Kolmogorov's two-series theorem

A theorem stating that a series of independent random variables converges almost surely when the series of their expectations and the series of their variances converge.

Independent random variables: Random variables whose joint probabilities factor into products of their individual probabilities. Independence is the theorem’s key hypothesis, allowing variance control of partial sums.
Series of random variables: A sequence of random variables added term by term, producing random partial sums. The theorem concerns convergence of these partial sums almost surely.
Kolmogorov's three-series theorem: A criterion characterizing almost-sure convergence of series of independent random variables through three series conditions. It handles general independent terms by truncation, beyond the direct moment assumptions here.
Rademacher series: A random series whose terms are fixed coefficients multiplied by independent random signs. Its centered terms have variances equal to squared coefficients, giving a direct summability test.
Expected value: The probability-weighted average of a random variable’s possible values. Convergence of the expectations controls the deterministic drift of the partial sums.
Convergence in probability: Convergence in which the probability of an error exceeding any fixed threshold tends to zero. The theorem gives almost-sure convergence, a stronger claim than convergence in probability.
Kolmogorov's strong law of large numbers: A theorem giving almost-sure convergence of normalized sums of independent random variables to their common mean under suitable conditions. It also uses independence and variance control, but concerns averages rather than an unnormalized series.
Random Fourier series: A Fourier series whose coefficients or phases include random variables. Independent random coefficients can make variance summability a tool for proving pointwise convergence.
Variance: The expected squared deviation of a random variable from its mean. Summable variances bound the cumulative fluctuations around the means.
Cauchy criterion: A criterion stating that a sequence converges in a complete space exactly when its terms become arbitrarily close. Proofs establish that random partial sums are almost surely Cauchy.

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