Berkes, István

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Strong approximation and a central limittheorem for St. Petersburg sumsI. Berkes AbstractThe St. Petersburg paradox (Bernoulli 1738) concerns the fair entry fee in a gamewhere the winnings are distributed as P (X = 2k) = 2 k, k = 1; 2; : : :. The tails of Xare not regularly varying and the sequence Sn of accumulated gains has a remark-able asymptotic behavior: as Martin-Lof (1985) and Csorg}o and Dodunekova (1991)showed, Sn=n  log2 n has a class of semistable laws as subsequential limit distribu-tions. This has led to a clarication of the paradox and an interesting and unusualasymptotic theory in past decades. In this paper we prove that Sn can be approxi-mated by a semistable Levy process fL(n); n  1g with a.s. error O(pn(log n)1+")and, surprisingly, the error term is asymptotically normal, exhibiting an unexpectedcentral limit theorem in St. Petersburg theory.MSC 2010. 60E07, 60F05, 60F17.Keywords. St. Petersburg sums, semistable process, strong approximation, centrallimit theorem.1 IntroductionLet X;X1; X2; : : : be i.i.d. r.v.'s withP (X = 2k) = 2 k; (k = 1; 2; : : : ) (1.1)and let Sn =Pnk=1Xk. The asymptotic behavior of the sequence fSn; n  1ghas attracted considerable attraction in the literature in connection with the St.Petersburg paradox concerning the 'fair' entry fee in a game where the winnings areA. Renyi Institute of Mathematics, Realtanoda u. 13-15, 1053 Budapest, Hungary, e-mail:berkes.istvan@renyi.mta.hu. Research supported by FWF grant P24302-N18 and NFKIH grant K108615.1distributed as X. We refer to Csorg}o and Simons [9] for the history and bibliographyof the problem. Feller [10] proved thatlimn!1Snn log2 n= 1 in probability(where log2 denotes logarithm with base 2) and Martin-Lof [14] showedS2k=2k   k d ! Gwhere G is the innitely divisible distribution function with characteristic functionexp(g(t)), whereg(t) =0Xl= 1(eit2l   1  it2l)2 l +1Xl=1(eit2l   1)2 l: (1.2)LettingG denote the distribution with characteristic function exp(g(t=) it log2 )and n = n=2[log2 n]+1, Csorg}o [6] proved thatsupxP Snn   log2 n  x Gn(x)  ! 0 as n!1 (1.3)and determined the precise convergence rate. In particular,Snk=nk   log2 nk d ! G (1.4)if and only if nk ! , a result obtained earlier in [8]. Relation (1.3) shows that theclass of subsequential limit distributions of Sn=n  log2 n is the classG = fG : 1=2   < 1g:If n runs through the interval [2k; 2k+1], then Gn moves through the distributionsGj=2k+1 ; 2k  j  2k+1 representing, in view of G1 = G2, a "circular" path in G. Inview of (1.3), the distribution of Sn=n  log2 n also describes approximately a circularpath, a remarkable asymptotic behavior called in [6] merging.Using a decomposition idea of Le Page, Woodroofe and Zinn [13], in [3] a newrepresentation of the limiting semistable variable of Petersburg sums was given, sim-plifying the theory considerably and leading to new asymptotic information. Let	(x) denote the function on (0;1) which grows linearly from 1 to 2 on any interval[2k; 2k+1), (k 2 Z), let 1; 2; : : : be independent exponential random variables withmean 1 and let Zk =Pkj=1 k. In [3], Lemma 2 it was proved that for any 1   < 2the seriesY () =1Xj=11Zj	Zj  1j	j(1.5)2converges absolutely with probability 1 and the limit distribution G above is iden-tical with the distribution of Y () + c , wherec =1Xk=1f2kg2k  log2 :Note that (1.5) gives an expansion of the semistable variable Y () itself, not of itsdistribution. A similar decomposition, implicit in [3], holds for the partial sums Sn,namely1nSn   an;n d= (1 + "n)nXj=11Zj	Zj(1 + "n)n  1j	jn+ "nan;n (1.6)where "n = Zn+1=n 1, n = n=2[log2 n]+1 is the dyadic location parameter introducedabove andan; =nXj=1	(j=)j: (1.7)For the simple proof, see Section 2. Formula (1.6) expands St. Petersburg sums