Consider a sequence of $(X^{n})_{n \in \mathbb N}$ as a sequence of independent continuous-time Markov chains on state space $\{0,1\}$ for every $ n \in \mathbb N$ with $Q$ matrices:
$\begin{pmatrix} -\beta_{n} & \beta_{n} \\ \delta_{n} & -\delta_{n}\end{pmatrix}$
such that $\sum\limits_{n}\frac{\beta_{n}}{\beta_{n}+\delta_{n}}<\infty.$
Now consider the process $\textbf{X}$ where for $ t \geq 0$ we have $\textbf{X}(t):=(X^{n}(t))_{n \in \mathbb N}$
Now consider the the set $S:=\{x\in \{0,1\}^{\mathbb N}: \sum\limits_{j} x_{j} < \infty\}$.
It is stated that $\mathbb P( \textbf{X}(t) \in S\mid \textbf{X}(0) \in S)=1$ where $(X^n)_{n \in \mathbb N}$ and I am having trouble seeing why this is the case.
Clearly, if $X(0)\in S$, then we know that for $N$ large enough, we have that $X^n(0)=0$ for any $n \geq N$. I need to use the Markov property to show that there exists $M\in \mathbb N$ such that for all $n \geq M: \; X^n(t)=0$.
I assume I have to use the independence of the markov chains in some way. I'm stuck, any ideas?
Given that for all $n>N$ we have $X^n(0)=0$, we can infer that for all $n>N$ and $t>0$, we have $E(X^n(t)) =P(X^n(t)=1) \le \pi_n= \beta_n/(\beta_n+\delta_n)$. Sum this over all $n>N$ and deduce that $$E[\sum_{n=1}^\infty X^n(t) ]\le N+\sum_{n>N} \pi_n < \infty \,.$$