11.1 Convergence almost surely

A sequence converges almost surely if the probability that the convergence occurs is one.

11.2 Convergence in probability

A sequence converges in probability if the probability of deviating from a limit approaches zero, for any given threshold value.

11.3 Sequence which convergences in probability but not almost surely

Some sequences converge in probability but diverge almost surely.

11.4 Weak law of large numbers

Empirical averages of uncorrelated square-integrable random numbers with equal mean and variance converge in probability to their mathematical expectation.

11.5 Strong law of large numbers

Empirical averages of mutually independent and identically distributed integrable random numbers converge to their mathematical expectation.


Alternative reading material

  • [Jacod & Protter, Chapters 17 and 20]
  • [Williams, Chapter 7]

Last modified: Monday, 15 February 2021, 1:16 PM