Snin a similar way as the Edgeworth expansions in Csorg}o [7], Pap [15] expand thedistribution function of Sn . However, (1.6) provides a pointwise expansion and thismakes the formula easier to apply. In particular, (1.6) makes the asymptotic theoryof St. Petersburg sums very transparent. By the law of the iterated logarithm wehave "n = O(n 1=2(log log n)1=2) a.s. and an easy calculation shows that replacing "nby 0 in (1.6) results in an error of oP (1) on the right hand side, and thus we get theresult1nSn   an;n = Y (n) + oP (1); (1.8)which is meant in the sense that for each xed n the variables Sn and Y(n) can bedened on a common probability space such that (1.8) holds. Relation (1.8) thusyields a pointwise version of the merging result (1.3). The purpose of the presentpaper is to prove that actually much more is valid: the partial sum process of (Xn) canbe approximated by a semistable Levy process fL(t); t  0g with L(1) d= Y (1) witha.s. error O(pn(log n)1+") and an asymptotically normal error term, establishing anunexpected central limit theorem in St. Petersburg theory.Theorem 1.1 Let fL(t); t  0g be the Levy process dened byE(exp(iuL(t)) = exp(tg(u)): (1.9)where g is the function in (1.2). Then on a suitable probability space one can denethe St. Petersburg sequence (Xn) and the process fL(t); t  0g jointly such thatnXk=1Xk = L(n) +O(pn (log n)1+") a.s. (1.10)3for any " > 0 anda 1n nXk=1Xk   L(n)!d ! N(0; 1); (1.11)where an  (n log n)1=2.Here cn  dn means that the ratio cn=dn lies between positive constants. Dueto the irregular tail behavior of the random variables in our construction (see theproof of Lemma 2.1), it seems likely that an  (n log n)1=2 in Theorem 1.1 cannot bereplaced by an  c(n log n)1=2 with a constant c.The process L(t) was introduced by Martin-Lof [14] who proved the scaling rela-tiong(2mt) = 2m(g(t)  imt):From this it follows that the transformation t  ! 2t does not change the distributionof the processfL(t)=t  log2 t; t > 0g: (1.12)In particular, L(2)=2   1 d= L(1), and since L(2) d= L(1) ? L(1), the distribution ofL(1) is semistable. In view of the atomic Levy measure in the characteristic functionof Z(1), its distribution is not stable. It also follows thatL(n)=n  log2 n d= L(n)=n   log2 n d= Gn ; (1.13)showing that L(n)=n   log2 n exhibits the merging behavior (1.3) in an ideal way,with zero error. Thus Theorem 1.1 gives an invariance principle for the mergingresult (1.3) and actually, for a class of further limit theorems for (Xn). It showsalso the surprising fact that the partial sum process of (Xn) can be represented as asemistable Levy process with an asymptotically normal perturbation.In a previous paper [1], a strong approximation of St. Petersburg sums with theweaker remainder term O(n5=6+") and without the asymptotic normality of the errorterm was proved by a standard blocking argument. The proof in [1] works for alarge class of i.i.d. sequences (Xn) in the domain of geometric partial attraction of asemistable law G. In contrast, the proof of Theorem 1.1 uses the structure of the St.Petersburg sequences in a substantial way and whether Theorem 1.1 remains validfor a larger class of i.i.d. sequences remains open.Weak and strong approximation of partial sums of i.i.d. random variables (Xn)in the domain of attraction of stable laws were proved in Stout [19], Simons andStout [18], Berkes and Dehling [2]. The remainder terms there are given in termsof the function (x) = xjP (X1 < x)   G(x)j, where G is the limit distribution,and are rather complicated. In the case when (x) is a slowly varying functiontending to 0, lower bounds for the remainder term (valid for any construction) arealso given in [2], leaving only a small gap between the upper and lower bounds.However, in the case of the stable analogue of St. Petersburg sums, when G is astable