Inequality for General random walks

Question

Let $S_n=(S_n^1,...,S_n^d)$ be a random walk with distribution $p(y,z):=\mathbb{P}\{S_{n+1}=z\mid S_n =y\}=p(z-y)$. Then $\mathbb{P}\{S_{2n}=0\}\geq(\mathbb{P}\{S_2=0\})^n\geq p(x)^{2n}\forall x\in \mathbb{Z}^d$. How do I get the second inequality?

Note that such a random walk is defined on a collection of generating sets $V$ and a function $k:V\rightarrow (0,1]$ s.t.$\sum_{x\in V}{k(x)}\leq 1$ and $p(x)=p(-x)=\frac{1}{2}k(x), \ x\in V$.

Thanks

Answer 1

Start at the origin, i.e., $S_0=0$. Since the random walk is symmetric, $p(x)^2=p(x)p(-x)$ is the probability that you choose the increment $x$ followed by increment $-x$. This implies $S_2=0$, so $p(x)^2\leq \mathbb{P}\{S_2=0\}$.