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Kempner Colloquium
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\begin{Huge}\textsc{Applications of Cramer�s formula\\ \vskip .2in 
 to probability theory}
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\begin{Large}\textbf{Daniel Stroock}
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MIT
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The theory of Markov chains on a finite state space is an elaborate story whose main character is a transition probability matrix $P$, a matrix with non-negative entries each of whose rows sums to $1$. Given $P$, a corresponding Markov chain is a family $\{X_n : n\ne 0\}$ of random variables with the property that, conditional on $(X_0, \ldots , X_n)$, the probability that $X_{n+1} = j$ is $P_{X_nj}$. Thus, at least in theory, every problem that one can pose about the chain can be solved in terms of $P$ once one knows its initial distribution, the distribution of $X_0$. One such problem is that of finding the stationary distribution $\pi$ for the chain. That is, the initial distribution with the property that $(X_0, \ldots , X_n, \ldots )$ has the same distribution as $(X_1 , \ldots , X_{n+1} , \ldots )$. In this lecture, I will use Cramer�s formula for determinants to derive an expression for $\pi$ in terms of the determinants of submatrices of $I - P$ . If there is time, I will also discuss how the same ideas lead to an elegant proof of Wilson's algorithm for putting the uniform distribution on the set of spanning trees in a connected graph and, as a corollary, derive Kirchhoff's formula for the number of spanning trees.


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Monday, September 30, 2013

4:00 p.m.

MATH 350

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