Renewal function with discrete interarrival times evaluated at non-integer points

Question

Of this question, how do I do approach part d)? I do not recall any continuity results which I could use for this..

Answer 1

Since the interarrival time $X_i$ are positive integers, there will be jumps on integral time point only, and thus

$$ N_{t+\varepsilon} = N_t $$

To write this out more explicitly, note that the relationship between the counting process and interarrival time

$$ N_t \leq n \iff \sum_{i=1}^{n+1} X_i > t $$

Therefore

$$ N_{t+\varepsilon} \leq n \iff \sum_{i=1}^{n+1} X_i > t + \epsilon $$

Moreover,

$$ \Pr\left\{\sum_{i=1}^{n+1} X_i > t\right\} = \Pr\left\{t < \sum_{i=1}^{n+1} X_i \leq t + \varepsilon \right\} + \Pr\left\{\sum_{i=1}^{n+1} X_i > t + \varepsilon \right\}$$

As $X_i$ are integer, $\displaystyle \sum_{i=1}^{n+1} X_i$ is also an integer, So for $t \in \mathbb{N}$,

$$ \Pr\left\{t < \sum_{i=1}^{n+1} X_i \leq t + \varepsilon \right\} = 0$$

As a result, for $t \in \mathbb{N}$,

$$ \Pr\left\{\sum_{i=1}^{n+1} X_i > t\right\} = \Pr\left\{\sum_{i=1}^{n+1} X_i > t + \epsilon\right\} $$

$$ \Pr\{N_t \leq n\} = \Pr\{N_{t+\epsilon} \leq n\}$$

$$ E[N_t] = E[N_{t+\epsilon}] $$