I am interested in the ergodicity of the Poisson shot noise process
$$X(t) = \sum_{i=0}^{N(t)} k \ h(t-T_i)$$
where $k,\tau>0$, $N(t)$ is a Poisson process with rate $\lambda$, and
$$ h(t) = \begin{cases} e^{-t/\tau}, & \text{if $t \geq 0$} \\ 0, & \text{if $t<0$} \end{cases}$$
This process can be shown to be a Markov process. Furthermore, according to [1]:
A Markov process is ergodic if there exists a unique invariant probability distribution and, for any state $x$, the transition probabilities $P(X_t ∈ ·|X_0 = x)$ converge to that distribution in total variation.
The authors then show that a sufficient and necessary condition for $X(t)$ to be ergodic is $log(1 \lor k) < \infty$.
However, I am confused about this result. I thought that for a stochastic process to be ergodic, the process must also be stationary. But for $X(t)$, it can be shown that
$$\mathbb{E}[X(t)] = \lambda k \int_{t_0}^t e^{-(t-y)/\tau} \ dy = \lambda k \tau \ (1-e^{-(t-t_0)/\tau}) $$
$$ \text{Var}[X(t)] = \lambda k \int_{t_0}^t e^{-2(t-y)/\tau} \ dy = \frac{\lambda k \tau}{2} \ (1-e^{-(t-t_0)/\tau})$$
which are not time-independent unless $t_0 \to - \infty$.
Am I missing something? Thanks a lot in advance.
- Borovkov, K., and A. Novikov. "On a piece-wise deterministic Markov process model." Statistics & probability letters 53.4 (2001): 421-428.