I want to show that if Zn has the binomial distribution with parameters n and λ/n with λ fixed, then Zn converges in distribution to the Poisson distribution, parameter λ as n→∞. How do I do this using characteristic functions?
Edit: i think the characteristic function of the binomial distribution is (peit+(1−p))n and that of the Poisson is eλ(eit−1), but i dont know which limit to take.
Answer
So the characteristic function of B(n,λ/n) is
((1−λ/n)+λ/neit)n=(1+1nλ(eit−1))n.
Now use that
lim
Then the convergence and uniqueness theorems for characteristic functions imply that the distribution is \text{Po}(\lambda).
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