On the Palm distribution of superposition of point processes

Abstract

Palm distributions are critical in the study of point processes. In the present paper we focus on a point process $\Phi$ defined as the superposition, i.e., sum, of two independent point processes, say $\Phi = \Phi_1 + \Phi_2$, and we characterize its Palm distribution. In particular, we show that the Palm distribution of $\Phi$ admits a simple mixture representation depending only on the Palm distribution of $\Phi_j$, as $j=1, 2$, and the associated moment measures. Extensions to the superposition of multiple point processes, and higher order Palm distributions, are treated analogously.