distribution with parameters  = 1,  =  1 (see e.g. [12], p. 164), we have4(x) = O(x ) for some  > 0 and no lower bounds for the remainder term have beenfound. For the same reason, we do not have universal lower bounds for the remainderterm in the St. Petersburg game and thus, even though Theorem 1.1 determines theprecise stochastic order of magnitude of the error term for a specic construction,the question whether other constructions can give a better error term remains open.2 ProofsWe rst prove (1.6). ClearlyP (X1 > x) = 	(x)=x (x  1): (2.1)Let F denote the distribution function of X1 and let F 1(x) = infft : F (t)  xg beits (generalized) inverse. ThenF 1(x) = 2k for x 2 (1  2 (k 1); 1  2 k]; k = 1; 2; : : :and thusF 1(1  x) = x 1	(x) for 0 < x < 1: (2.2)We also have	(2 k x) = 	(x) for all x 2 R; k 2 Z:As in the Introduction, let 1; 2; : : : ; be independent exp(1) random variables, Zk =Pkj=1 j , k = 1; 2; : : :, putXj;n = F 11  ZjZn+1; 1  j  nand let X1;n  : : :  Xn;n be the decreasing ordered sample of X1; : : : ; Xn. Bythe well known representation of ordered samples (see e.g. [5], page 285 or [17],p. 335), the vector (Z1=Zn+1; : : : ; Zn=Zn+1) is distributed as the ordered sampleUn;1  : : :  Un;n of i.i.d. uniform r.v.'s U1; : : : ; Un in (0; 1) and thus the vectors(X1;n; : : : ; Xn;n) and (X1;n; : : : ; Xn;n) have the same distribution. By (2.2)Xj;n = F 11  ZjZn+1=Zn+1Zj	ZjZn+1= (1 + "n)nZj	Zjn(1 + "n) 1(2.3)where "n = Zn+1=n   1. Now if 2k  n < 2k+1, then n = n=2k+1 and thus from(2.3) we getXj;nn= (1 + "n)1Zj	Zjn 2k+1(1 + "n) 1= (1 + "n)1Zj	Zj(1 + "n)n(2.4)which implies (1.6) immediately.5We turn now to the proof of Theorem 1.1, which uses, as in [2], [18], [19], atermwise approximation of partial sums. As it turns out (see Lemma 2.1 below),the termwise error in this approximation is determined by the second term of theexpansion (1.5) whose tails were shown in [3] to be  x 2. This implies that thetermwise error is in the domain of attraction of the normal law, explaining relation(1.11) in Theorem 1.1. The crucial inuence of the second term of the expansion (1.5)in our approximation problem is similar to the convergence of Markov chains to thestationary distribution whose speed is determined by the second largest eigenvalueof the transition matrix.Lemma 2.1 A St. Petersburg variable X with distribution (1.1) and a random vari-able Y distributed as Y (1) in (1.5) can be jointly dened on a suitable probabilityspace such thatc1x 2  P (jX   Y j > x)  c2x 2 (x  x0) (2.5)for some positive constants c1; c2.Proof. PutW1 =	(Z1)Z1  1; W2 = 	(1  e Z1)1  e Z1 ; W3 =1Xk=1	(Zk)Zk  	(k)k: (2.6)We show that (2.5) holds with X = W2, Y = W3. Clearly, the distribution functionof Z1 is G(x) = 1   e x, (x  0) and thus U = G(Z1) = 1   e Z1 has distributionU(0; 1). Next we observe that for any k 2 Z the function 	(u)=u equals 2k foru 2 [2 k; 2 k+1) and thus for a xed x 2 [2`; 2`+1), ` 2 Z, the inequality 	(u)=u > xholds i u < 2 `. Therefore for x 2 [2`; 2`+1) we haveP (W2 > x) = P (	(U)=U > x) = P (U < 2 `): (2.7)If x  1, then `  0 and thus the last probability in (2.7) equals 2 `; otherwise ` < 0and the last probability in (2.7) equals 1. Thus W2 is a St. Petersburg variable. Onthe other hand, W3 has distribution Y(1) in (1.5) and thus to prove Lemma 2.1 itsuces to show thatc1x 2  P (jW2  W3j > x)  c2x 2 (x  x0): (2.8)We rst prove thatc3x 2  (jW1  W2j > x)  c4x 2 (x  x0) (2.9)with some positive constants c3; c4. As already noted, for any k 2 Z the function	(x)=x equals 2k on the interval Ik = [2 k; 2 k+1). Let now k  0 and assumeZ1 2 Ik. Then 0 < Z1 < 2, 0 < 1  e Z1 < Z1 and1  e Z1 = Z1   12Z21 +O(Z31 ) (2.10)6where the constant in the O is absolute. Thus if Z1 2 Ik, then for k  k0 we have1  e Z1 2 Ik or 1  e Z1 2 Ik+1 and thus = jW1  W2 + 1j =	(Z1)Z1   	(1  e Z1)1  e Z1 (2.11)equals 0 or 2k+1  2k = 2k according as 1  e Z1 belongs to Ik or Ik+1 and in view of(2.10) the second alternative can occur only if Z1 is closer to the left endpoint of Ikthan C2 2k, where C is a constant. But then we have Z1  2 k, 12Z21 O(Z31 )  122 2kand thus the second alternative occurs i Z1 2 Jk, where Jk = [2 k; 2 k + uk) withuk  122 2k. Since the density e x of Z1 is 1 +O(2 k) on Jk, we have P (Z1 2 Jk) 122 2k as k ! 