On Sums of Products of Bernoulli Variables and Random Permutations

A study of counts of Bernoulli strings via conditional Poisson processes

“…an unpublished manuscript of Diaconis, ChernHwang-Yeh [4] (which derives approximations via several probability distances), Móri [11] (which uses generating functions), Joffe-Marchand-Perron-Popadiuk [8] (which gives a formula for the 1-string count in a general finite independent Bernoulli sequence in terms of a nonhomogeneous Markov chain and which uses generating functions), and references therein in these and the above papers.…”

A note on records in a random sequence

“…Consequences of this are, that the d-marked T k 's is a Poisson process Π d with intensity Hahlin [6] derived the distribution of M 1 . In a number of papers M 1 and related random variables have been studied by different methods, see for example [4], [7], [8], [12], [14] and [16], and the references in these papers. The random variables M 1 , M 2 , ... also occur in connection with random permutations and Ewens Sampling Formula, see [1], [2] and [5].…”

On the number of runs for Bernoulli arrays

“…In Section 2 we derive a general recurrence (Lemma 1) for the probability generating function (PGF) of S n using a conditioning argument (as in [7,Proposition 1]). We then proceed to more explicit representations (Corollary 1) for the particular case of multinomial rows (as in Example 1), which will lead to an explicit expression (Theorem 1) for the probability mass function of S n in the identically distributed case.…”

“…r = 1 or as pertaining to the marginal distribution of Z j ) has generated much recent interest, analysis, and interpretation; see [4], [5], [6], [7], [9], and [10], among others. Although our main interest relates to cases where the Z j s are not independent, it is worthwhile here summarizing results that may be inferred in cases where the columns are independent.…”