I got stuck today trying to understand an argument of the Frank den Hollander Book's. The problem is described below.
Let $S_n=\sum_{i=1}^n X_i$ be the simple random walk in $\mathbb{Z}^d$, that is $ \mathbb{P}(X_i=x)= \left\{ \begin{array}{ll} \frac{1}{2d}&\text{if}\ \|x\|=1;\\ &&\\ 0&\text{otherwise.} \end{array}\right. $ I would like to know how to prove that $ \mathbb{P}(S_{2n}=0)\sim 2\left(\frac{d}{4\pi n}\right)^{\frac{d}{2}}, \qquad n\to\infty. $
I learn from the Gregory Lawler book's that this is a consequence of the Local Central Limit Theorem. But I would like to know if one can prove this fact without use this result. I tried to Taylor Expand $ \hat{p}(k)=\frac{1}{d}\sum_{j=1}^d \cos k_j $ $k=(k_1,\dots,k_d)\in [-\pi,\pi)^d$ and use that $ \mathbb{P}(S_{2n}=0)=\left(\frac{1}{2\pi}\right)^d\int_{[-\pi,\pi)^d} [\hat{p}(k)]^{2n} dk. $ But It is not working. Any help or reference is welcome. Thanks.