1. We thus proved that the dierence  in (2.11) equals 2k on a aset Ak in the probability space, where the Ak are disjoint for k  k0, P (Ak)  122 2kand otherwise  = 0. ThusP ( > x) X2k>x122 2k  234 k0 as x!1where k0 denotes the smallest integer such that 2k0 > x. This proves (2.9) and wealso see that x2P ( > x) and thus x2P (jW1  W2j > x) uctuate between positiveconstants, without a limit.Next we observe thatW3  W1 =1Xk=2	(Zk)Zk  	(k)kis a tail sum of the series representing Y (1) in (1.5) whose tail behavior is describedby Theorem 5 of [3]; in particular we havec5x 2  P (W3  W1 > x)  P (jW3  W1j > x)  c6x 2 (2.12)with suitable positive constants c5; c6. Theorem 5 of [3] also shows that x2P (jW1  W3j > x) has no limit as x!1. Now (2.9) and (2.12) implyP (jW2  W3j > x)  P (jW2  W1j > x=2) + P (jW1  W3j > x=2)  c7x 2; (2.13)proving the upper half of (2.8). To prove the lower half, we note thatP (jW2  W3j > x)  P (W3  W2 > x)  P (W3  W1 > 3x=2; W1  W2 >  x=2)= P (W3  W1 > 3x=2)  P (W3  W1 > 3x=2; W1  W2   x=2) (2.14) P (W3  W1 > 3x=2)  P (W3  W1 > 3x=2; jW1  W2j  x=2):For any t  0, setVt =1Xk=2	(t+ Zk)t+ Zk  	(k)k; (2.15)7where Zk =Pkj=2 j for k  2. We claim that there exists a positive constant Csuch that for any 0  t  1 we haveEjVtj  C: (2.16)Since the sequence (Zk) has the same distribution as (Zk), for t = 0 relation (2.16)follows from Lemma 2 of [3]. As inspection shows, the properties of (Zk) used in theproof in [3] remain valid for the sequence (Zk + t) for any xed t  0 and moreover,the inequalities in [3] hold uniformly for 0  t  1, proving (2.16). Now, conditionallyon Z1 = t, W3  W1 becomes Vt in (2.15) which is independent of 1 = Z1 and thusof  = jW1  W2 + 1j in (2.11) and consequently for x  x0P (W3  W1 > 3x=2; jW1  W2j  x=2 j Z1 = t) P (W3  W1 > 3x=2;   x=4 j Z1 = t)= P (Vt > 3x=2)If(t)  x=4g 23xEjVtjIf(t)  x=4g  Cx 1If(t)  x=4g;(2.17)where (t) is the expression in (2.11) with Z1 replaced by t. If Z1 is bounded awayfrom 0, then jW1 W2j is bounded above, or putting dierently, if jW1 W2j is large,then Z1 is near 0. Thus integrating (2.17) over 0  t  1 with respect to P (Z1 2 dt)we getP (W3  W1 > 3x=2; jW1  W2j  x=2)  Cx 1P (jj > x=2)  C 0x 3 (2.18)for suciently large x, where in the last step we used (2.9). Now using (2.12), (2.14)and (2.18) we get the lower half of (2.8).Proof of Theorem 1.1. For the vector (X;Y ) in Lemma 2.1, let H denote thedistribution function of X   Y and putU(x) =Zjtjxt2dH(t):Using Lemma 2.1 and integration by parts we getU(x) =  x2(1 H(x) +H( x)) +Z x02t(1 H(t) +H( t))dt= O(1) +Z x02t(1 H(t) +H( t))dt (2.19)provided that x and  x are continuity points of H. Using Lemma 2.1 again for thelast integral it follows thatc8 log x  U(x)  c9 log x and U(2x)  U(x) = O(1) as x!1 (2.20)with suitable positive constants c8 and c9. Thus limx!1 U(2x)=U(x) = 1, i.e. thenondecreasing function U is slowly varying. Further, (2.5) implies that H has a nite8expectation. Let now (Xn; Yn) be i.i.d. copies of the vector (X;Y ) in Lemma 2.1.By the slow variation of U , X   Y is in the domain of attraction of the normal law,specically we have1annXk=1(Xk   Yk   c) d ! N(0; 1) (2.21)where c = E(X   Y ) and the norming sequence (an) satisesa2n  nU(an) as n!1: (2.22)(See e.g. [11], p. 580, Theorem 3.) Using (2.22) and the rst relation of (2.20), weget by a simple calculationc10(n log n)1=2  an  c11(n log n)1=2 (2.23)with suitable constants c10; c11. Recall now that