VarshamovTenengoltsCodes

Varshamov-Tenen Inequality is a mathematical inequality that provides an upper bound on the probability that the sum of a sequence of independent random variables exceeds a certain threshold. It is particularly useful in scenarios where the random variables have bounded variances.

Let X_1, X_2, ..., X_n be independent random variables with zero mean and finite variances. Let S_n = X_1 + X_2 + ... + X_n. The Varshamov-Tenen Inequality states that for any t > 0,

P(S_n >= t) <= exp(-t^2 / (2 * sum(Var(X_i)))).

This inequality is a generalization of Chebyshev's inequality and provides a tighter bound when the variances of the individual random variables are known. It finds applications in various fields, including statistical learning theory, randomized algorithms, and information theory, where bounding the deviation of sums of random variables is crucial. The inequality is named after its discoverers, and it is a fundamental tool for probabilistic analysis.