Stopping times and Wald identity for renewal processes

[3/100%]

Show

LaTeX

Show LaTeX

Problem 10.2.3 from Grimmett G.R., Stirzaker D.R. "Probability and Random Processes" [GrSt20]More problems in Section 10.2 "Limit theorems"

Show that $\mathbb {E}\left[T_{N(t)+k}\right] = \mu (m(t)+k)$ for all $k \geq 1$ , but that it is not generally true that $\mathbb {E}\left[T_{N(t)}\right] = \mu m(t)$ .

Problem 10.3.5 from Grimmett G.R., Stirzaker D.R. "Probability and Random Processes" [GrSt20]More problems in Section 10.3 "Excess life"

(a)(b)

Let $M(t) = N(t)+1$ , and suppose that $X_{1}$ has finite non-zero variance $\sigma^{2}$ .

(a)

Show that $\operatorname {Var}\left[T_{M(t)}-\mu M(t)\right] = \sigma^{2}(m(t)+1)$ .

(b)

In the non-arithmetic case, show that $\operatorname {Var}\left[M(t)\right] / t \rightarrow \sigma^{2} / \mu^{3}$ as $t \rightarrow \infty$ .

Problem 12.4.2 from Grimmett G.R., Stirzaker D.R. "Probability and Random Processes" [GrSt20]More problems in Section 12.4 "Stopping times"

Let $X_{1}, X_{2}, \ldots$ be a sequence of non-negative independent random variables and let $N(t) = \max \left\{ n: X_{1}+X_{2}+\cdots +X_{n} \leq t\right\}$ . Show that $N(t)+1$ is a stopping time with respect to a suitable filtration to be specified.

References

[GrSt20] Grimmett G.R., Stirzaker D.R. Probability and Random Processes. 4th ed., Oxford University Press, 2020.