The rencontre problem

03/08/2020

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by   F. Thomas Bruss, et al.

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Let {X^1_k}_k=1^∞, {X^2_k}_k=1^∞, ..., {X^d_k}_k=1^∞ be d independent sequences of Bernoulli random variables with success-parameters p_1, p_2, ..., p_d respectively, where d ≥ 2 is a positive integer, and 0<p_j<1 for all j=1,2,...,d. Let S^j(n) = ∑_i=1^n X^j_i = X^j_1 + X^j_2 + ... + X^j_n, n =1,2 , .... We declare a "rencontre" at time n, or, equivalently, say that n is a "rencontre-time," if S^1(n) = S^2(n) = ... = S^d(n). We motivate and study the distribution of the first (provided it is finite) rencontre time.