Almost sure convergence

A sequence of random variables converges almost surely if, with probability one, its values converge pointwise to a limit random variable.

Random variable: A measurable function from a probability space to a space of possible values. Each term in the sequence assigns a value to every outcome.
Convergence in probability: Random variables converge in probability when the chance of an error exceeding any fixed threshold tends to zero. Almost sure convergence implies this weaker mode, but the converse can fail.
Borel–Cantelli lemma: A pair of probability results linking event probabilities to whether infinitely many events occur. Summable probabilities of large errors ensure that only finitely many such errors occur almost surely.
Strong law of large numbers: A theorem stating that sample averages of suitable independent random variables converge almost surely to their expected value. It is a canonical theorem whose conclusion is almost sure convergence.
Almost surely: A property holds almost surely when the event on which it fails has probability zero. The convergence requirement permits failure only on a null event.
Convergence in distribution: Random variables converge in distribution when their probability laws converge weakly. This mode concerns distributions rather than pointwise behavior on shared outcomes.
Tail event: An event whose occurrence is unchanged by altering any finite initial segment of a sequence. The event that a sequence converges is determined by its eventual behavior.
Kolmogorov three-series theorem: A criterion for almost sure convergence of a series of independent random variables. It turns almost sure convergence into three tests on truncated summands.
Pointwise convergence: A sequence of functions converges pointwise when it converges separately at every point in its domain. Almost sure convergence is pointwise convergence after excluding a null set.
L^p convergence: A sequence converges in Lp when the expected pth power of its error tends to zero. Moment control gives a distinct route to convergence, sometimes implying almost sure convergence with extra conditions.

Linked from 19 pages