Kolmogorov's inequality

In probability theory, Kolmogorov's inequality is a so-called "maximal inequality" that gives a bound on the probability that the partial sums of a finite collection of independent random variables exceed some specified bound. The inequality is named after the Russian mathematician Andrey Kolmogorov.

Statement of the inequality

Let X1, ..., Xn : Ω → R be independent random variables defined on a common probability space (Ω, F, Pr), with expected value E[Xk] = 0 and variance Var[Xk] < +∞ for k = 1, ..., n. Then, for each λ > 0,

\Pr \left(\max_{1\leq k\leq n} | S_k |\geq\lambda\right)\leq \frac{1}{\lambda^2} \operatorname{Var} [S_n] \equiv \frac{1}{\lambda^2}\sum_{k=1}^n \operatorname{Var}[X_k],

where Sk = X1 + ... + Xk.

Proof

The following argument is due to Kareem Amin and employs discrete martingales. As argued in the discussion of Doob's martingale inequality, the sequence S_1, S_2, \dots, S_n is a martingale. Without loss of generality, we can assume that S_0 = 0 and S_i \geq 0 for all i. Define (Z_i)_{i=0}^n as follows. Let Z_0 = 0, and

Z_{i+1} = \left\{ \begin{array}{ll} S_{i+1} & \text{ if } \displaystyle \max_{1 \leq j \leq i} S_j < \lambda \\ Z_i & \text{ otherwise} \end{array} \right.

for all i. Then (Z_i)_{i=0}^n is also a martingale. Since S_i-S_{i-1} is independent and mean zero,

\begin{align} \sum_{i=1}^n \text{E}[ (S_i - S_{i-1})^2] &= \sum_{i=1}^n \text{E}[ S_i^2 - 2 S_i S_{i-1} + S_{i-1}^2 ] \\ &= \sum_{i=1}^n \text{E}\left[ S_i^2 - 2 (S_{i-1} + S_{i} - S_{i-1}) S_{i-1} + S_{i-1}^2 \right] \\ &= \sum_{i=1}^n \text{E}\left[ S_i^2 - S_{i-1}^2 \right] - 2\text{E}\left[ S_{i-1} (S_{i}-S_{i-1})\right]\\ &= \text{E}[S_n^2] - \text{E}[S_0^2] = \text{E}[S_n^2]. \end{align}

The same is true for (Z_i)_{i=0}^n. Thus

\begin{align} \text{Pr}\left( \max_{1 \leq i \leq n} S_i \geq \lambda\right) &= \text{Pr}[Z_n \geq \lambda] \\ &\leq \frac{1}{\lambda^2} \text{E}[Z_n^2] =\frac{1}{\lambda^2} \sum_{i=1}^n \text{E}[(Z_i - Z_{i-1})^2] \\ &\leq \frac{1}{\lambda^2} \sum_{i=1}^n \text{E}[(S_i - S_{i-1})^2] =\frac{1}{\lambda^2} \text{E}[S_n^2] = \frac{1}{\lambda^2} \text{Var}[S_n] \end{align}

by Chebyshev's inequality.

This inequality was generalized by Hájek and Rényi in 1955.

See also

References

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