Connections Between Bernoulli Strings and Random Permutations

A sequence of random variables, each taking only two values “0” or “1,” is called a Bernoulli sequence. Consider the counts of occurrences of strings of the form {11}, {101}, {1001}, … in Bernoulli sequences. Counts of such Bernoulli strings arise in the study of the cycle structure of random permutations, Bayesian nonparametrics, record values etc. The joint distribution of such counts is a problem worked on by several researchers. In this paper, we summarize the recent technique of using conditional marked Poisson processes which allows to treat all cases studied previously. We also give some related open problems.