Probabilistic Convergence

Probabilistic convergence, also called convergence in probability, of a sequence of real-valued random variables to a random variable , all defined on a common probability space, means that, for every ,

The notation is commonly used. If converges almost surely, then it converges in probability, and convergence in probability implies distributional convergence. Neither converse holds without additional hypotheses.

For a constant , distributional convergence to also implies convergence in probability to (Massachusetts Institute of Technology 2018, p. 12). The weak law of large numbers is a basic example of convergence in probability.

A sequence converges in probability to iff every subsequence has a further subsequence converging almost surely to (Massachusetts Institute of Technology 2018, p. 4). Convergence in probability is the special case of convergence in measure in which the underlying measure is a probability measure.