$\left\{\xi_{n}\right\}_{n\in\ \mathbb{N}_{+}}$ is a sequence of independently and identically distributed random variables, each taking a finite number of integer values.$\mathbf{E}(\xi_1)\ne 0,$ For any $n\in \mathbb{N}_{+},$define $S_{n}:=\sum_{i=1}^{n}\xi_{i}.$ Show that $\left\{S_{n}\right\}_{n\in\ \mathbb{N}_{+}}$ is a Markov chain, with each state being transient.
My question is how to prove whose each state is transient.I attempt to demonstrate $\sum_{n=1}^{\infty}\mathbf{P}(S_{n}=i\mid S_1=i)<\infty $ for each $i$ in the state space $S$, by Kolmogorov's strong law of large numbers and Borel–Cantelli lemma. Maybe we can use this equivalence $$\frac{S_n}{n} \xrightarrow[]{a.s.}\mathbf{E}(\xi_1)\Longleftrightarrow \displaystyle \lim_{ n\to \infty}\mathbf{P}\left(\bigcup_{k=n}^{\infty}\left\{\left|\frac{S_k}{k} -\mathbf{E}(\xi_1)\right|\ge\varepsilon\right\}\right)=0, \forall \varepsilon>0.$$to create a paradox to support that starting from some positive integer $n(i)$, all subsequent terms in $\sum_{n=1}^{\infty}\mathbf{P}(S_{n}=i\mid S_1=i)$ i.e. $\mathbf{P}(S_{n(i)}=i\mid S_1=i),\mathbf{P}(S_{{n(i)+1}}=i\mid S_1=i),......$ are zeros.
$\left\{S_{n}\right\}_{n\in\ \mathbb{N}_{+}}$ is a time-homogeneous Markov chain. Someone have objections, and the reasons are as follows:
From Chapman-Kolmogorov equation, $k+m$-step transition probability \begin{align} \mathbf{P}(S_{n+k+m}=j\mid S_{n}=j)&=:\mathbf{P}^{(k+m)}(j,j)\\ &=\sum_{r\in S}\mathbf{P}^{(k)}(j,r)\cdot\mathbf{P}^{(m)}(r,j) \\ &\ge\mathbf{P}^{(k)}(j,j)\cdot\mathbf{P}^{(m)}(j,j).\end{align}
If both $\mathbf{P}^{(k)}(j,j)>0$ and $\mathbf{P}^{(m)}(j,j)>0$, then $\mathbf{P}^{(k+m)}(j,j)>0.$ That will lead to $\mathbf{P}(S_{n+k+m}=j)>0,$ further steps will result in infinitely many $\mathbf{P}(S_{n}=j)>0.$What is your opinion on this?
I thought from the definition of time-homogeneous, $\mathbf{P}(S_{n+1}=j|S_{n}=i)=\mathbf{P}(S_{m+1}=j|S_{m}=i),\forall i,j\in S \& \forall n,m\in \mathbb{N},$ both $\mathbf{P}(S_{n})>0$ and $\mathbf{P}(S_{m})>0$ seem to be needed.
$S_n/n$ converges to some nonzero quantity a.s., so $|S_n|$ converges to infinity a.s., so for any fixed finite $a$, $|S_n|=|a|$ a finite number of times a.s, so $a$ is transient for any $a$.
Note that when the mean is $0$, the above approach doesn’t work, and it might not be transient everywhere, so you have to be a little careful in generalizing this approach.