Jayaram Sethuraman

Department of Statistics

Floriday State University

Distribution of Frequencies of Patterns in Bernoulli Sequences

Consider a sequence of independent Bernoulli random variables X1, X2, . . . with P(Xn =1) = 1 - P(Xn = 0) = pn = 1/(B +n), n = 1, 2, . . .. Such sequences arise in record value data and in problems concerning random permutations. Let Z1, Z2, Z3, . . . be the frequencies of patterns {1,1}, {1,0,1} {1,0,0,1}, . . . , etc. in the sequence X1, X2, . . . . Persi Diaconis showed that the distribution of Z1 is Poisson(1) when B = 0. Later, (Z1, Z2, . . .) was shown to be independent with Poisson distributions with various parameters, when B = 0. We present four different proofs for the joint distribution of (Z1, Z2, . . .) for the general case with B ≤ 0. The joint distribution is not independent, but there is a random variable V, and given V, the frequencies (Z1, Z2, . . .) are independent with Poisson distributions with parameters dependent on V. The first proof is combinatorial in character and involves using recurrence equations to obtain factorial moments. The other proofs are more probabilistic and use a different model to generate the Bernoulli sequence. When the Bernoulli probabilities pn are not of the form 1/(B + n), the joint distribution of (Z1, Z2, . . .) is an open problem. Keywords: Random permutations, Record values.

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