Note on generalizations of Mecke's formula and extensions of H = λG

Abstract

Mecke's formula is concerned with a stationary random measure and shift-invariant measure on a locally compact Abelian group, and relates integrations concerning them to each other. For, we generalize this to a pair of random measures which are jointly stationary. The resulting formula extends the so-called Swiss Army formula, which was recently obtained as a generalization for Little's formula. The generalized Mecke formula, which is called GMF, can be also viewed as a generalization of the stationary version ofH = λG. Under the stationary and ergodic assumptions, we apply it to derive many sample path formulas which have been known as extensions ofH = λG.This will make clear what kinds of probabilistic conditions are sufficient to get them. We also mention a further generalization of Mecke's formula.