Strong law of large numbers

The strong law of large numbers states that averages of repeated random observations converge almost surely to their expected value under specified conditions.

Almost-sure convergence: A sequence of random variables converges almost surely when it converges with probability one. This is the mode of convergence asserted for the sample averages.
Expected value: The probability-weighted average of a random variable's possible values, when the expectation exists. It is the target value approached by the sample averages.
Weak law of large numbers: A theorem stating that sample averages converge in probability to the expected value under specified conditions. Unlike the strong law, it guarantees small error probabilities at each large sample size, not almost-sure convergence.
Émile Borel: A French mathematician whose work helped establish modern measure theory and probability. Borel proved an early strong law for independent trials with two possible outcomes.
Sample mean: The arithmetic average of observations in a finite sample. The theorem tracks this statistic as the number of observations grows.
Variance: A measure of the average squared deviation of a random variable from its expected value. Variance conditions help control how far partial averages stray from their target.
Central limit theorem: A theorem describing the limiting distribution of suitably normalized sums of random variables. It describes the fluctuations around the average's limit rather than convergence to that limit alone.
Francesco Cantelli: An Italian mathematician who contributed foundational results to probability theory. Cantelli established an early general strong law for independent random variables.
Independent and identically distributed random variables: Random variables are independent and identically distributed when they are mutually independent and share one probability distribution. This standard setting supplies a familiar version of the theorem's assumptions.
Random variable: A measurable function that assigns a numerical value to each outcome of a random experiment. The theorem applies to sequences of random variables and their averages.

Linked from 14 pages