In the revision lectures we shall discuss some questions from 2007-2015 papers; you are strongly encouraged to attempt these papers. As a natural supplement, you should revise the homework and tutorial problems. You might also find the following checklist helpful.

• state and apply the continuity property of probability measures along monotone sequences of events;
• state and prove the Borel-Cantelli lemma; apply it to infinite sequences of events.

• define convergence in L^r, verify whether a given sequence of random variables converges in L^r;
• define convergence in probability, verify whether a given sequence of random variables converges in probability;
• explain the relation between convergence in L^r and convergence in probability (Lemma 2.8);
• state and apply the criterion of convergence in L² (Theorem 2.10);
• define almost sure convergence, verify whether a given sequence of random variables converges almost surely;
• state and apply the criterion of almost sure convergence (Lemma 2.15);
• state and apply the Strong Laws of Large Numbers.

• describe the main steps in the construction of the Lebesgue integral and compare it to the Riemann integral;
• state the main properties of the Lebesgue integral;
• state and apply the Monotone Convergence theorem;
• state and apply the Dominated Convergence theorem.

• define ordinary, probability, and moment generating functions;
• derive the value of the n^th term of a sequence from the corresponding generating function;
• state and apply the theorem about generating functions of convolutions;
• use probability generating functions to compute moments of random variables;
• state and apply the uniqueness and continuity theorems for generating functions;
• use generating functions in solving recurrent relations;
• explain the role of generating functions in proving convergence in distribution.

• define a Markov chain, verify whether a given process is a Markov chain;
• compute the n-step transition probabilities p_ij⁽ⁿ⁾ for a given Markov chain;
• identify classes of communicating states, explain which classes are closed and which are not; identify absorbing states;
• check whether a given Markov chain is irreducible;
• determine the period of every state for a given Markov chain;
• compute absorption probabilities and expected hitting times;
• define transient and (positive/null) recurrent states;
• compute the first passage probabilities f_ij and use them to classify states into transient and recurrent;
• use the n-step transition probabilities p_ij⁽ⁿ⁾ to classify states into transient and recurrent;
• find all stationary distributions for a given Markov chain;
• state and apply the convergence to equilibrium theorem for Markov chains;
• state and apply the Ergodic theorem;
• state and use the relation between the stationary distribution and the mean return times;
• define reversible measures and state their main properties;
• check whether a given Markov chain is reversible;
• find stationary distributions for random walks on graphs.