Some students say that the notation is one of the most difficult things about this course. I recommend that you make for yourself a one-page crib sheet of all the notation:

\begin{align*}

&(X_n)_{n\geq 0},\ I,\ \text{Markov}(\lambda,P),\ P = (p_{ij}),\ P(n) = (p_{ij}^{(n)}),\\[12pt]

&h_i^A,\ k_i^A,\ H_i,\ T_i,\ V_i,\ V_i^{(n)},\ f_i,\ \lambda,\ \pi,\ \gamma_i^k,\ m_i,\ m_{ij} .

\end{align*}

Write a little explanation for yourself as to what each notation means, and how it used in our theorems about right-hand equations, recurrence/transience, left-hand equations, existence/uniqueness of invariant measure, aperiodicity/periodicity, positive/null recurrence and detailed balance. It should all seems pretty straightforward and memorable once you summarise it on one page and make some notes to place it in context.

Of course I could easily typeset a page like this for you — but I think that you'll learn more, and it will be more memorable for you personally, if you create this crib sheet yourself!

\begin{align*}

&(X_n)_{n\geq 0},\ I,\ \text{Markov}(\lambda,P),\ P = (p_{ij}),\ P(n) = (p_{ij}^{(n)}),\\[12pt]

&h_i^A,\ k_i^A,\ H_i,\ T_i,\ V_i,\ V_i^{(n)},\ f_i,\ \lambda,\ \pi,\ \gamma_i^k,\ m_i,\ m_{ij} .

\end{align*}

Write a little explanation for yourself as to what each notation means, and how it used in our theorems about right-hand equations, recurrence/transience, left-hand equations, existence/uniqueness of invariant measure, aperiodicity/periodicity, positive/null recurrence and detailed balance. It should all seems pretty straightforward and memorable once you summarise it on one page and make some notes to place it in context.

Of course I could easily typeset a page like this for you — but I think that you'll learn more, and it will be more memorable for you personally, if you create this crib sheet yourself!