Discrete-time Markov chain

Question

Consider the following simplistic model of transitions between social classes as defined by sociologists. Only males are considered and by assumption every male has exactly 1 son. Let $X_n$ denote the social class of the individual at generation $n$, (and $X_{n+1}$ the social class of his son). We assume $X_n$ forms a discrete-time markov chain., with states $\{1\dots s\}$ and 1-step transition matrix; $\ p_{ij} = \theta + (1-\theta)\phi_j$ for $\ i = j $

$\ p_{ij} = (1-\theta)\phi_j$ for $\ i \neq j $

where $\ i,j = 1,\dots,s $

$\phi_j > 0 $

$\ \sum \phi_j= 1 $

Let state $s$ denote the highest social class "toffs". What is the expected number of generations taken by a family starting in social class "toffs" to next be in this class?

I do not know how to proceed with this question. We are told to separate the markov chain into toffs and not toffs ( i.e. with 2 states instead of s states) but I am unsure what to do beyond this.

Thanks for any help

Answer 1

Your transition matrix can be written $P=\theta I+(1-\theta)Q$, where $I$ is the identity matrix and $Q$ has identical rows $(\phi_1,\phi_2,\dots,\phi_s)$. Provided $\theta<1$, the unique invariant measure $\pi$ is, in fact, $\pi=(\phi_1,\phi_2,\dots,\phi_s)$.

Basic Markov chain theory tells us that the expected return time to any state is the reciprocal of the invariant measure of that state, i.e., $\mathbb{E}(T_s\,|\, X_0=s)={1/\phi_s},$ where $T_s=\inf(n\geq 1: X_n=s)$.

Answer 2

Following the hint, define $Y_n := 1_{X_n = s}$, so $Y_n = 1$ if and only if $X_n$ is a toff, and $Y_n = 0$ if and only if $X_n$ is not a toff. Then $(Y_n)_n$ forms a Markov chain with transition probabilities which you can compute. Use the Markov property to calculate $P(Y_n = 1, Y_k = 0\quad \forall\, 1 \leq k \leq n-1)$.