4.1 Counting Processes
Let \(N(t)\) be the number of events of some kind (arrivals, failures, claims) that occur in the time interval \((0, t]\). Then \(\{N(t) : t\geq 0\}\) is a counting process: \(N(t)\) takes non-negative integer values, \(N(s)\leq N(t)\) for \(s<t\), and for \(s<t\) the increment \(N(t) - N(s)\) is the number of events in \((s, t]\).
Definition 4.1.1. A counting process has
- independent increments if the numbers of events in disjoint time intervals are independent (figure 4.1);
- stationary increments if the distribution of the number of events in \((s, s + t]\) depends only on the length \(t\), not on \(s\).
Definition 4.1.2. A function \(f\) is \(o(h)\) (“little-o of \(h\)”) if \(\lim _{h\to 0}\frac {f(h)}{h} = 0\).
For example \(f(h) = h^2\) is \(o(h)\), while \(f(h) = h\) is not. Sums of \(o(h)\) terms and constant multiples of them are again \(o(h)\), so in the derivations below all such terms can be lumped together.
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