r-Lah Distribution: Properties, Limit Theorems and an Application to Compressed Sensing

TL;DR

This paper introduces the r-Lah distribution, explores its properties including expectation, variance, and limit theorems, and applies these findings to compressed sensing, particularly in signal recovery and threshold phenomena.

Contribution

It presents the first comprehensive study of the r-Lah distribution, linking it to convex geometry and compressed sensing applications.

Findings

Derived expectation and variance of r-Lah distribution

Proved log-concavity and limit theorems for the distribution

Applied results to threshold phenomena and sparse signal recovery

Abstract

We introduce and study the r-Lah distribution whose definition involves r-Stirling numbers of both kinds. We compute its expectation and variance, show its log-concavity and prove limit theorems for this distribution. We use these results to prove threshold phenomena for convex cones generated by random walks and to analyze the probability of unique recovery of sparse monotone signals from linear measurements.