Kolmogorov's inequality

The Kolmogorow inequality , also called Kolmogorow's maximum inequality , is an inequality from stochastics . It was proved by the Russian mathematician Andrei Nikolajewitsch Kolmogorow in the late 1920s and serves to prove a strong law of large numbers for random variables that do not necessarily have to be identically distributed , but meet the Kolmogorov condition . The Doobsche inequality is a generalization of Kolmogorov's inequality for martingale .

Your testimony

The Kolmogorow inequality states that the following maximum inequality holds for a sequence of independent random variables with for each : ${\ displaystyle (X_ {n}) _ {n} \;}$ ${\ displaystyle E [X_ {n}] = 0 \;}$ ${\ displaystyle \ varepsilon> 0 \;}$

{\ displaystyle P \ left (\ max _ {0 \ leq k \ leq n} | S_ {k} |> \ varepsilon \ right) \ leq {\ frac {1} {\ varepsilon ^ {2}}} \ sum _ {k = 0} ^ {n} {\ mbox {Var}} (X_ {k})}

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${\ displaystyle S_ {k} \;}$ denotes the -th partial sum . ${\ displaystyle k}$ ${\ displaystyle \ sum _ {i = 0} ^ {k} X_ {i}}$

literature

Heinz Bauer : Probability Theory. De Gruyter, Berlin 2002, ISBN 3110172364 .

Individual evidence

↑ David Meintrup, Stefan Schäffler: Stochastics . Theory and applications. Springer-Verlag, Berlin Heidelberg New York 2005, ISBN 978-3-540-21676-6 , pp. 154 , doi : 10.1007 / b137972 .

[1] David Meintrup, Stefan Schäffler: Stochastics . Theory and applications. Springer-Verlag, Berlin Heidelberg New York 2005, ISBN 978-3-540-21676-6 , pp. 154 , doi : 10.1007 / b137972 .