Kolmogorov's inequality

Kolmogorov's inequality bounds the probability that partial sums of independent, mean-zero random variables ever reach magnitude at least λ by their total variance divided by λ².

Partial sum: A sum of the first several terms of a sequence. The inequality controls the maximum magnitude among these successive sums.
Independent random variables: Random variables whose joint probabilities factor into products of their individual probabilities. Independence prevents cross terms from inflating the variance of a partial sum.
Kolmogorov's convergence criterion: A theorem giving conditions under which a series of independent random variables converges almost surely. It uses maximal-probability bounds to turn summable variance tails into almost-sure convergence.
Andrey Kolmogorov: A Russian mathematician who formalized probability theory using measure theory. The inequality bears his name and reflects his foundational work in probability.
First-passage time: The first time a stochastic process reaches a specified set or threshold. Stopping at the first threshold crossing organizes the maximal-probability argument.
Expected value: The probability-weighted average value of a random variable. The mean-zero condition centers each increment before the sums are controlled.
Strong law of large numbers: A theorem stating that sample averages converge almost surely to their expected value under suitable assumptions. Maximal inequalities help establish almost-sure control across many sample sizes.
Foundations of the Theory of Probability: Kolmogorov's 1933 book axiomatizing probability with measure theory. Its framework made rigorous treatment of random sequences and their limits possible.
Variance: A measure of the spread of a random variable around its mean. Independence makes the partial sum's total variance the bound's numerator.
Finite variance: The condition that a random variable's squared deviation from its mean has finite expectation. Finite variances make the inequality's total-variance bound meaningful.

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