Random walk with drift

Question

Let $X_1,X_2,...$ be independent and identically distributed $\mathbb{Z}-$valued bounded random variables with mean $a=\mathbf{E}[X_1]$, and let $S_n = X_1+\cdots+ X_n$ be the associated random walk. Is it true that $\mathbb{P}(S_n = \lceil{an}\rceil) \sim \frac{C}{\sqrt{n}}$ in the same way that $\mathbf{P}(S_n=0) \sim \frac{C}{\sqrt{n}}$ if the $X_i$'s were centered ?

Answer 1

This is true. You can check out McDonald's "The Local Limit Theorem: A Historical Perspective" for an interesting discussion of a group of related ideas. In Theorem 2.1 he gives the local limit theorem for $P(X_1 = 1) = p \neq 1/2$, $P(X_1 = 0) = 1-p$. (In the paper there is an obvious typo directly above the theorem stating that $p=1/2$, which should be ignored.) By linearity, this covers the case when your random walk has only two possible increments. This is from way back.

For your problem you'll need a more general result. McDonald gives the scenic route and contains applicable theorems, but for a big hammer to swing I'd suggest looking at Bhattacharya and Rao's text, "Normal Approximations and Asymptotic Expansions". In Chapter 5 they give local limit theory for integer valued random walks like yours (see Theorem 22.1). In particular, if $k = \lceil an \rceil$ and we assume Var$(X_1) = \sigma^2$ is finite, and we write $\phi$ for the standard normal distribution, then \begin{eqnarray*} P(S_n = k) &= & P\left({S_n - an \over n\sigma} = {k - an \over n\sigma}\right) \\ & = & n^{-1/2} \phi\left( {k - an \over n\sigma} \right) + O(n^{-1}). \end{eqnarray*} As $k - an < 1$, the last expression converges to $(2\pi n)^{-1/2}$ like $1/n$ or faster.

Local limit theory tends to give results that hold uniformly over all the space, which is pretty darn powerful, and much more than you need. The theorem cited also makes minimal assumptions on your variable $X_1$ (existence of a second moment).