en>Excirial: Reverted edits by 138.195.23.88 (talk) to last revision by 138.195.23.88 (HG 3)

2013-12-11T21:05:39Z

Reverted edits by 138.195.23.88 (talk) to last revision by 138.195.23.88 (HG 3)

← Older revision		Revision as of 23:05, 11 December 2013
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	~~Wilber Berryhill~~ is ~~what his wife loves to call him and he totally enjoys this name~~. ~~Alaska~~ is ~~where he~~'~~s usually been residing~~. ~~Invoicing is what I do~~. It's ~~not~~ a ~~typical thing but what she likes performing~~ is to ~~perform domino but she doesn~~'~~t have~~ the ~~time lately~~.<br><br>~~my blog :: psychic readers~~ (~~[http~~://~~appin~~.co.kr/~~board_Zqtv22~~/~~688025 http~~:/~~/appin~~.co.~~kr/board_Zqtv22/688025~~])		In [[probability theory]], '''Kolmogorov's inequality''' is a so-called "maximal [[inequality (mathematics)\|inequality]]" that gives a bound on the probability that the [[partial sum]]s of a [[Finite set\|finite]] collection of [[independent random variables]] exceed some specified bound. The inequality is named after the [[Russia]]n [[mathematician]] [[Andrey Kolmogorov]].{{Citation needed\|date=May 2007}}

			==Statement of the inequality==
			Let ''X''<sub>1</sub>, ..., ''X''<sub>''n''</sub> : Ω → '''R''' be [[Statistical independence\|independent]] [[random variable]]s defined on a common [[probability space]] (Ω, ''F'', Pr), with [[expected value]] E[''X''<sub>''k''</sub>] = 0 and [[variance]] Var[''X''<sub>''k''</sub>] < +∞ for ''k'' = 1, ..., ''n''. Then, for each λ > 0,

			:<math>\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], </math>

			where ''S''<sub>''k''</sub> = ''X''<sub>1</sub> + ... + ''X''<sub>''k''</sub>.

			==Proof==
			The following argument is due to [[Kareem Amin]] and employs discrete [[Martingale (probability theory)\|martingale]]s.
			As argued in the discussion of [[Doob's martingale inequality]], the sequence <math>S_1, S_2, \dots, S_n</math> is a martingale.
			[[Without loss of generality]], we can assume that <math>S_0 = 0</math> and <math>S_i \geq 0</math> for all <math>i</math>.
			Define <math>(Z_i)_{i=0}^n</math> as follows. Let <math>Z_0 = 0</math>, and
			:<math>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.
			</math>
			for all <math>i</math>.
			Then <math>(Z_i)_{i=0}^n</math> is also a martingale. Since <math>S_i-S_{i-1}</math> is independent and mean zero,
			:<math>\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}
			</math>
			The same is true for <math>(Z_i)_{i=0}^n</math>. Thus
			:<math>\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}
			</math>
			by [[Chebyshev's inequality]].

			This inequality was generalised by Hájek and Rényi in 1955.

			==See also==
			* [[Chebyshev's inequality]]
			* [[Etemadi's inequality]]
			* [[Landau–Kolmogorov inequality]]
			* [[Markov's inequality]]
			* [[Bernstein inequalities (probability theory)]]

			==References==
			* {{cite book \| last=Billingsley \| first=Patrick \| title=Probability and Measure \| publisher=John Wiley & Sons, Inc. \| location=New York \| year=1995 \| isbn=0-471-00710-2}} (Theorem 22.4)
			* {{cite book \| last=Feller \| first=William \| authorlink=William Feller \| title=An Introduction to Probability Theory and its Applications, Vol 1 \| edition=Third Edition \| origyear=1950 \| year=1968 \| publisher=John Wiley & Sons, Inc. \| location=New York \| isbn=0-471-25708-7 \| nopp=true \| page=xviii+509 }}

			{{PlanetMath attribution\|id=3687\|title=Kolmogorov's inequality}}

			[[Category:Probability theory]]
			[[Category:Stochastic processes]]
			[[Category:Probabilistic inequalities]]
			[[Category:Articles containing proofs]]

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Wilber Berryhill is what his wife loves to call him and he totally enjoys this name. Alaska is where he's usually been residing. Invoicing is what I do. It's not a typical thing but what she likes performing is to perform domino but she doesn't have the time lately.<br><br>my blog :: psychic readers ([http://appin.co.kr/board_Zqtv22/688025 http://appin.co.kr/board_Zqtv22/688025])

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