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