits [8] by a applicatio of the Rosethal iequality (see Lemma 1). We may derive Theorem 4 from Theorem 3 by repeatig the argumets of Lifshits [8]. However, a applicatio of a aalogue of the Rosethal iequality for sums of idepedet o-egative radom variables (see Lemma 2) yields a simpler proof. We do ot try to optimize the depedece of costats o the dimesio d. It is importat that it is a power-type depedece. Fixig α, say, α = 1/2, we could rewrite iequality (1.12) i the form a4 ( σmax /σ mi ) γ E ξ γ, for all, (1.14) where a 4 is a positive quatity depedig o γ ad d oly. I [5], it was oted that Theorem 2 implies the validity of (1.14) for sufficietly large fixed. I this paper we show that there exists a costructio such that (1.14) holds for all simultaeously. If oe or more eigevalues of the operator cov ξ are such that their sum is ifiitesimally small i compariso with the rest of eigevalues, the Theorems 3 ad 4 could provide stroger results if oe applies them to sums of coordiates with large variaces. The other coordiates may be costructed i a arbitary way. The momets of their sums could be estimated by iequalities (1.15) ad (1.16), by aalogy with (2.16). The existece of costructio is esured by the well-kow Berkes Philipp lemma. Sice iequalities (1.15) ad (1.16) are valid for Hilbert-valued radom vectors too, these argumets could be used for obtaiig comprehesive results eve i ifiite dimesioal situatio. For the proof we eed the followig Lemmas 1 3. Lemma 1. Let ξ 1,...,ξ be idepedet radom vectors which have mea zero ad assume values i R d. The E ( ξ γ j a5 E ξ j γ + with a positive quatity a 5 depedig o γ oly. E ξ j 2 γ/2 ), for γ 2, (1.15) This multidimesioal versio of the Rosethal iequality follows easily from a result of de Acosta [2]. I the i.i.d. case, the secod summad i the right-had side of (1.15) grows faster tha the first oe as. Iequality (1.14) shows that this growth correspods to the growth of momets of sums of Gaussia approximatig vectors. Lemma 2 below is proved by Rosethal [10], see also Johso, Schechtma ad Zi [6].

6 6 F. GÖTZE AND A.YU. ZAITSEV Lemma 2. Let ξ 1,...,ξ be idepedet radom variables which are o-egative with probability oe. The E with a 6 depedig o γ oly. ( ξ γ j a6 E ξ j γ + E ξ j γ ), for γ 1, The followig Lemma 3 is a particular case of Theorem from the moograph of de la Peña ad Gié [3]. Lemma 3. Let ξ 1,...,ξ be i.i.d. radom vectors which assume values i R d. The, for all x 0, P max 1 k k o ξ j > x 9 P ξ j > x/30 o. Combied with the well-kow equality E η γ = γ 0 x γ 1 P { η > x } dx, γ > 0, which is valid for ay radom variable η, Lemma 3 allows oe to estimate the momets E max 1 k k ξ γ j a7 E ξ j γ, γ > 0, (1.16) i the case of i.i.d. radom vectors ξ 1,...,ξ, where a 7 is a quatity depedig o γ oly. 2. Proofs The symbols c,c 1,c 2,... will be used for absolute positive costats. The letter c may deote differet costats if we are ot iterested i their umerical values. The same otatio is used for positive quatities which deped oly o α ad γ ivolved i the coditios of Theorem 3. This meas, i fact, that we cosider α ad γ as absolute positive costats. We shall write A B, if there exists a c such that A cb. We shall also use otatio A B, if A B A. Proof of Theorem 3. Without loss of geerality, we assume that γ > 2 ad covξ = I d. Ideed, if γ = 2, the iequality (1.10) is evidet. It is clear that we ca assume that the operator B = cov ξ is o-degeerate. If B I d, we ca cosider the vector B 1/2 ξ istead of ξ.

