Ascending runs of sequences of geometrically distributed random variables: A probabilistic analysis
The authors consider geometrically distributed random variables (RVs), sequences obtained by such independent RVs, and ascending runs (i.e. continuously increasing sequences) on the invoked sequences. The aim of this paper is to study several asymptotic properties of the sequences of ascending runs of geometrically distributed RVs. The paper comprises the combinatorial structure of polyomino, a set of points made up of integer pairs such that every point can be reached from another by a sequence of steps in the lattice plane (up to translation). Using a description of the ascending runs either as a (Markov chain) stochastic process or as a polyomino, the authors derive several new asymptotic probability distributions of RVs and processes such as asymptotic Markov chain and limiting trajectories, number of runs and run length distribution, hitting time to a length \(k\) run and maximum run length. The obtained results are mainly based on the \(q\)-series theoretical framework.
geometrically distributed random variables
scientific article; zbMATH DE number 1965390