Student Seminar Talk THE PROBABILISTIC METHOD IN ANALYSIS

The law of large numbers says that the sum of independent random variables is close to its expected value with high probability. In this talk, I will present a quantitative statement of this sort and demonstrate how this concentration phenomenon can be applied to problems in analysis. 1. Concentration around the expected value The most important statistical measure of a (real) random variable X is its expected value E X, also called the mean of X. How likely is it for X to deviate from its expected value? Theorem 1.1 (Markov’s inequality). Let X be a non-negative random variable. Then for any α > 0, P(X ≥ α) ≤ E X α . Markov’s inequality is a quantitative statement on the concentration ofX around its expected value : it says that X is unlikely to greatly exceed its mean. For example, the probability for a non-negative X to become twice as large as its expected value is less than half. The variance V (X) = E (X − E X) is a measurement of the dispersion of X around its expected value. A knowledge of the variance makes it possible to say more on the concentration of X. Theorem 1.2 (Chebyshev’s inequality). For any α > 0, P(|X − E X| ≥ α) ≤ V (X) α2 . Chebyshev’s inequality is more or less the best possible without any additional assumptions on X. 2. Sum of independent random variables Let X1, . . . , XN be independent random variables, and let S be their sum S = X1 + · · ·+XN . What can be said on the concentration of S around its expected value? Nothing will be changed if Xj is replaced with Xj − E Xj , and so we may assume that E Xj = 0 and then also E S = 0. This is an expository talk delivered on October 16, 2006 in the Student Seminar, Tel-Aviv university (thanks to Shahaf Nitzan-Hahamov for suggesting the topic of this talk). 1