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.Fact|date=May 2007

tatement of the inequality

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

:$Pr\; left(max\_\{1leq\; kleq\; n\}\; |\; S\_k\; |geqlambda\; ight)leq\; frac\{1\}\{lambda^2\}\; operatorname\{Var\}\; [S\_n]\; equiv\; frac\{1\}\{lambda^2\}sum\_\{k=1\}^n\; operatorname\{Var\}\; [X\_k]\; ,$

where "S"_"k" = "X"₁ + ... + "X"_"k".

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:for all $i$.Then $(Z\_i)\_\{i=0\}^n$ is a also a martingale. Since $ext\{E\}\; [S\_\{i\}]\; =\; ext\{E\}\; [S\_\{i-1\}]$ for all $i$ and $ext\{E\}\; [\; ext\{E\}\; [X|Y]\; =\; ext\{E\}\; [X]$ by the law of total expectation,:The same is true for $(Z\_i)\_\{i=0\}^n$. Thus:by Chebyshev's inequality.

ee also

* Chebyshev's inequality
* Doob's martingale inequality
* Etemadi's inequality
* Landau-Kolmogorov inequality
* Markov's inequality

References

* (Theorem 22.4)
*

----