Bernstein inequalities (probability theory)

In probability theory, Bernstein inequalities give bounds on the probability that the sum of random variables deviates from its mean. In the simplest case, let X1, ..., Xn be independent Bernoulli random variables taking values +1 and −1 with probability 1/2, then for every positive $\varepsilon$,

$\mathbf{P} \left (\left|\frac{1}{n}\sum_{i=1}^n X_i\right| > \varepsilon \right ) \leq 2\exp \left (-\frac{n\varepsilon^2}{ 2 (1 + \frac{\varepsilon}{3})} \right).$

Bernstein inequalities were proved and published by Sergei Bernstein in the 1920s and 1930s.[1][2][3][4] Later, these inequalities were rediscovered several times in various forms. Thus, special cases of the Bernstein inequalities are also known as the Chernoff bound, Hoeffding's inequality and Azuma's inequality.

This distribution is also known as the Rademacher distribution.

Some of the inequalities

1. Let X1, ..., Xn be independent zero-mean random variables. Suppose that |X i| ≤ M almost surely, for all i. Then, for all positive t,

$\mathbf{P} \left (\sum_{i=1}^n X_i > t \right ) \leq \exp \left ( -\frac{\tfrac{1}{2} t^2}{\sum \mathbf{E} \left[X_j^2 \right ]+\tfrac{1}{3} Mt} \right ).$

2. Let X1, ..., Xn be independent random variables. Suppose that for some positive real L and every integer k > 1,

$\mathbf{E} \left [|X_i^k|\right ] \leq \tfrac{1}{2} \mathbf{E} \left[X_i^2\right] L^{k-2} k!$

Then

$\mathbf{P} \left ( \sum_{i=1}^n X_i \geq 2 t \sqrt{\sum \mathbf{E} \left [ X_i^2 \right ]} \right ) < \exp (-t^2), \qquad \text{for } 0 < t \leq \tfrac{1}{2L}\sqrt{\sum \mathbf{E} \left[X_j^2\right ]}.$

3. Let X1, ..., Xn be independent random variables. Suppose that

$\mathbf{E} \left[|X_i^k|\right ] \leq \frac{k!}{4!} \left(\frac{L}{5}\right)^{k-4}$

for all integer k > 3. Denote

$A_k = \sum \mathbf{E} \left [ X_i^k\right ].$

Then,

$\mathbf{P} \left( \left| \sum_{j=1}^n X_j - \frac{A_3 t^2}{3A_2} \right|\geq \sqrt{2A_2} \, t \left[ 1 + \frac{A_4 t^2}{6 A_2^2} \right] \right ) < 2 \exp (- t^2), \qquad \text{for } 0 < t \leq \frac{5 \sqrt{2A_2}}{4L}.$

4. Bernstein also proved generalizations of the inequalities above to weakly dependent random variables. For example, inequality (2) can be extended as follows. Let X1, ..., Xn be possibly non-independent random variables. Suppose that for all integer i > 0,

\begin{align} \mathbf{E} \left [ X_i | X_1, \dots, X_{i-1} \right ] &= 0, \\ \mathbf{E} \left [ X_i^2 | X_1, \dots, X_{i-1} \right ] &\leq R_i \mathbf{E} \left [ X_i^2 \right ], \\ \mathbf{E} \left [ X_i^k | X_1, \dots, X_{i-1} \right ] &\leq \tfrac{1}{2} \mathbf{E} \left[ X_i^2 | X_1, \dots, X_{i-1} \right ] L^{k-2} k! \end{align}

Then

$\mathbf{P} \left( \sum_{i=1}^n X_i \geq 2 t \sqrt{\sum_{i=1}^n R_i \mathbf{E}\left [ X_i^2 \right ]} \right) < \exp(-t^2), \qquad \text{for } 0 < t \leq \tfrac{1}{2L} \sqrt{\sum_{i=1}^n R_i \mathbf{E} \left [X_i^2 \right ]}.$

More general results for martingales can be found in Fan et al. (2012).[5]

Proofs

The proofs are based on an application of Markov's inequality to the random variable

$\exp \left ( \lambda \sum_{j=1}^n X_j \right ),$

for a suitable choice of the parameter λ > 0.