Lecture 12

The last lecture was on Tuesday November 15. I have enjoyed giving this course and I hope you have enjoyed it too.

The notes are now finalized and probably safe for you to download as a final version. I made a change to page 48 around 9am today. If there are any more changes then I will put a remark on the course page. You can now also look at the overhead projector slides that I sometimes used in lectures, such as those today that I used to summarise four Part II courses that you may like to study next year.

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: 

(X_n)_n≥0,   I,   Markov(λ,P),   P = (p_ij),   P⁽ⁿ⁾ = (p_ij⁽ⁿ⁾),   h_i^A,   k_i^A,   H_i,   T_i,   V_i,   V_i(n),   f_i,   λ,   π,   γ_i^k,   m_i .

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!


In my discussion of random walk and electrical networks in Section 12.4 I appealed to Rayleigh's Monotonicity Law: " if some resistances of a circuit are increased (decreased) the resistance between any two points of the circuit can only increase (decrease)." A proof of this "obvious" fact can be constructed by (i) proving Thomson's Principle: "Flows determined by Kirchhoff's Laws minimize energy dissipation", and then (ii) showing that Thomson's Principle implies Rayleigh's Monotonicity Law. You can read the details of this in Doyle and Snell Random walks and electric networks, pages 51–52.


In Section 12.2 I mentioned Burke's output theorem (1956) which says that the output process of a M/M/1 queue in equilibrium is a Poisson process with the same rate as the input. In writing "M/M/1" the "M"s mean Markovian (i.e. a Poisson input process of rate λ and i.i.d. exponentially distributed service times with parameter μ (where μ>λ), and the "1" means a single server. In queueing theory this very useful notation is known as Kendall's notation.)


I remarked (just for fun) that queueing is the only common word in the OED with five vowels in a row. Obscure words are ones like "miaoued" (what the cat did).


I once proved a generalization of Burke's output theorem that holds even when the queue has not reached equilibrium (see: The interchangeability of ·/M/1 queues in series, Weber, 1979). Suppose we have two single-server queues in series, which we might write as /M/1 → /M/1. The customers' service times in the first queue are i.i.d. exponentially distributed with parameter λ and in the second queue they are i.i.d. exponentially distributed with parameter μ. On finishing service in the first queue a customer immediately joins the second queue. Suppose the system starts with N customers in the first (upstream) queue and no customers in the second (downstream) queue. My theorem says that all statistics that we might measure about the departure process from the second queue are the same if λ and μ are interchanged. Thus by observing the process of departures from the second queue we cannot figure out which way around the two /M/1 servers are ordered. For example, the time at which we see the first departure leave the second queue has expected value 1/λ + 1/μ (which is symmetric in λ and μ). All other statistics are also symmetric in λ and μ. Burke's theorem is a corollary of this that can be obtained by thinking about N tending to infinity (can you see how?)

Lecture 11

I started the lecture by mentioning the second law of thermodynamics, which says that entropy is a nondecreasing function of time.

For a distribution p=(p1, ..., pk) the entropy, H(p) = –sum_i p_i log(p_i), is a measure of disorder, or how surprising on average would be an outcome that is chosen according to this distribution. For example, the outcome of the toss of a biased coin is less surprising on average than the outcome of a toss of a fair coin, and this is expressed by the inequality

– p log p – (1–p) log(1–p) ≤ – (1/2) log(1/2) – (1/2) log(1/2).

When the log is taken base 2 then H(p) is a lower bound on the average number of binary bits that would be required to communicate an outcome that is chosen as one of k possible outcomes according to this distribution. The bound can be achieved if and only if every component of p is one of 1/2, 1/4, 1/8, ... . If that is not the case then we might consider taking m i.i.d. samples from this distribution, and then try to communicate these m results optimally as one block. There exists a coding that does this and needs only m·H(p) bits, asymptotically as as m tends to infinity.

I gave an example of a 4 state Markov chain which starts in state 1, and thus having entropy H((1,0,0,0))=0. As n increases the distribution given by the first row of P^n tends to (1/4,1/4/,1/4,1/4), and the entropy increases monotonically to 2 (using log base 2). The point of this example was to motivate the idea that reversibility is only going to make sense once our Markov chain has reached its equilibrium. Otherwise the process will look different when reversed in time because it will appear that entropy is decreasing.

In fact, I cheated a little here, because it is not always the case that the entropy of p_1^{(n)}=(p_{1j}^{(n)}, j=1, ....,k) is monotonically increasing in n. This is true if and only if the equilibrium distribution of the Markov chain is the uniform distribution (as it was in my example). Nonetheless the same point is valid. Unless we start the chain in equilibrium we will be able to detect some difference between the process run forward in time and the process run backwards in time.

There is a Part II course called Coding and Cryptography in which you can learn more about entropy as it relates to efficient communications. Not surprisingly, it also crops up in the Cosmology and Statistical Physicscourses.

Following the lecture I made a change in the notes to the proof of Theorem 11.4, and a small correction in Example 11.5.

Lecture 10

We only proved the first part of Theorem 10.2 (Ergodic theorem). The second part is a simple corollary of the first part. For details you can look at page 3 in Section 1.10 of Jame's Norris's notes.

The material on the random target lemma and Kemeny's constant is non-examinable, but I have presented because I think it is fun. It is surprising (don't you think?) that the expected time to reach equilibrium (in the sense of this lemma) is independent of the starting state. (As well being a co-author with Laurie Snell of the book Finite Markov Chains, John Kemeny was President of Dartmouth College, and one of the inventors of the BASIC programming language.)

Of course there are other ways to think about the time that is required for a Markov chain to reach equilibrium. One is the mixing time, τ(ε), which is defined for ε>0 as the the least time such that

max_i ∑ _j | p_ij⁽ⁿ⁾ - π_j | < ε for all n ≥ τ(ε).

This is closely related to the magnitude of the second-largest eigenvalue of P, say λ₂. The smaller is |λ₂| then the smaller is the mixing time. In fact, one can prove bounds such as

|λ₂| log(1/(2 ε))/(2(1−|λ₂|)) ≤ τ(ε) ≤ log(1/(π* ε))/(1−|λ₂|)

where π* is the smallest component of π (the invariant distribution).

Other quantities that are interesting are the coupling time (the time for two independent copies of the Markov chain to couple) and the cover time (the time for the Markov chain to visit every state at least once).

I concluded the lecture with a demonstation how Engel's probabilistic abacus can be used to calculate the invariant distribution of a finite-state Markov chain in which all p_{ij} are rational numbers. The puzzle is "why does this algorithm always terminate in a finite number of steps?" There is more about this algorithm in Example Sheet 2, #15 and Section 12.5 of the notes. I first heard about this problem from Laurie Snell circa 1975. At that time he said that the questions as why it works was still open.