Let $X_i$'s be iid and define $X_1+\ldots+X_n=S_n$. I was trying to show that if $S_n$ is recurrent, then $S_{2n}$ is also recurrent. Assume these walks are in $\mathbb{R}^d$.
Using Chung-Fuchs theorem, this boils down to showing that
$$\int_{(-\delta,\delta)^d} \textrm{Re}\frac{dx}{1-\phi(x)} = \infty$$ implies
$$\int_{(-\delta,\delta)^d} \textrm{Re}\frac{dx}{1-\phi^2(x)} = \infty$$
for some $\delta > 0$. It's here that I'm stuck. Does anyone have an idea how to go from here? By the way, $\phi$ denotes the characteristic function for a single increment $X_i$.
Also someone told me that $S_{kn}$ will be recurrent for all $k\in \mathbb{N}$ but I figure I could show it for $S_{2n}$ first.