Abstract
Symmetry properties of the random grid spanned on the set of records from the planar Poisson point process imply bizarre multivariate distributional identities for certain rational functions of independent exponential and uniform random variables.
Highlights
Let E1, E2, . . . and U1, U2, . . . be two independent sequences of independent rateone exponential and [0, 1]-uniform random variables, respectively
We consider a rectangular grid induced by the south-west records from the planar Poisson point process in R2+
A random symmetry property of the matrix whose entries are the areas of tiles of the grid implies cute multivariate distributional identities for certain rational functions of independent exponential and uniform random variables
Summary
Let E1, E2, . . . and U1, U2, . . . be two independent sequences of independent rateone exponential and [0, 1]-uniform random variables, respectively. We will show that (1.1) along with more general identities for matrix functions in the exponential and uniform variables follow from symmetry properties of the set of records ( known as Pareto-extremal points [2]) from the planar Poisson process in the positive quadrant. This continues the line of [5], where it was argued that the planar Poisson process is a natural framework for two gems of combinatorial probability: Ignatov’s theorem [6] and the Arratia-Barbour-Tavaré lemma on the scale-invariant Poisson processes on R+ [1]. (1.2) appears by a row summation in (1 − U1)E1
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