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# Kolmogorov’s inequality

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 $\lambda>0$,

$P\left(\max_{{1\leq k\leq n}}|S_{k}|\geq\lambda\right)\leq\frac{1}{\lambda^{2}% }\operatorname{Var}[S_{n}]=\frac{1}{\lambda^{2}}\sum_{{k=1}}^{n}\operatorname{% Var}[X_{k}],$ |

where $S_{k}=X_{1}+\cdots+X_{k}$.

Related:

ChebyshevsInequality2, MarkovsInequality, ChebyshevsInequality

Type of Math Object:

Theorem

Major Section:

Reference

## Mathematics Subject Classification

60E15*no label found*

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new correction: Error in proof of Proposition 2 by alex2907

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new correction: Error in proof of Proposition 2 by alex2907

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new question: A good question by Ron Castillo

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new question: A trascendental number. by Ron Castillo

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new question: Banach lattice valued Bochner integrals by math ias

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new question: young tableau and young projectors by zmth

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new question: binomial coefficients: is this a known relation? by pfb

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new question: difference of a function and a finite sum by pfb

## Comments

## Regarding standard notation

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.

## Re: Regarding standard notation

I definitely agree with Koro. Similar things have happened to me. For instance, I received a similar correction regarding my usage of Landau notation. I rejected the correction and told the person who filed it the following:

"I feel like you have the mentality that every single member of PM needs to approach mathematics from the exact same standpoint and use the exact same notation. Mathematics will never work this way. We are individuals and therefore think differently. I will say this again: I definitely feel that there needs to be room for individuality here on PM."

Of course, PM would not fulfill its purpose if no one can understand the notation and terminology used within it. On the other hand, I believe that PM cannot function if everyone must be forced to use the exact same notation and terminology. I feel the same way as Koro: If I were forced to adhere to notation and terminology that I did not find natural, I would not create an entry.

Also, throughout history, people have had to use "nonstandard" notation and terminology to introduce new concepts. Imagine if the person or people who came up with idea of using sigma notation for sums were told, "You can't do that! No one else uses it! Use '...+...+...' instead like everyone else!" It is through experimenting with notation and terminology that mathematics has become what it is today. If people are discouraged from tinkering around with these things, mathematics would not be able to progress.

Those are my thoughts for what they are worth.

Warren

## Re: Regarding standard notation

I also agree with everything that has been said here.

And I also want to add that different subfields in mathematics often use different notation for the same underlying concepts, sometimes for good reasons (e.g. X in probability versus f in measure theory). So there might not even be one single 'standard notation'.

Also, I find that I sometimes use different (nicer) notation when typesetting math on computer versus the notation that I use for quick hand-writing. (e.g. bold vectors versus arrows, blackboard bold for expectation versus none, etc.) Of course, some writers on PM have different preferences here too.

I think the onus to familiarize the reader with the different common notations in use should be on the definition entry for a concept.

// Steve