Convergence in probability

A sequence of random variables converges in probability to a limit when, for every positive error tolerance, the probability of exceeding that error tends to zero.

Random variable: A measurable function that assigns a numerical value to each outcome of a random experiment. The sequence consists of random variables, whose deviations from the limit are measured probabilistically.
Convergence in mean square: A sequence converges in mean square when the expected squared distance from its limit tends to zero. Mean-square convergence implies convergence in probability through Markov's inequality.
Consistency of an estimator: An estimator is consistent when it converges in probability to the parameter it estimates as sample size increases. This is the central statistical use of convergence in probability.
Convergence in mean: A sequence converges in mean when the expected absolute distance from its limit tends to zero. Mean convergence implies probability convergence, but probability convergence need not control expected error.
Convergence of a sequence: The property that terms of a sequence approach a specified limit. Convergence in probability adapts ordinary limiting behavior to random variables.
Markov's inequality: For a nonnegative random variable, its probability of exceeding a positive threshold is bounded by its expectation divided by that threshold. It turns vanishing expected error measures into bounds on large-error probabilities.
Law of large numbers: A family of theorems stating that sample averages approach an expected value under specified assumptions. The weak law expresses sample-average convergence in probability.
Uniform integrability: A family of integrable random variables has uniformly controlled expected tails. Together with probability convergence, it can yield convergence in mean.
Probability measure: A function assigning probabilities to events, with total probability one. The definition asks whether events of large error have probabilities tending to zero.
Continuous mapping theorem: A theorem stating that continuous functions preserve convergence in probability when applied to random variables. It lets convergent random variables pass through continuous transformations.

Linked from 16 pages