On the distribution of the longest success-run in Bernoulli trials

Abstract

The success-run in a sequence of Bernoulli trials was considered in a large number of papers. W. Feller [4] concentrates his attention on the number of runs what find the principal application in the theory of test of randomness and tests of homogeneity. Many papers deal with random variable Zn introduced by Erdos and Renyi [2] (see also Erdos and Revesz [3]), defined as the length of longest head-run during n coin tossings. They give the asymptotic estimations of that random variable if n tends to infinity. The asymptotic estimation of the distribution of Zn was given by Antonia Foldes [5]. Note that in the theory of extremes in random sequences it is proved that in a sequence of random variables geometrically distributed the maximum linearly standarized do not have a limiting distribution (see [7], p. 26). In [1] the multivariate extension of the problem of the largest cube filled up by successes which may be found in a random lattice in a cube of range n. The problem has very much of practical implications. Our purpose in this paper is to show the recurrent formulas useful in the calculation of the distribution of the random variable Zn and in the method of calculation of its expected value and also in the test of the accuracy of the limiting estimations.