Tuesday, 20 November 2018

limits - Find limntoinftysumnk=0fracennkk!





We need to find out the limit of,



limnnk=0ennkk!




One can see that ennkk! is the cdf of Poisson distribution with parameter n.



Please give some hints on how to find out the limit.


Answer



It's a good start to try to solve it in a probabilistic way: notice that the Poisson random variable has the reproducibility property, that is, if XkPoisson(1), k=1,2,,n independently, then
Sn=nk=1XkPoisson(n),


whose distribution function FSn satisfies:
FSn(n)=P[Snn]=nk=0ennkk!,

which is exactly the expression of interest. Hence this suggests linking this problem to central limit theorem.




By the classic CLT, we have
SnnnN(0,1).


Hence
P[Snn]=P[Snnn0]P[Z0]=12

as n.


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