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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 bedifferent from the driving increment distribution. We call this process\emph{last progeny modified branching random walk (LPM-BRW)}. Depending on thevalue of a parameter, $\theta$, we classify the model in three distinct cases,namely, the boundary case, below the boundary case, and above the boundarycase. Under very minimal assumptions on the underlying point process of theincrements, we show that at the \emph{boundary case}, $\theta=\theta_0$, where$\theta_0$ is a parameter value associated with the displacement point process,the maximum displacement converges to a limit after only an appropriatecentering, which is of the form $c_1 n - c_2 \log n$. We give an explicitformula for the constants $c_1$ and $c_2$ and show that $c_1$ is exactly thesame, while $c_2$ is $1/3$ of the corresponding constants of the usual BRWAidekon (2013). We also characterize the limiting distribution. We further showthat below the boundary, $\theta < \theta_0$, the logarithmic correction termis absent. For above the boundary case, $\theta > \theta_0$, the logarithmiccorrection term is exactly the same as that of the classical BRW. For $\theta\leq \theta_0$, we further derive Brunet-Derrida -type results of point processconvergence of our LPM-BRW to a Poisson point process. Our proofs are based ona novel method of coupling the maximum displacement with a \emph{linearstatistic} associated with a more well-studied process in statistics, known asthe \emph{smoothing transformation}.

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