Reference request: $\mathbb{E}|X_t| \to \infty$ as $t \to \infty$ when $\{X_t\}_{t\geq 0}$ is a continuous-time (symmetric) random walk

Question

Let $\{X_t\}_{t\geq 0}$ be a one dimensional continuous-time (symmetric) random walk on $\mathbb Z$ defined via $$X_t = X_0 + \sum_{i=1}^{N_t} Y_i,$$ where $X_0 \in \mathbb{Z}_+$ is a non-negative integer-valued random variable with mean $\mu > 0$, $Y_i \in \{-1,1\}$ is a sequence of independent Rademacher random signs and $N_t$ being a Poisson clock running at unit rate. I am pretty sure somewhere in the literature contains the proof of the following asymptotic: $$\mathbb{E}|X_t| \to \infty ~~\textrm{as}~~ t \to \infty$$ However, I can not find a reference paper or literature which contains such intuitive result. I would appreciate anyone who can point out (hopefully easy-to-read) relevant reference regarding the aforementioned estimate.

Answer 1

Apparently, the condition that your "clock" is independent of the $Y_i$'s is missing in your post. If so, it is unlikely to find a proof of your desired conclusion in respectable literature. Instead, regarding such a conclusion, one can likely find there something like "it is obvious that" or "it is easy to see that".

Indeed, let $S_n:=X_0+\sum_{i=1}^nY_i$, so that $X_t=S_{N_t}$. By the central limit theorem, $S_n/\sqrt n\to Z\sim N(0,1)$ in distribution. So, by the Fatou lemma, $\liminf_{n\to\infty}E|S_n|/\sqrt n\ge E|Z|>0$, and hence $$\text{$E|S_n|\to\infty$ as $n\to\infty$.}\tag{1}\label{1}$$
For any natural $m$, $$E|X_t|=E|S_{N_t}|=\sum_{n=0}^\infty P(N_t=n)E|S_n| \\ \ge\sum_{n\ge m} P(N_t=n)\inf_{n\ge m}E|S_n| =P(N_t\ge m)\inf_{n\ge m}E|S_n|.$$ Also, $P(N_t\ge m)\to1$ as $t\to\infty$. So, $$\liminf_{t\to\infty}E|X_t| \ge \inf_{n\ge m}E|S_n|$$ for each $m$. Letting finally $m\to\infty$ and recalling \eqref{1}, we get $\liminf_{t\to\infty}E|X_t|\ge\infty$, that is, $\lim_{t\to\infty}E|X_t|=\infty$. $\quad\Box$