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.
The approach is taken on pages 78 and 79 of Principles of Random Walk (2nd edition) by Frank Spitzer. I was able to see these pages using Google Books.
Spitzer first translates $[-\pi,\pi)^d$ by the vector $(\pi/2,\pi/2,\dots,\pi/2)$ which doesn't change the value of the integral. Then he argues that the bulk of the integral is concentrated at two points, the origin and $(\pi,\pi,\dots,\pi)$ both contributing the same value asymptotically.
The Taylor's series expansion ${1\over d}\sum_{j=1}^d \cos k_j\approx \exp(-|k|^2/2d)$ near the origin finishes the result.