A consequence of the law of large numbers

Question

Let $(X_k)_{k=1}$ be Poisson random variables with expectation $\mu$, let $Y_n = \sum_{k=1}^{n} X_k$. The weak law of large numbers states that, $$ \forall \delta>0, \forall \epsilon>0 \, \, \exists N>0\, \,\, s.t.\, \forall n >N \, \, P ( |Y_n/n - \mu| > \delta ) < \epsilon, $$ From this statement, is there a simple way to prove that there exists a constant $q>0$ which does not depend on $n$ such that $\forall n$, $$ P (\, Y_n/n - \mu \geq 0 \, )\, > q \,? $$ Probably the fact that the random variables are Poisson is not essential.

Answer 1

The result holds and is a simple consequence of central limit theorem.

To see this, let $Z_n=\frac1{\sqrt{n}}(Y_n-n\mu)$ and $A_n=[Y_n/n-\mu\geqslant0]$. Then $A_n=[Z_n\geqslant0]$ and $Z_n\to Z$ in distribution where $Z$ is a nondegenerate centered normal random variable hence $P[A_n]\to P[Z\geqslant0]=\frac12$ when $n\to\infty$.

Keep from this convergence the fact that $P[A_n]\geqslant\frac14$ for every $n\geqslant n_0$, then $P[A_n]\gt0$ for every $n\lt n_0$ hence $\inf\{P[A_n]\,;\,n\geqslant1\}$ is positive, QED.