Factorial moments of the critical Markov branching process with geometric reproduction of particles

Assen Tchorbadjieff,Penka Mayster

doi:10.15559/22-vmsta201

Abstract

The factorial moments of any Markov branching process describe the behaviour of its probability generating function $F(t,s)$ in the neighbourhood of the point $s=1$. They are applied to solve the forward Kolmogorov equation for the critical Markov branching process with geometric reproduction of particles. The solution includes quickly convergent recurrent iterations of polynomials. The obtained results on factorial moments enable computation of statistical measures as shape and skewness. They are also applicable to the comparison between critical geometric branching and linear birth-death processes.

Highlights

A branching process model driven by geometric reproduction of particles was introduced in [12]
The factorial moments of any Markov branching process describe the behaviour of its probability generating function F (t, s) in the neighbourhood of the point s = 1
The obtained results on factorial moments enable computation of statistical measures as shape and skewness. They are applicable to the comparison between critical geometric branching and linear birth-death processes

Summary

Introduction

A branching process model driven by geometric reproduction of particles was introduced in [12]. The factorial moments in the subcritical case are computed in [12] They are used to obtain the probability mass function, central moments and variance-to-mean ratio (VMR) for the limiting random variable. The graphs of F (t, s) computed with the Lambert-W function at the point x = eKt C(s) (see (5), (6)) for different values of Kt (computation details can be found in [12]). The values of F (t, s) for s > 1 are computed with the W−1 branch of the Lambert-W function major statistical moments as variance and skewness remain undefined. The results follow from the solution to the forward Kolmogorov equation and rely on the factorial moments of the critical infinitesimal geometric branching reproduction.

Backward Kolmogorov equation for critical geometric branching mechanism

Forward Kolmogorov equation and factorial moments

Summation by diagonals

Solution of the Kolmogorov equation by the diagonal summation

Statistical inferences

Conclusion

A Appendix