Null-recurrent random walks with symmetric step distribution are mapped to give null-recurrent Markov chains on the interval (0, 1). These chains are used to construct stationary Wold processes (point processes with Markov-dependent intervals) with infinite intensity, which processes are simple (i.e., almost surely orderly) but not (analytically) orderly. As examples of point processes with infinite intensity, they are constructed without any intermediate step that relies on using point processes of finite intensity.

The probability generating functional (p.g.fl.) of a non-homogeneous Poisson cluster process is characterized in Ammann and Thall (1977) via a decomposition of the KLM measure of the process. This p.g.fl. representation is utilized in the present article to show that the family 𝒟 of Poisson cluster processes with a.s. finite clusters is invariant under a class of cluster transformations. Explicit expressions for the finite-dimensional count distributions, product moment measures, and the distribution of clusters are derived in terms of the KLM measure. It is also shown that an element of 𝒟 has no multiple events iff the points of each cluster are a.s. distinct.

Some relationships are derived between the asynchronous and partially synchronous counting and interval processes associated with a multivariate stationary point process. A few examples are given to illustrate some of these relationships.

Simple definitions and derivations of elementary properties are given for the various intensities and Palm-Khinchin functions associated with a multivariate point process.