# Point Stochastic Processes

**DOI:**https://doi.org/10.1007/978-1-4419-1153-7_762

## Introduction

A point process is a stochastic process {*N*(*t*), *t* ≥ 0}, where *N*(*t*) = number of occurrences by time *t*, which describes the appearance of a sequence of instant random events in time. Usually (though not always) intervals between two neighboring events are considered to be independently distributed. A process of this type is called a point process with restricted memory. If times between occurrences are a sequence of independent and identically distributed (i.i.d.) random variables, the point process is called a renewal or recurrent point process. The Poisson process represents a particular case of a renewal process in which the intervals between occurrences are exponentially distributed (Cox and Isham, 1980; Daley and Vere-Jones, 2002, 2007; Franken et al. 1981).

A special type of point process can be formed by two independent subsequences of random variables that alternate, as in the sequence *X*_{1}, *Y*_{1}, *X*_{2}, *Y*_{2},.... Such a process is called an alternating point process, and...

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