## Zentralblatt MATH

Publications of (and about) Paul Erdös

Zbl.No:  362.60044
Autor:  Erdös, Paul; Revesz, P.
Title:  On the length of the longest head-run. (In English)
Source:  Top. Inf. Theory, Keszthely 1975, Colloq. Math. Soc. Janos Bolyai 16, 219-228 (1977).
Review:  [For the entire collection see Zbl 349.00030.]
Let X1,X2,... be a sequence of i.i.d.r.v.'s with P(X1 = 0) = P(X1 = 1) = ½, further put Sn = X1+X2+...+Xn, I(N,k) = max0 \leq i \leq N-k(Si+k-Si) and let ZN be the largest integer for which I(N,ZN) = ZN that is ZN is the length of the longest head run. The main result says: for any \epsilon > 0 there exists a r.v. N0 = N0(\epsilon,\omega) such that ZN \geq [ log N- log log log N+ log log e-2-\epsilon] for all N \geq N0 but for any \epsilon > 0 there exists a sequence N1 = N1(\omega ,\epsilon) < N2 = N2(\omega,\epsilon) < ... such that ZNi \leq [ log Ni- log log log Ni+ log log e-1+\epsilon] (i = 1,2,...). The base of the log is 2.
Classif.:  * 60F15 Strong limit theorems
60C05 Combinatorial probability

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