7 By the Lyapuov iequality, MULTIDIMENSIONAL INVARIANCE PRINCIPLE 7 1 d = E ξ 2 (E ξ γ ) 2/γ. (2.1) Let ξ 1,...,ξ be idepedet copies of the vector ξ ad let L γ be defied i (1.3). Thus, L γ = E ξ γ. Set y = L 2/γ γ = 1 2/γ( E ξ γ ) 2/γ. (2.2) Let us assume that y c 1 d γ/2 (log d) γ+1, (2.3) where the quatity c 1 is as large as ecessary for the validity of the argumets below. Deote u = L2/γ γ log 2 y = (see (2.2)). By (2.1) (2.4), we have ` y log 2 y = 2/γ E ξ γ 2/γ log 2 y (2.4) 1 y2/(γ 2) log 2 y u /4, (2.5) if c 1 is large eough. Let us choose a iteger s 2 ad a strictly icreasig sequece of o-egative itegers m 0 = 0, m 1,...,m s = such that m j m j 1 = u, for j = 1,...,s 1 ad u m s m s 1 2 u. Let ζ k, B k ad λ k,γ be defied i equalities (1.4) ad (1.7). The B k = (m k m k 1 ) I d, k = 1,...,s. (2.6) Accordig to (2.3) (2.5), ad, hece, s Therefore, by (2.4), (2.7) ad (2.8), u u = y log2 y d γ/2 (log d) γ+3 (2.7) log y log s. (2.8) m k m k 1 u u s L2/γ γ, k = 1,...,s. (2.9) log 2 s Now, usig (2.6) ad (2.9), we ca fid C 1 = c 2 ad C 2 = c 3 such that relatios (1.5) ad (1.6) are satisfied. Moreover, by (2.9), λ k,γ = (m k m k 1 )E ξ γ E ξ γ s = L γ s, k = 1,...,s. (2.10)

8 8 F. GÖTZE AND A.YU. ZAITSEV Set Let us show that ε = 1 3 log d 1 3. (2.11) d γ/2 s ε 1 (log s) γ+3 1. (2.12) If s d γ, the (2.12) is evidet. If s < d γ, the, by (2.7), (2.8) ad (2.11), log d log s log y, s ε 1, ad (2.12) follows easily from (2.7). Usig (2.10) ad (2.12), we obtai d γ/2 s ε (log s) γ+3 L 1 γ max λ k,γ 1. (2.13) 1 k s Coditio (1.8) is ow satisfied with C 3 = c 4. Takig ito accout (2.11) ad applyig the statemet of Theorem 2, we see that oe ca costruct o a probability space a sequece of idepedet radom vectors X 1,...,X ad a correspodig sequece of idepedet Gaussia radom vectors Y 1,...,Y such that L(X j ) = L(ξ), EY j = 0, cov Y j = cov X j = I d, j = 1,...,, ad Cosider ow the case, where ( d 21/2+α (log d) 2) γ Lγ. (2.14) y c 1 d γ/2 (log d) γ+1. (2.15) The, for ay costructio o a probability space of a sequece of idepedet radom vectors X 1,...,X ad a correspodig sequece of idepedet Gaussia radom vectors Y 1,...,Y such that L(X j ) = L(ξ j ), EY j = 0, cov Y j = cov X j = I d, j = 1,...,, the followig iequality is valid E max 1 k k X γ j E ξ γ + (d) γ/2, ad E max 1 k k Y γ j (d) γ/2, E max 1 k k E ξ γ + (d) γ/2 X j γ + E max 1 k k Y j γ d γ(γ+2) 4 (log d) γ(γ+1) 2 E ξ γ (2.16)

