A Last Progeny Modified Branching Random Walk

Abstract: In this work, we consider a modification of the usual Branching Random Walk (BRW), where at the n-th step we give certain i.i.d. displacements to each individuals, which may be different from the driving increment distribution. Depending on the value a parameter, we classify the model in three distinct cases, namely, the boundary case, below the boundary case and above the boundary case. Under very minimal assumptions on the underlying point process of the increments, we show that at the boundary case, the maximum displacement converges to a limit after only an appropriate centering, which is of the form c1 n - c2log n. We give explicit formula for the constants c1 and c2 and show that c1 is exactly same, while c2 is 1/3 of the corresponding constants of the usual BRW. We also characterize the limiting distribution. We further show that below the boundary the logarithmic correction term is absent. For above the boundary case, we have only partial result which indicates a possible existence of the logarithmic correction in the centering with exactly same constant as that of the classical BRW. Our proofs are based on a novel method of coupling with a more well studied process known as the smoothing transformation, which is used in various non-parametric statistical methods.

probability

Audience: researchers in the topic

Bangalore Probability Seminar

Series comments: The link to zoom meeting can be found on the seminar's google calendar - www.isibang.ac.in/~d.yogesh/BPS.html