  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
\emph{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 \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 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
Aidekon (2013). We also characterize the limiting distribution. We further show
that below the boundary, $\theta < \theta_0$, the logarithmic correction term
is absent. For above the boundary case, $\theta > \theta_0$, the logarithmic
correction 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 process
convergence of our LPM-BRW to a Poisson point process. Our proofs are based on
a novel method of coupling the maximum displacement with a \emph{linear
statistic} associated with a more well-studied process in statistics, known as
the \emph{smoothing transformation}.
