Classical and Spatial Stochastic Processes by Rinaldo B. Schinazi

Author:Rinaldo B. Schinazi
Language: eng
Format: epub
Publisher: Springer New York, New York, NY

Since N(t + h) − N(t) has the same distribution as N(h) we get

Hence,

Observe now that by (c) and (d)

Since a linear combination of o(h) is an o(h) (why?) we get

(2.1)

Therefore,

and

Since , by letting h go to 0 we get

Integrating this differential equation yields

where C is a constant. Note that p 0(0) = P(N(0) = 0) = 1 by (a). Hence, C = 1 and

We now compute p n for n ≥ 1. If N(t + h) = n, then N(t) = n, N(t) = n − 1 or N(t) < n − 1. We now compute the corresponding probabilities.

Using (b) we have

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