Math Problems Vault

Problem 10.3.1

Suppose that the distribution of the excess lifetime $E(t)$ does not depend on $t$ . Show that the renewal process is a Poisson process.

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Problem 10.3.2

(a)(b)

Show that the following are Markov processes:

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(a)

the excess lifetime process $E(t)$ ,

(b)

the current lifetime process $C(t)$ .

Problem 10.3.3

(a)(b)(c)

Suppose that $X_{1}$ is non-arithmetic with finite mean $\mu$ .

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(a)

Show that $E(t)$ converges in distribution as $t \rightarrow \infty$ , the limit distribution function being

H(x) = \int _{0}^{x} \frac{1}{\mu }[1-F(y)] d y

(b)

Show that the $r$ th moment of this limit distribution is given by

\int _{0}^{\infty } x^{r} d H(x) = \frac{\mathbb {E}\left[X_{1}^{r+1}\right]}{\mu (r+1)}

assuming that this is finite.

(c)

Show that

\mathbb {E}\left[E(t)^{r}\right] = \mathbb {E}\left[\left\{ \left(X_{1}-t\right)^{+}\right\} ^{r}\right]+\int _{0}^{t} h(t-x) d m(x)

for some suitable function $h$ to be found, and deduce by the key renewal theorem that $\mathbb {E}\left[E(t)^{r}\right] \rightarrow \mathbb {E}\left[X_{1}^{r+1}\right] /\left\{ \mu (r+1)\right\}$ as $t \rightarrow \infty$ , assuming this limit is finite.

Problem 10.3.4

Find an expression for the mean value of the excess lifetime $E(t)$ conditional on the event that the current lifetime $C(t)$ equals $x$ .

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Problem 10.3.5

(a)(b)

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

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(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 $t \rightarrow \infty$ .