Theoretical Statistics and Mathematics Unit, ISI Delhi
In this talk, we consider a modification of the usual *Branching Random Walk (BRW)*, where at the last step we give certain displacements which may be different from the increments. Under very minimal assumption on the underlying point process we show that the maximum displacement converges to a limit after only an appropriate centering. We further show that the centering term is $c_1 n + c_2 \log n$ and give explicit formula for the constants $c_1$ and $c_2$. We show that $c_1$ is exactly same and $c_2$ is $1/3$-of the corresponding constants of the usual BRW. Under some additional assumptions we further show that a large deviation result holds for the modified BRW, which essentially matches with that of the usual BRW. Our proofs are based on a novel method of coupling with a more well studied process known as the *smoothing transformation, *which has a lot of application in Statistics, particularly in the context of non-parametric regression.
[Some parts are joint work with Antar Bandyopadhyay]