In the process of working on a project, I am faced with the following question:
let us say our transition matrix $P$ is given by: \begin{bmatrix} P_{1,1} & P_{1,2} & P_{1,3} & P_{1,4} \\ P_{2,1} & P_{2,2} & P_{2,3} & P_{2,4} \\ P_{3,1} & P_{3,2} & P_{3,3} & P_{3,4} \\ P_{4,1} & P_{4,2} & P_{4,3} & P_{4,4} \end{bmatrix} Where each state in our state space corresponds to a log level of income in the following vector: \begin{bmatrix} a & b & c&d \\ \end{bmatrix} i.e $a, b, c, d \in \mathbb{R}$ are the four levels of income.
Given this info, what is the general process of finding the variance of the period-to-period change in income over some $ n \in \mathbb{N}$ number of periods? Suppose it is known that the process begins in a given state (i.e begins in state 1 (income = $a$)). How might one begin this calculation? I'm confused how to approach this problem since we don't technically have a probability distribution for the initial state since we are simply assuming it starts in a given state.
If your initial state is definitely state $1$, then you do have a probability distribution over the initial state, $\ X_0\ $, namely, $$ P\big(X_0=i\big)=\cases{1&if $\ i=1\ $,\\ 0&otherwise,} $$ which, for the purposes of calculating subsequent state probability distributions, is used in the form of the probability row vector $\ \pi_0^T=\begin{bmatrix}1&0&0&0\end{bmatrix}\ $. After $\ n\ $ periods, the state probability distribution will be $$ \pi_n^T=\pi_0^TP^n\ . $$ If $\ \alpha=\begin{bmatrix}a\\b\\c\\d\end{bmatrix}\ $ is the column vector of incomes, and $\ \beta=$$\,\begin{bmatrix}0\\(b-a)^2\\(c-a)^2\\(d-a)^2\end{bmatrix}\ $ the column vector of squared income changes, then the mean change in income over the first $\ n\ $ periods is $$ \pi_n^T\alpha-a\ , $$ and the variance is $$ \pi_n^T\beta-\big(\pi_n^T\alpha-a\big)^2\ . $$