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We introduce and study the r-Lah distribution whose definition involvesr-Stirling numbers of both kinds. We compute its expectation and variance, showits log-concavity and prove limit theorems for this distribution. We use theseresults to prove threshold phenomena for convex cones generated by random walksand to analyze the probability of unique recovery of sparse monotone signalsfrom linear measurements.

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