Let $S_n$ be a random walk on $Z$ by step $X_1$. Let's suppose $P (X_1 = -1) = P (X_1 = 1) = P (X_1 = 2) = 1/3$ Check what kind of walk is $S_n$: recursive or transitive.
I tried to proceed with this well known result that random walk is transient iff:
$\mathbb{E}N = \lim_{t \to 1^-} \int_{[-\pi , \pi]^d}\frac{ds}{(2 \pi)^d(1-t\varphi(s))} < \infty$, where $N = \sum_{n \geq 0}\mathbb{1}_{\{S_n = S_0\}}$.
I was stuck on figuring out the $\varphi(t)$, any help there and in the next steps will be appreciated.
I got to the moment of $$ \varphi(t) = \frac{2}{3}\cos(t) + \frac{1}{3}e^{2it} $$ and I got lost with the imaginary part of it ($\frac{1}{3}e^{2it}$) in the limitation of $1-\varphi(s)$ the integral.
No such complicated criterion is needed. The answer is immediate from SLLN.
$\frac {S_n} n \to EX_1=\frac 2 3 $ a.s. by SLLN's. Hence $S_n \to \infty$ a.s., which shows that $(S_n)$ is transient. [$P(S_n=0 \, i.o.]=0$].