## Zentralblatt MATH

Publications of (and about) Paul Erdös

Zbl.No:  632.60027
Autor:  Deheuvels, P.; Erdös, Paul; Grill, K.; Révész, P.
Title:  Many heads in a short block. (In English)
Source:  Mathematical statistics and probability theory, Vol. A, Proc. 6th Pannonian Symp., Bad Tatzmannsdorf/Austria 1986, 53-67 (1987).
Review:  [For the entire collection see Zbl 623.00015.]
Consider an i.i.d. sequence X1,X2,... with P(X1 = -1) = P(X1 = 1) = ½. Denote the partial sums by S0 = 0, Sn = X1+...+Xn, and set

I(N,K) = max0 \leq n \leq N-K(Sn+k-Sn),  1 \leq K \leq N,  N = 1,2,....

For KN/ log N bounded away from zero and infinity, the authors provide a precise characterization of the strong limiting behaviour of I(N,KN) in terms of UUC, ULC, LUC and LLC classes.
These results are based upon sharp probability inequalities on I(N,K), which are developed first. The cases of KN = [C log N] with C > 1 or KN = log N+o(log N) are studied in more detail. A summary of further results on I(N,KN) for Kn \leq C log N or log N << KN \leq N completes the picture.
Reviewer:  J.Steinebach
Classif.:  * 60F15 Strong limit theorems
60F10 Large deviations
Keywords:  Erdös-Rényi law; strong theorems; large deviations; number of heads; law of iterated logarithm; sharp probability inequalities
Citations:  Zbl 623.00015

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