9 MULTIDIMENSIONAL INVARIANCE PRINCIPLE 9 (we have used Lemma 1 ad relatios (1.1), (1.16), (2.1), (2.2) ad (2.12)). Usig (1.11), (2.14) ad (2.16), we obtai the statemet of Theorem 3. Proof of Theorem 4. Without loss of geerality, we assume agai that γ > 2 ad cov ξ = I d. By Theorem 3, there exists a costructio such that the joit distributio of { X j } ad { Yj } satisfies the relatio where δ m = Eδ γ m 2 m A E ξ γ, m = 1,2,..., (2.17) max 2 m 1 <k 2 m k j=2 m 1 +1 X j k j=2 m 1 +1 Y j. (2.18) Of course, we ca assume that the costructios described above are joitly idepedet for differet m s. By the Lyapuov iequality, By Lemma 2 ad by relatios (2.17) (2.19), E ( 2 k(x,y ) ) γ E ( k Eδ m ( Eδ γ m) 1/γ. (2.19) m=1δ m)γ A( k m=1 ( k 2 m E ξ γ + 2 m/γ) ) γ γ E ξ m=1 2 k AE ξ γ, (2.20) for k = 1,2,... Thus, the statemet of Theorem 4 is proved for = 2 k, k = 1,2,... If 2 k 1 < 2 k, the the statemet of Theorem 4 follows from already proved iequality (2.20). The correspodig costat should be multiplied ot more tha by two. Refereces 1. Borovkov, A. A., O the rate of covergece i the ivariace priciple, Theor. Probab. Appl. 18 (1973), o. 2, de Acosta, A., Iequalities for B-valued radom vectors with applicatios to the strog law of large umbers, A. Prob. 9 (1981), de la Peña, V. H. ad Gié, E., Decouplig. From depedece to idepedece. Radomly stopped processes. U-statistics ad processes. Martigales ad beyod, Probability ad its Applicatios (New York). Spriger-Verlag, New York, Eimahl, U., Extesios of results of Komlós, Major ad Tusády to the multivariate case, J. Multivar. Aal. 28 (1989), Götze, F. ad Zaitsev, A. Yu., Bouds for the rate of strog approximatio i the multidimesioal ivariace priciple, Theor. Probab. Appl. 53 (2008), o. 1, Johso, W. B., Schechtma, G. ad Zi, J., Best costats i momet iequalities for liear combiatios of idepedet ad exchageable radom variables., A. Probab. 13 (1985),

10 10 F. GÖTZE AND A.YU. ZAITSEV 7. Komlós, J., Major, P. ad Tusády, G., A approximatio of partial sums of idepedet RV - s ad the sample DF. I; II, Z. Wahrscheilichkeitstheor. verw. Geb. 32 (1975), ; 34 (1976), Lifshits, M. A., Lectures o the strog approximatio, St. Petersburg Uiversity Press, St. Petersburg,, 2007, pp (i Russia) 9. Major, P., The approximatio of partial sums of idepedet r.v. s, Z. Wahrscheilichkeitstheor. verw. Geb. 35 (1976), Rosethal, H. P., O the subspaces of L p (p > 2) spaed by sequeces of idepedet radom variables, Israel J. Math. 8 (1970), Rosethal, H. P., O the spa i L p of sequeces of idepedet radom variables. II, Proceedigs of the Sixth Berkeley Symposium o Mathematical Statistics ad Probability (Uiv. Califoria, Berkeley, Calif., 1970/1971), Vol. II: Probability theory, Uiv. Califoria Press, Berkeley, Calif., 1972, pp Sakhaeko, A. I., Estimates i the ivariace priciples, I: Trudy Ist. Mat. SO AN SSSR, vol. 5, Nauka, Novosibirsk, 1985, pp (Russia) 13. Zaitsev, A. Yu., Rate for the strog approximatio i the multidimesioal ivariace priciple, Zapiski Nauchykh Semiarov POMI 364 (2009), (Russia); Eglish trasl. i J. Math. Sci. Friedrich Götze Fakultät für Mathematik Uiversität Bielefeld Postfach Bielefeld 1 Germay address: goetze@math.ui-bielefeld.de Adrei Zaitsev St. Petersburg Departmet of Steklov Mathematical Istitute, Fotaka 27, St. Petersburg , Russia address: zaitsev@pdmi.ras.ru