# Derivation condition to obtain the correct asymptotic link

Suppose that $$X sim text {Bin} (n, theta)$$. By the local limit theorem (Theorem 7 here) for discrete random variables,
$$P (X = t) = frac {1} { sqrt {2 pi n theta (1- theta)}} exp left (- frac {(tn theta) ^ 2} {2n theta (1- theta)} right) + o (n ^ {- 1/2})$$
for all $$n geq 1$$ and uniformly in the integers $$t$$.

I'm interested in the form. $$t = n theta + sqrt {2 theta (1- theta) n log m + O (1)}$$ and the relationship between $$n$$ Y $$m$$. In particular, $$m >> n >> 1$$. For example, in my application, $$m approx. 20000$$ Y $$n approx. 200$$ it seems to work well

I am interested in finding a theoretical relationship between $$n$$ Y $$m$$ such amount $$mP (X = t) = O (1)$$. Now if the $$o (n ^ {- 1/2})$$ the rest was not there, so I can reason the following,

begin {align *} mP (X = t) & = O (mn ^ {- 1/2} exp (-O ( log m))) \ & = O left ( exp left ( log m- log n ^ {1/2} -O ( log m) right) right) \ & = O left ( exp left ( log m ^ {r} – log n ^ {1/2} right) right) text {for some constant r> 0 } \ & = O left ( frac {m ^ {r}} {n ^ {1/2}} right) end {align *}

This suggests that $$m leq n ^ gamma$$, where $$gamma = frac {1} {2r}> 0$$ for $$r> 0$$ It can be a reasonable relationship.

How do I manage that rest? $$o (n ^ {- 1/2})$$? Intuitively, $$m$$ should grow much faster than $$n$$.