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

Title Kolmogorov’s inequality KolmogorovsInequality 2013-03-22 13:13:15 2013-03-22 13:13:15 Koro (127) Koro (127) 9 Koro (127) Theorem msc 60E15 ChebyshevsInequality2 MarkovsInequality ChebyshevsInequality