Let say I have some discrete time Markov chain $(X_n)_{n \in \mathbb{N}}$ taking value on the finite state space $\{1,2, \ldots N\}$. The chain is irreducible and I know that the $n$-step transition probabilities from $N$ to any $j \neq N$ are increasing in $n$, that is: $\big(\mathbb{P} (X_n = j \mid X_0 = N)\big)_{n \in \mathbb{N}}$ is an increasing sequence.
Now let say that I consider a (homogeneous) continuous-time Markov chain $(Y_t)_{t \in \mathbb{R_+}}$ and that $(X_n)_{n \in \mathbb{N}}$ is the skeleton (or embedded chain) of $(Y_t)_{t \in \mathbb{R_+}}$. Assuming that for any $i \in \{1,2, \ldots N\}$ the intensity $\lambda_i$ corresponding to the rate of the exponential random variable that gives the time I wait in state $i$ before making a transition to another state only satisfies $0 < \lambda_i < \infty$, can I conclude that for any $j \neq N$ the function $P_{N,j}(t) = \mathbb{P} (Y_t = j \mid Y_0=N)$ is increasing in $t$?
If yes how would you prove it? If no can you give a counter-example?
The reason I want such a result is because I would like to claim that $P_{N,j}(t) \leq \mu(j)$ for any $t$ (and $j \neq N$), where $\mu$ is the invariant measure of the continuous-time Markov chain.
Any hint or comment is welcome, thank you!