Right-Most Position of a Last Progeny Modified Branching Random Walk

In this work, we consider a modification of the usual Branching Random Walk (BRW), where we give certain independent and identically distributed (i.i.d.) displacements to all the particles at the $n$-th generation, which may be different from the driving increment distribution. We call this process last progeny modified branching random walk (LPM-BRW). Depending on the value of a parameter, $theta$, 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, when $theta$ takes a particular value $theta_0$, the maximum displacement converges to a limit after only an appropriate centering, which is of the form $c_1 n - c_2 log n$. We give an explicit formula for the constants $c_1$ and $c_2$ and show that $c_1$ is exactly the same, while $c_2$ is $1/3$ of the corresponding constants of the usual BRW. We also characterize the limiting distribution. We further show that below the boundary (that is, when $theta < theta_0$), the logarithmic correction term is absent. For above the boundary case (that is, when $theta > theta_0$), we have only a partial result, which indicates a possible existence of the logarithmic correction in the centering with exactly the same constant as that of the classical BRW. For $theta leq theta_0$, we further derive Brunet--Derrida-type results of point process convergence of our LPM-BRW to a decorated Poisson point process. Our proofs are based on a novel method of coupling the maximum displacement with a linear statistics associated with a more well-studied process in statistics, known as the smoothing transformation.