Equalities involving Exponential Distributions and Markov process

Question

I am trying to understand Remark 17.26 of the book Probability Theory by A. Klenke (3rd version), where the author is showing how a condition on the Q-matrix of a discrete Markov process in continuous time is crucial to guarantee that the process does not explode.

He defines:

• $T_1,T_2,...$ as independent exponential random variables with parameter $n^2$
• $S_n=T_1+...+T_{n-1}$
• $X_t=\text{sup}\{ n \in \mathcal{N_0}: S_n\leq t\}$

I do not understand how he concludes that

$$\lim_{s\downarrow 0} s^{-1}P[X_{t+s}=n+1|X_t=n]=n^2$$

and

$$\lim_{s\downarrow 0} s^{-1}(P[X_{t+s}=n|X_t=n]-1)=-n^2$$

from this result

$$P[X_{t+s}\geq n+1|X_t=n]= \\ P[S_{n+1}\leq t+s|S_n\leq t, S_{n+1}>t]= \\P[T_n\leq s+t-S_n|S_n\leq t, T_n > t- S_n] = \\P[T_n \leq s]=1-\exp(-n^2s) $$

In particular, I do not understand:

1. Why the event $\{X_{t+s}\geq n+1|X_t=n\}$ is equivalent to $\{S_{n+1}\leq t+s |X_t=n \}$ in the first above equality. Where are the events $X_{t+s}=n+2, n+3, ...$ and so on?
2. How to calculate $P[X_{t+s}=n+1|X_t=n]$ and $P[X_{t+s}=n|X_t=n]$ from the result.

Thanks for the help. Let me know if more contest is needed.

Answer 1

1. Why the event $\{X_{t+s}\geq n+1|X_t=n\}$ is equivalent to $\{S_{n+1}\leq t+s |X_t=n \}$ in the first above equality. Where are the events $X_{t+s}=n+2, n+3, ...$ and so on?

Conditioning is not a thing that is defined for sets. However,

$$ \{X_{t+s}\geq n+1\}\cap \{X_t=n\} = \{S_{n+1}\leq t+s \}\cap \{X_t=n\} $$ because they both mean, "the sum of $n$ durations is less than or equal to $t$ and the sum of $n+1$ durations is at least $t+s$."

2. How to calculate $P[X_{t+s}=n+1|X_t=n]$ and $P[X_{t+s}=n|X_t=n]$ from the result.

You can use the formula that you have for $P[X_{t+s}\geq n+1|X_t=n]$ along with additivity:

$$ P[X_{t+s}=n+1|X_t=n] = P[X_{t+s}\geq n+1|X_t=n] - P[X_{t+s}\geq n+2|X_t=n]. $$