On the Palm distribution of superposition of point processes

Palm distributions are critical in the study of point processes. In the present paper we focus on a point process $\mathrm{\Phi }$ defined as the superposition, i.e., sum, of two independent point processes, say $\mathrm{\Phi }={\mathrm{\Phi }}_1+{\mathrm{\Phi }}_2$, and we characterize its Palm distribution. In particular, we show that the Palm distribution of $\mathrm{\Phi }$ admits a simple mixture representation depending only on the Palm distribution of ${\mathrm{\Phi }}_j$, 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.