Probabilistic Convergence

Probabilistic convergence, also called convergence in probability, of a sequence of random variables to a random variable 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.

If converges in probability to a constant, then it also converges in distribution to that constant. The weak law of large numbers is a basic example of convergence in probability.