Self-Similar Random Measures, their Carrying Dimension, and Application to Records

Abstract

What structure possesses the set Ξ = {t: X_t = sup{X_s:s ≤ t }} of instants at which a random process X on T = [0,∞) takes its record values. A simple answer in terms of the Hausdorff dimension can be given using an axiomatic approach to statistically self-similar random measures. Let 0 < H ≤ 1 and X_t be a continuous “record-H-self-similar” process with X₀ = 0. Define a “natural” random measure ξ, concentrated on Ξ, by ξ[0,t) = sup(X_s:s ≤ t). ξ is H-self-similar, meaning that (a) ξ[0,t) = r^Hξ[0,t/r), (b) ξ is distributed according to a Palm distribution. The theory of self-similar random measures tells then that ξ is carried by sets of Hausdorff dimension H.