After clarifying possible misunderstandings concerning covariances of thinned random variables, we propose a refined definition of the first-order integer-valued autoregressive model for count random fields. We provide a comprehensive derivation of its autocorrelation structure, which also covers some former results. Moreover, we expand the refined model to higher-order autoregressions and study its stochastic properties.

This brief paper identifies the Palm distribution of a linear Hawkes process. The textbook example for Palm distributions is the Palm version of a stationary Poisson process that corresponds to the original process plus a point in zero. The present result generalizes this example in a more complex but nevertheless tractable way. As a next step, we derive the intensity measure of the Palm version of a Hawkes process and show how it could be used for estimation. Finally, we discuss further possible applications to the theory of Hawkes processes. • We identify the Palm version of a Hawkes process in Theorem 1. • As a possible application, we show how the result could be used in estimation. • In the discussion, we sketch how the result could yield further theoretical developments of the linear Hawkes process, and we formulate a conjecture on a possible application for simulation without edge effects. • From a pedagogical point of view, the brief paper provides a novel example for the introduction of Palm distributions—adding some complexity to the standard Poisson example, while still being quite explicit and tractable.