Today's lecture was a high point in our course: the proof by coupling that for an ergodic Markov chain P(X_n=i) tends to the equilibrium probability pi_i as n tends to infinity.

In response to a good question that a student asked during the lecture I have slightly modified in the notes the wording to the proof of Theorem 9.1, for the part (iii) implies (i).

I mentioned Vincent Doblin (1915-40) [also known as Wolfgang Doeblin] to whom is due the coupling proof of Theorem 9.8. There is a good article about his life in a 2001 article in

*The Telegraph*:*Revealed: the maths genius of the Maginot line*. Some of Doblin's work was only discovered in summer 2000, having been sealed in an envelope for 60 years.
I quoted J. Michael Steele on coupling:

Coupling is one of the most powerful of the "genuinely probabilistic" techniques. Here by "genuinely probabilistic'' we mean something that works directly with random variables rather than with their analytical co-travelers (like distributions, densities, or characteristic functions.

I like the label "analytic co-travelers". The coupling proof should convince you that Probability is not merely a branch of Analysis. The above quote is taken from Mike's blog for his graduate course on Advanced Probability at Wharton (see here). Mike has a fascinating and very entertaining web site, that is full of "semi-random rants" and "favorite quotes" that are both funny and wise (such as his "advice to graduate students"). I highly recommend browsing his web site. It was by reading the blogs for his courses that I had the idea of trying something similar myself this year. So if you have been enjoying this course blog - that is partly due to him. Here is a picture of Mike and me with Thomas Bruss at the German Open Conference in Probability and Statistics 2010, in Leipzig.

The playing cards for the magic trick that I did in the lecture were "dealt" with the help of the playing card shuffler at random.org. By the way, there is a recent interview with Persi Diaconis in the October 27 (2011) issue of

*Nature,*entitled "The Mathemagician".

*The following is a little sidebar for those of you who like algebra and number theory*.

In the proof of Lemma 9.5 we used the fact that if the greatest common divisor of n1,..., nk is 1 then for all sufficiently large n there exist some non-negative integers a1,..., ak such that

n = a1 n1 +···+ ak nk. (*)

**Proof**. Think about the smallest positive integer d that can be expressed as d = b1 n1 +

**···**+ bk nk for some integers b1,..., bk (which may be negative as well as non-negative). Notice that the remainder of n1 divided by d is also of this form, since it is r =n1 − m (b1 n1 +

**···**+ bk nk) for m=⌊n1 /d⌋. If d does not divide n1 then r<d, and d fails to be the smallest integer that can be expressed in the form d = b1 n1 +

**···**+ bk nk. Thus we must conclude that d divides n1. The same must be true for every other nj, and so d=gcd(n1,...,nk)=1. So now we know that it is possible to write 1 = b1 n1 +

**···**+ bk nk, and so also we know we can write j = j (b1 n1 +

**···**+ bk nk), for all j=1,..., n1. Finally, we can leverage this fact to conclude that for some large N we can write all of N, N+1, N+2,..., N+n1 in the required form (*), and hence also we can also express in form (*) all integers N + m n1 + j, where m and j are non-negative integers, i.e. we can do this for all integers n≥ N.

*(This is a proof that I cooked up on the basis of my rather limited expertise in algebra. Perhaps one of you knows a quicker or more elegant way to prove this little fact? If so, please let me know.*)