along the sequence n = 2k we have1nnXk=1Xk   log2 n d ! G;1nnXk=1Yk   log2 n d ! G (2.24)where G = G1=2 is the semistable distribution dened after (1.2). The rst relationhere follows from (1.4) and the second from (1.13), sincePnk=1 Ykd= L(n). Relation(2.23) shows that replacing 1=an by 1=n in (2.21), the left hand side will converge to0 in probability and adding the second relation of (2.24) yields1nnXk=1Xk   log2 n  c d ! Gwhich, together with the rst relation of (2.24), implies c = 0 and thus (2.21) yields1annXk=1(Xk   Yk) d ! N(0; 1): (2.25)Since the process fPnk=1 Yk; n  1g has the same distribution as fL(n); n  1gwhere L is the Levy process dened by (1.9), relation (1.11) is proven.To prove (1.10), let bn =pn(log n)1+", " > 0. Then using Lemma 2.1, (2.20) andintegration by parts one can easily verify the relations1Xn=11b2nZjxj<bnx2dH(x) < +1; 1bnnXk=1Zjxj<bnxdH(x)  ! 0 (2.26)and 1Xn=1P (jX   Y j  bn) < +1:9(In the case of the second relation of (2.26) replace the domain fjxj < bng of inte-gration by fjxj  bng in view of E(X   Y ) = 0.) Thus using Theorem 6.8 in Petrov[16], p. 211 we get1bnnXk=1(Xk   Yk)  ! 0 a.s.;yielding (1.10).References[1] Berkes, I.: Strong approximation of the St. Petersburg game. Statistics, to ap-pear.[2] Berkes, I. and Dehling, H.: Almost sure and weak invariance principles forrandom variables attracted by a stable law. Prob. Theory Rel. Fields 83 (1989),331{353.[3] Berkes, I., Gy}or, L. and Kevei, P.: Tail probabilities of St. Petersburg sums,trimmed sums, and their limit. J. Theor. Probability, to appear.[4] Bernoulli, D.: Specimen theoriae novae de mensura sortis. Comm. Acad. Sci.Imp. Petropolitanae 5 (1738), 175-192. German translation: A. Pringsheim:Versuch einer neuen Theorie der Wertbestimmung von Glucksfallen, Verlag vonDuncker & Humblot, Leipzig, 1896.[5] Breiman, L.: Probability. Second edition, SIAM, 1992.[6] Csorg}o, S. Rates of merge in generalized St. Petersburg games. Acta Sci. Math.(Szeged) 68 (2002), 815{847.[7] Csorg}o, S.: Merging asymptotic expansions in generalized St. Petersburg games.Acta Sci. Math. (Szeged) 73 (2007), 297{331.[8] Csorg}o, S. and Dodunekova, R.: Limit theorems for the Petersburg game, in:Trimmed sums and extremes (M. G. Hahn, D. M. Mason, D. C. Weiner, eds.)pp. 285{315. Progress in Probability, Vol. 23, Birkhauser, Boston 1991.[9] Csorg}o, S., Simons, G.: A bibliography of the St. Petersburg paradox. Universityof Szeged, 2014.[10] Feller, W.: Note on the law of large numbers and "fair" games. Ann. Math.Statist. 16 (1945), 301{304.[11] Feller, W.: An introduction to probability theory and its applications, Vol II,second edition. Wiley, 1971.[12] Gnedenko, B.V. and Kolmogorov, A.N.: Limit distributions of sums of indepen-dent random variables. Addison-Wesley, 1954.[13] LePage, R., Woodroofe, M. and Zinn, J.: Convergence to a stable distributionvia order statistics. Ann. Probab. 9 (1981), 624{632.10[14] Martin-Lof, A.: A limit theorem which claries the `Petersburg paradox'.J. Appl. Probab. 22 (1985), 634{643.[15] Pap, Gy.: The accuracy of merging approximation in generalized St. Petersburggames. J. Theoret. Probab. 24 (2011), 240{270.[16] Petrov, V.V.: Limit theorems of probability theory. Oxford University Press,1995.[17] Shorack, G. and Wellner, J.: Empirical processes with applications to statistics.Wiley, 1986.[18] Simons, G. and Stout, W.: A weak invariance principle with applications todomains of attraction. Ann. Probab. 6 (1978), 294{315.[19] Stout, W.F: Almost sure invariance principles when EX21 = 1. Z. Wahrsch.Verw. Gebiete 49 (1979), 23{32.11

Strong approximation and a central limit theorem for St. Petersburg sums

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Strong approximation and a central limit theorem for St. Petersburg sums

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