Convergence of Random Variables - Convergence in Mean

Convergence in Mean

We say that the sequence Xn converges in the r-th mean (or in the Lr-norm) towards X, for some r ≥ 1, if r-th absolute moments of Xn and X exist, and

\lim_{n\to\infty} \operatorname{E}\left( |X_n-X|^r \right) = 0,

where the operator E denotes the expected value. Convergence in r-th mean tells us that the expectation of the r-th power of the difference between Xn and X converges to zero.

This type of convergence is often denoted by adding the letter Lr over an arrow indicating convergence:

The most important cases of convergence in r-th mean are:

  • When Xn converges in r-th mean to X for r = 1, we say that Xn converges in mean to X.
  • When Xn converges in r-th mean to X for r = 2, we say that Xn converges in mean square to X. This is also sometimes referred to as convergence in mean, and is sometimes denoted

Convergence in the r-th mean, for r > 0, implies convergence in probability (by Markov's inequality), while if r > s ≥ 1, convergence in r-th mean implies convergence in s-th mean. Hence, convergence in mean square implies convergence in mean.

Read more about this topic:  Convergence Of Random Variables