# Bernstein inequality

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2010 Mathematics Subject Classification: Primary: 60E15 Secondary: 26D05 [MSN][ZBL]

$\newcommand{\expect}{\mathbb{E}} \newcommand{\prob}{\mathbb{P}} \newcommand{\abs}{\left|#1\right|}$

Bernstein's inequality in probability theory is a more precise formulation of the classical Chebyshev inequality in probability theory, proposed by S.N. Bernshtein [Be2] in 1911; it permits one to estimate the probability of large deviations by a monotone decreasing exponential function. In fact, if the equations $\expect X_j=0,\quad \expect X_j^2=b_j,\quad j=1,\ldots,n,$ hold for the independent random variables $X_1,\ldots,X_n$ with $\expect\abs{X_j}^l \leq \frac{b_j}{2}H^{l-2}l!$ (where $l>2$ and $H$ is a constant independent of $j$), then the following inequality of Bernstein (where $r>0$) is valid for the sum $S_n=X_1+\cdots+X_n$: \begin{equation}\label{eq1} \prob\left( \abs{S_n} > r \right) \leq 2\exp\left( - \frac{r^2}{2(B_n + Hr)} \right), \end{equation} where $B_b = \sum b_j$. For identically-distributed bounded random variables $X_j$ ($\expect X_j = 0$, $\expect X_j^2 = \sigma^2$ and $\abs{X_j}\leq L$, $j=1,\ldots,n$) inequality \ref{eq1} takes its simplest form: \begin{equation}\label{eq2} \prob\left( \abs{S_n} > t\sigma\sqrt{n} \right) \leq 2\exp\left( - \frac{t^2}{2(1 + a/3)} \right), \end{equation} where $a = Lt/\sqrt{n}\sigma$. A.N. Kolmogorov gave a lower estimate of the probability in \ref{eq1}. The Bernstein–Kolmogorov estimates are used, in particular, in proving the law of the iterated logarithm. Some idea of the accuracy of \ref{eq2} may be obtained by comparing it with the approximate value of the left-hand side of \ref{eq2} which is obtained by the central limit theorem in the form $\frac{2}{\sqrt{2\pi}}\int_t^\infty \mathrm{e}^{-u^2/2}\,\mathrm{d}u = \frac{2}{\sqrt{2\pi}\,t} \left( 1-\frac{\theta}{t^2} \right) \mathrm{e}^{-t^2/2},$ where $0<\theta<1$. Subsequent to 1967, Bernstein's inequalities were extended to include multi-dimensional and infinite-dimensional cases.