I know that $P^{2n}(0,0) = C^{2n}_{n}2^{-2n}$ where $C^{2n}_{n}$ is $2n$ choose $n$. Using Sterling's approximation, $n! \sim \sqrt{2\pi}n^{n+\frac{1}{2}}e^{-n}$, we get $$P^{2n}(0,0) = C^{2n}_{n}2^{-2n} \sim \frac{C}{\sqrt{n}}$$ for some constant $C$. Now I am trying to get the same formula using the inversion formula for mass functions:
$$P(X = x) = \frac{1}{2\pi}\int_{-\pi}^{\pi}e^{-itx}\phi(t)dt$$ where $\phi$ is the characteristic function of $X$. Thus for the SRW on $\mathbb{Z}$, $$P(S_{2n} = 0) = \frac{1}{2\pi}\int_{-\pi}^{\pi}\phi(t)dt = \frac{1}{2\pi}\int_{-\pi}^{\pi}E[e^{itS_{2n}}]dt$$ Taylor's approximation: $$E[e^{itS_{2n}}] \approx 1 - \frac{t^2}{2}ES_{2n}^{2} = 1 - 2n\frac{t^2}{2} $$
$$\therefore P(S_{2n} = 0) \approx \frac{1}{2\pi}\int_{-\pi}^{\pi}(1-nt^2)dt$$ which does not give me a $\frac{C}{\sqrt{n}}$ kind of an expression? Thanks and appreciate a hint.
The problem is that $E[e^{itS_{2n}}] \approx 1 - \frac{t^2}{2}ES_{2n}^{2} $ is too crude an approximation. You need to use the exact form of the characteristic function of $S_n$, then approximate the integral by other methods.
Since a single $\text{Unif}(\pm1)$ random variable has c.f. $(e^{it}+e^{-it})/2=\cos t$, it follows that $S_n$ has c.f. $\cos^n t$, so when $n$ is even,
$$P(S_n=0)=\frac1{2\pi}\int_{-\pi}^{\pi}\cos^nt\,dt=\frac1{\pi}\int_{-\pi/2}^{\pi/2}\cos^n t\,dt.$$ Writing $\cos^nt=e^{n\log(\cos t)}$, the last integral can be approximated using Laplace's method.