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| Kolmogorov's inequality | |
|---|---|
| Name | Kolmogorov's inequality |
| Field | Probability theory |
| Named after | Andrey Kolmogorov |
| First published | 1928 |
Kolmogorov's inequality Kolmogorov's inequality is a classical result in Probability theory that gives an upper bound on the probability that the maximum of partial sums of independent zero-mean random variables exceeds a threshold. It plays a central role in the theories developed by Andrey Kolmogorov, Paul Lévy, William Feller, Aleksandr Khinchin, and Émile Borel and is frequently invoked alongside results by Kolmogorov (1933), Sergio Fubini, Édouard Vitali, and André Weil in foundational work on convergence, martingales, and measure-theoretic probability.
Let X1, X2, ..., Xn be independent real-valued random variables with E[Xi] = 0 for each i. Define partial sums Sk = X1 + X2 + ... + Xk. Kolmogorov's inequality asserts that for any ε > 0, P(max_{1 ≤ k ≤ n} |Sk| ≥ ε) ≤ Var(Sn)/ε^2. This formulation connects to variance estimates used by Sergey Bernstein, Harald Cramér, Norbert Wiener, Kurt Gödel, and Emil Artin in studies of limit theorems, and it complements inequalities attributed to Chebyshev, Markov (Alexander Markov), Paul Erdős, and Mark Kac.
A standard proof uses orthogonality and projection techniques familiar from works of Andrey Kolmogorov, John von Neumann, Stefan Banach, Alfred Tarski, and Stefan Banach's collaborators. Introduce stopping time T = min{k ≤ n: |Sk| ≥ ε} with the convention T = n+1 if none occurs. Consider truncated sums S_{k ∧ T} and note that E[S_{k ∧ T}] = 0 by independence and zero means, an approach paralleling arguments in texts by Doob, Joseph L. Doob, Khinchin, Paul Lévy, and Paul Halmos. Compute E[S_{n ∧ T}^2] = ∑_{i=1}^n E[(X_i)^2 1_{i ≤ T}] and use nonnegativity to bound ε^2 P(T ≤ n) ≤ E[S_{n ∧ T}^2] ≤ E[S_n^2] = Var(Sn). Rearranging yields the inequality. Variants of this argument appear in expositions by William Feller, Claude Shannon, Norbert Wiener, André Weil, and Kolmogorov's successors.
Kolmogorov's inequality is applied in proofs of the strong law of large numbers by Andrey Kolmogorov, Émile Borel, S. N. Bernstein, Vyacheslav Stepanov, and Feller; in maximal inequality derivations by Joseph Doob, Donald Burkholder, Richard Durrett, K. L. Chung, and Paul Lévy; and in martingale convergence results used in work by Frederick Mosteller, Paul Erdős, Mark Kac, William Feller, and Andrey Kolmogorov. It is also instrumental in empirical process theory developed by Aldous, David Pollard, Vladimir Vapnik, Alexey Chervonenkis, and Lucien Le Cam, and in concentration inequalities associated with Joel Spencer, Michel Talagrand, Elliott Lieb, and Murray Gell-Mann.
Generalizations and relatives include Kolmogorov's inequality for martingales by Joseph L. Doob, exponential bounds by Chernoff (Herman Chernoff), Hoeffding (Wassily Hoeffding), and Bernstein (Sergey Bernstein), Rosenthal-type inequalities associated with Bennett (George Bennett), Rosenthal (H. P. Rosenthal), and Burkholder–Davis–Gundy inequalities developed by Donald Burkholder, Roger Davis, and Bruce Gundy. Further connections appear with maximal inequalities by Stefan Banach, Andrey Kolmogorov, Paul Lévy, and martingale transforms investigated by Gundy and Burkholder. Links to functional limit theorems studied by Donsker (Monroe Donsker), invariance principles by Prokhorov (Yuri Prokhorov), and empirical process bounds by Talagrand and Vapnik–Chervonenkis theory are standard.
Examples illustrating the bound include symmetric Bernoulli sums treated by Jean-Pierre Kahane, Paul Erdős, and Alfréd Rényi; Gaussian sums connected to work by Norbert Wiener, Joseph Doob, and Kolmogorov; and heavy-tailed examples studied by William Feller, Paul Lévy, and Benoît Mandelbrot, where the inequality can be nonsharp. Counterexamples highlighting limits of the inequality are provided by dependent sequences examined by Bruno de Finetti, Kolmogorov (dependency studies), René Gateaux, and mixing counterexamples analyzed by Ibragimov (Ildar Ibragimov). Modifications are necessary for arrays and dependent structures studied by David Aldous, James Propp, Mikhail Gordin, and M. Iosifescu.