Maxima on trees

Maxima on trees

Probability Seminar

Mariana Olvera-Cravioto (UNC, Statistics and Operations Research)

Thursday, January 23, 2020 -
4:15pm to 5:15pm
at UNC, 125 Hanes Hall

This talk will focus on the study of the endogenous solution to the high-order Lindley equation: $$ W =_d \max\{ Y , \max_{1 \leq i \leq N} (W_i + X_i) \} $$ where the $W_i$ are i.i.d.~copies of $W$, independent of $(N, Y, {X_i})$. The solution we are interested in corresponds to the maxima of a “perturbed” branching random walk, and it includes as a special case the unique solution to the well-known Lindley equation. Under a condition analogous to the so-called Cram\’er condition for the standard random walk, the tail distribution $P(W > t)$ is known to decay exponentially. This result can be established using a powerful technique known as implicit renewal theory, which was originally described by C.M. Goldie (1991) and later extended to the tree setting by P. Jelenkovi\’c and M. Olvera-Cravioto (2012). However, the implicit nature of the approach does not lead to efficient computational methods nor provides clear insights into the most likely ways in which rare events occur. Our current work establishes analternative analysis of $P(W > t)$ which leads to a more explicit representation of its asymptotic behavior as well as to an efficient importance sampling algorithm to compute rare events. This is joint work with Michael Conroy, Bojan Basrak and Zbigniew Palmowski.

Last updated: 2020/02/20 - 2:34pm