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Kolmogorov's inequality (Theorem)

Let $ X_1,\dots, X_n$ be independent random variables in a probability space, such that $ \operatorname{E}[X_k]=0$ and $ \operatorname{Var}[X_k] <\infty$ for $ k=1,\dots, n$. Then, for each $ \lambda>0$,

$\displaystyle P\left(\max_{1\leq k\leq n} \vert S_k\vert\geq\lambda\right)\leq ... ...ratorname{Var}[S_n] = \frac{1}{\lambda^2}\sum_{k=1}^n \operatorname{Var}[X_k], $

where $ S_k = X_1 +\cdots + X_k$.



"Kolmogorov's inequality" is owned by Koro.
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See Also: Chebyshev's inequality, Markov's inequality, Chebyshev's inequality


Attachments:
proof of Kolmogorov's inequality (Proof) by kshum
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Cross-references: probability space, random variables, independent
There is 1 reference to this entry.

This is version 6 of Kolmogorov's inequality, born on 2002-12-08, modified 2006-09-15.
Object id is 3687, canonical name is KolmogorovsInequality.
Accessed 4997 times total.

Classification:
AMS MSC: 60E15 (Probability theory and stochastic processes :: Distribution theory :: Inequalities; stochastic orderings)
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Regarding standard notation by Koro on 2006-09-15 06:11:31
This post is about the correction I received telling me to use the notation \operatorname{E}[X] instead of \operatorname[E]X.

I understand the need for some standarization in the entries of planetmath. But I disagree with the fact that the choice made by the author of the definition entry of a given concept should determine the standards. If only, he should mention different possible notations.
I wasn't asked what notation I preferred, I was imposed the one used in the definition of expectation, which by the way is a poor and incomplete entry.

If we are going to be so picky about notation, it would be good to know who decides the standards and how.

There is notation being used in some entries that I would never use. If I have to choose between using that notation and not writing an entry, I'd probably decide not to write the entry. And I think I'm not the only one, so we should ponder allowing certain flexibility, or at least trying to reach some consensus before forcing standarization.

By the way, there are entries that use some notation and that were intoduced prior to the entries defining the concept associated to that notation.

Maybe this was already discussed and a decision was taken, since I'm not up to the date. If so I'd like to know what's the agreement.
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