homogeneous one-dimensional random walk

Question

I have another question from Theory of Probability and Random Processes book Prove that the spatially homogeneous one-dimensional random walk with $p_1 = 1− p_{−1} \neq 1/2 $is non-recurrent.

Answer 1

Let $Y_n\sim Bin(n,p_{1})$ be the number of steps right within the first $n$ steps and $X_n=Y_n-(n-Y_n)$ the position after $n$ steps.

$$E(X_n)= 2np_1 - n = n(2p_1-1):=\mu_n$$ and $$V(X_n)=4np_1p_{-1}:=\sigma_n^2$$ Suppose $p_1>0.5$, so $\mu_n>0$.

According to the central limit theorem (you can also use Chebyshev bound, if you prefer), for $n$ large enough: $$\Pr(X_n\leq 1)=\Pr(Z\leq \tfrac{1-\mu_n}{\sigma_n})=\Phi(\tfrac{1-n(2p_1-1)}{2\sqrt{np_1p_{-1}}})\xrightarrow{n\to\infty}0 $$ So the probability of returning to the initial place vanishes with time and the process is non-recurent.