lunedì, luglio 21, 2008

Bernoulli vs Poisson

A Bernoulli process with window h and intensity i, shortly BP[h,i] is a renewal process in discrete time.

We prove that BP[h*i,i] converges to a Poisson process with intensity i (PP[i]) if h goes to 0. More precisely, we prove that a Poisson process with intensity i can be approximated in distribution by a sequence of Bernoulli process.

I - Renewal property

Since BP[h*i,i] are renewal, it suffices to study the interevent distribution. For example, we can characterize BP[h*i,i] by their survival function

s(t):= P[Interevent > t]

II - Euler Formula

The exponential function is defined by

e^a:= \lim_{n \to \infty} (1+ \frac{a}{n})^n

III - Poisson process

The survival function of the Poisson process with intensity i is given by exp(-t*i).

IV - Limiting process


BP[h*i,i][ Waiting > t] = (1-ih)^t/h = (1+ h/t (-it))^t/h

Substituting n for t/h shows that BP[h*i,i][ Waiting > t] converges towards exp(-it) when h goes to 0. In other words, the distribution of BP[h*i,i] converges towards the one of PP[i] and the proof is complete.

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