How can i find, does $\lim_{n\to\infty}\frac{\eta_n}{n}$ where $\eta_n$ has poisson distribution with $\lambda = n$ exists?
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This really depends on what you mean by "exist". Since $\eta$ is random variable, it is really a function $\eta: \Omega \rightarrow \mathbb{R}$, where $\Omega$ is the underlying probability space.
If you mean "what is the distribution of $\lim_{n\rightarrow \infty}\frac{\eta_n}{n}$, if it exists", then notice the moment generating function of $\frac{\eta_n}{n}$ can be shown to be $$e^{n(e^{\frac{t}{n}} - 1)}$$
Taking the limit of the above as $n\rightarrow \infty$ gives us
$$\lim_{n\rightarrow \infty} e^{n(e^{\frac{t}{n}} - 1)} = e^t$$
which is the moment generating function of the degenerate distribution centered at $1$.
SO, we can say that $\frac{\eta_n}{n}$ converges to $1$ in distribution as $n\rightarrow \infty$
measure_theory
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And in probability! – Sangchul Lee Sep 30 '16 at 19:36
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2@SangchulLee I answered this question a few months ago; there is actually almost sure convergence. – Math1000 Sep 30 '16 at 21:46
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True to both! Just thought this was a pretty simple proof that it converges in some manner since the type of convergence wasn't specified. – measure_theory Sep 30 '16 at 21:51
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@Math1000 Sounds interesting, I will have a look on that! – Sangchul Lee Sep 30 '16 at 22:12