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In this article, we consider a Branching Random Walk on the real line. Thegenealogical structure is assumed to be given through a supercritical branchingprocess in the i.i.d. environment and satisfies the Kesten-Stigum condition.The displacements coming from the same parent are assumed to have jointlyregularly varying tails. Conditioned on the survival of the underlyinggenealogical tree, we prove that the appropriately normalized (normalizationdepends on the quenched size of the $n$-th generation) maximum among positionsat the $n$-th generation converges weakly to a scale-mixture of Frech\'{e}trandom variable. Furthermore, we derive the weak limit of the point processescomposed of appropriately scaled positions at the $n$-th generation and showthat the limit point process is a member of the randomly scaled scale-decoratedPoisson point processes. Hence, an analog of the predictions by Brunet andDerrida (2011) holds. We have obtained an explicit description of the limitpoint process. This description captures the influence of the environment inthe joint asymptotic behavior of the extreme positions. We show that the law ofthe clusters in the limit depends on the time-reversed environment. Theasymptotic distribution of the normalized rightmost position is derived as aconsequence. This approach (based on weak convergence of extremal processes)can not be adapted when the genealogical structure is given through asupercritical Branching Process in a time-Inhomogeneous Environment (BPIE) (dueto lack of structural regularity in the genealogical structure). We provide asimple example where the point processes do not converge weakly. The tightnessof the point processes holds in this example though the point processes do not(weakly) converge (due to having different subsequential weak limits). Thisphenomenon is not yet known in the literature.

Extreme positions of regularly varying branching random walk in random and time-inhomogeneous environment