##
**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 X_{1},X_{2},... with P(X_{1} = -1) = P(X_{1} = 1) = ½. Denote the partial sums by S_{0} = 0, S_{n} = X_{1}+...+X_{n}, and set I(N,K) = **max**_{0 \leq n \leq N-K}(S_{n+k}-S_{n}), 1 \leq K \leq N, N = 1,2,.... For K_{N}/ log N bounded away from zero and infinity, the authors provide a precise characterization of the strong limiting behaviour of I(N,K_{N}) 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 K_{N} = [C log N] with C > 1 or K_{N} = log N+o(log N) are studied in more detail. A summary of further results on I(N,K_{N}) for K_{n} \leq C log N or log N << K_{N} \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

© European Mathematical Society & FIZ Karlsruhe & Springer-Verlag