Stationary 1-dependent Counting Processes: from Runs to Bivariate Generating Functions

TL;DR

This paper derives a formula for the bivariate generating function of stationary 1-dependent counting processes, linking it to run probabilities and connecting to known distributions and combinatorial structures.

Contribution

It provides a new probabilistic formula for bivariate generating functions of 1-dependent processes, unifying different approaches and extending known results.

Findings

Derived a probabilistic formula for bivariate generating functions.

Connected the formula to Eulerian distribution and determinantal point processes.

Compared the new formula with existing combinatorial and probabilistic expressions.

Abstract

We give a formula for the bivariate generating function of a stationary 1-dependent counting process in terms of its run probability generating function, with a probabilistic proof. The formula reduces to the well known bivariate generating function of the Eulerian distribution in the case of descents of a sequence of indepependent and identically distributed random variables. The formula is compared with alternative expressions from the theory of determinantal point processes and the combinatorics of sequences.