Limits for Counting and Compound Processes

1. Counting Processes and the Inverse Relation A counting process is defined in §2.1 of Ross, right at the beginning. It is a generalization of a renewal (counting) process, which in turn is a generalization of a Poisson (counting) process, typically denoted by {N(t) : t ≥ 0}. For a counting process, there are no specific stochastic (distributional) assumptions. The random variable N(t) counts the number of events or points appearing in the time interval [0, t]. A counting process has an alternative representation through the sequence of partial sums of the intervals between successive points. This structure is used to discuss a renewal process in §3.1 of Ross, right at the beginning. There is an equivalence (one-to-one representation) between the stochastic processes {Sn : n ≥ 0} and {N(t) : t ≥ 0}, characterized by the inverse relation Sn ≤ t if and only if N(t) ≥ n, (1) which is given in (3.2.1) on page 99 of Ross. This inverse relation is easy to view from a plot of typical sample paths of the stochastic processes {Sn : n ≥ 0} and {N(t) : t ≥ 0}. If we plot a sample path of {N(t) : t ≥ 0}, we have t on the x axis and N(t) on the y axis. At the same time, i.e., in the same plot, we have n on the y axis and Sn on the x axis. Thus this inverse relation is naturally expressed as a mapping of one sample path (a function) into the other sample path (another function). It thus natural to view the relation as a map from a space of functions (sample paths) to itself.