The Chernoff Bound gives a good tail estimate for a Binomial Distribution, but only if the mean goes to infinity. However, for a constant mean, Chernoff bound does not help at all. Is there some General formula that will help here?
More precisely. Given: $X \sim Bin(n,p)$. We consider the case where $ n \rightarrow \infty$, but $p=O\left(\frac{1}{n}\right)$ is possible. Then $\mu:=E[X]=np$ can be finite.
I would like to Show now that $$P[X \in [(1-\delta) \mu,(1+\delta) \mu]] \rightarrow 1$$ for a $\delta=o(1)$.
Or stated differently, I would like to show for a Binomial random variable $X$, that its value is $\mu$ w.h.p. Is this possible or not? And if yes, how? And if no, is it possible with further assumptions?
With the conditions you stated, your distribution converges to a Poisson distribution with fixed parameter $\lambda = np$. Thus the mean and variance will always be approximately $\lambda$, so your convergence in probability statement cannot hold.