Studia Scientiarium Mathematicarum Hungarica 34. (1998)

1-3. szám - Kiesel R.-Stadtmüller U.: Erdős-Rényi-Shepp laws for dependent random variables

Studia Scientiarum Mathematicarum Hungarica 34 (1998), 253-259 ERDÖS-RÉNYI-SHEPP LAWS FOR DEPENDENT RANDOM VARIABLES R. KIESEL and U. STADTMULLER To the memory of Alfréd. Rényi Abstract We prove an Erdős Rényi-Shepp law for the partial sums of a uniform strong mixing stationary sequence. 1. Introduction and main results While there is a large amount of literature on versions of the Erdős- Rényi-Shepp law for sequences of independent, identically distributed (i.i.d.) random variables, see e.g. [11, 3, 4, 5, 6, 7, 12, 13], not much is known for dependent random variables (see [9] for a first result in this direction). Using a recent large deviation result by Bryc [1] we proceed to a more general setting. Let {Xn} be a stationary sequence. We define (FTM~o{Xk :n^k%m) the canonical cr-algebra generated by Xn,..., Xrn and the 0-mixing coefficient We say that a sequence {An} is 0-mixing if 0n —> 0 for n —> oo. We shall need the following hypergeometric rate of convergence (1.1)) eKn(j)n—>0 (n—>oo) for each K t 0. We have the following large deviation theorem by Bryc [1]. THEOREM B. Let {An} he a stationary 0-mixing sequence of random variables such that |Xi| ^ C < oo and (1.1) holds. Define Zn = (Xi + ... +Xn)/n, n't 1. Then the limit limn 1 log E (exp(nAZn)} = L(X) exists for each A G K and the function I: R —> [0, oo] defined by I(x) := snp{xA — L(A)} 1991 Mathematics Subject Classification. Primary 60F15; Secondary 60G50. Key words and phrases. Erdős-Rényi laws, mixing random variables. 0081-6906/98/$ 5.00 ©1998 Akadémiai Kiadó, Budapest

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