I am studying stochastic processes with the textbook written by Robert G. Gallager. Due to my short knowledge, I have been faced with some difficulties to solve....
Question: A town starts a mosquito control program and the rv $Z_{n}$ is the number of mosquitoes at the end of the $n$ th year $(n=0,1,2, \ldots) .$ Let $X_{n}$ be the growth rate of the mosquito population in year $n ;$ i.e., $Z_{n}=X_{n} Z_{n-1} ; n \geq 1 .$ Assume that $\left\{X_{n} ; n \geq 1\right\}$ is a sequence of IID rv s with the $\operatorname{PMF} \operatorname{Pr}\{X=2\}=1 / 2 ; \operatorname{Pr}\{X=1 / 2\}=1 / 4 ; \operatorname{Pr}\{X=1 / 4\}=$ $1 / 4 .$ Suppose that $Z_{0},$ the initial number of mosquitoes, is some known constant and assume for simplicity and consistency that $Z_{n}$ can take on non-integer values.
(a) Find $E\left[Z_{n}\right]$ as a function of $n$ and find $\lim _{n \rightarrow \infty} E\left[Z_{n}\right]$.
(b) Let $W_{n}=\log _{2} X_{n} .$ Find $\mathrm{E}\left[W_{n}\right]$ and $\mathrm{E}\left[\log _{2}\left(Z_{n} / \mathrm{Z}_{0}\right)\right]$ as a function of $n$
(c) There is a constant $\alpha$ such that $\lim _{n \rightarrow \infty}(1 / n)\left[\log _{2}\left(Z_{n} / Z_{0}\right)\right]=\alpha$ WP1. Find $\alpha$ and explain how this follows from the SLLN.
(d) Using (c), show that $\lim _{n \rightarrow \infty} Z_{n}=\beta$ WP1 for some $\beta$ and evaluate $\beta$.
(e) Explain carefully how the result in (a) and the result in (d) are compatible. What you should learn from this problem is that the expected value of the log of a product of IID rv s might be more significant than the expected value of the product itself.
I don't know how to answer the question (e). The result I calculated is $E\left(z_{n}\right)=z_{0}\left(\frac{19}{16}\right)^{n}$ from (a). The result of (d) is $\beta=Z_{0}$.