A seminar by Tom Körner

by Ian Short

Tom Körner gave a seminar here at the Open University a couple of weeks back about using the Baire Category Theorem to prove a quantitative form of a theorem of Rudin. Tom made some passing remarks on the Central Limit Theorem, which I discuss and develop in this post.

Central Limit Theorem. Let X1, X2,… be identically distributed, independent random variables with mean 0 and variance 1. Then, for a<b, $latex \displaystyle{P\left(\frac{X_1+X_2+\cdots + X_n}{\sqrt{n}}\in [a,b]\right)\to \int^b_a \frac{1}{2\pi}e^{-t^2/2}dt}.$

This theorem is well used in statistics, finance, and so forth, but as stated above it says nothing about the speed of convergence. It may be that convergence is so slow that the theorem is practically useless. Tom mentioned a related theorem (of Bernstein) which gives an idea of rates of convergence of sums of independent random variable.

Theorem. Let X1, X2,… be independent random variables, each with |Xi|≤1 and mean 0. Then, for k>0,
[latex]\displaystyle{P(X_1+X_2+\cdots+X_n\geq k)\leq \exp(-k^2/(4n)).}[/latex]

Sketch proof. First, by expanding out exp(tXi), we can check that

[latex]\displaystyle{E(\exp(tX_i))\leq \exp(t^2).}[/latex] Let [latex]S_n=X_1+X_2+\cdots X_n.[/latex] Then[latex]\displaystyle{E\left(\exp\left(tS_n\right)\right)\leq \exp(nt^2).}[/latex] Also,
[latex]\displaystyle{P(S_n\geq k)\exp(tk) \leq \int_{S_n\geq k} \exp(tS_n) dP \leq E(\exp(tS_n)).}[/latex] Choose t=k/(2n) and combine the past two equations for the result. ■

I’ll apply this theorem, with certain simplifying assumptions, to the football Premier League. I have written about the Premier League before, in Football Leagues. Each team in the Premier League plays 38 matches. I assume these matches are all independent, and there is no home bias. I will assume that each match is either won or lost (no draws), and 11/4 points are awarded for a win, and 0 for a loss. (The 11/4 points for a win is less than the usual 3, but this compensates well for ignoring draws.)

Let Y1, Y2, …, Y38 be the points scored by a particular team. Let us suppose this team has a 50% chance of winning, and a 50% chance of losing. Define Xi = (Yi-11/8)×8/11. Then the conditions of the theorem above are satisfied and, roughly, we find that

[latex]\displaystyle{P(Y_1+Y_2+\cdots +Y_{38}\geq 1.4k+52)\leq \exp(-k^2/152).}[/latex]
Choose k=30 and we find that

[latex]\displaystyle{P(Y_1+Y_2+\cdots +Y_{38}\geq 94)\leq 0.003.}[/latex]
In other words, there is a less than 0.3% chance that a team from this random league scores more than 94 points. In 2005 Chelsea amassed 95 points, which indicates that teams weren’t all of equal ability in the 2004–2005 season; Chelsea were better than most teams. That is, without knowledge of football, from analysing the distribution of points alone, we see that teams vary in ability.

This quick application of Bernstein’s Theorem was an experiment I carried out through curiosity. There are better ways of analysing football scores.