In the previous lectures, we have introduced several notions of convergence of
a sequence of random variables (also called
**modes of convergence**). There are several relations among the
various modes of convergence, which are discussed below and are summarized by
the following diagram (an arrow denotes implication in the arrow's
direction):.

Table of contents

If a sequence of random variables converges almost surely to a random variable , then also converges in probability to .

Proof

See, e.g., Resnick (1999).

If a sequence of random variables converges in probability to a random variable , then also converges in distribution to .

Proof

See, for example, Resnick (1999).

If a sequence of random variables converges almost surely to a random variable , then also converges in distribution to .

Proof

This is obtained putting together the previous relations (almost sure convergence implies convergence in probability, which in turn implies convergence in distribution).

If a sequence of random variables converges in mean square to a random variable , then also converges in probability to .

Proof

We can apply Markov inequality to a generic term of the sequence :for any strictly positive real number . Taking the square root of both sides of the left-hand inequality, we obtainTaking limits on both sides, we getwhere we have used the fact that, by the very definition of convergence in mean square,Since, by the definition of probability, it must be thatthen it must be that alsoNote that this holds for any arbitrarily small . By the definition of convergence in probability, this means that converges in probability to (if you are wondering about strict and weak inequalities here and in the definition of convergence in probability, note that implies for any strictly positive ).

If a sequence of random variables converges in mean square to a random variable , then also converges in distribution to .

Proof

This is obtained putting together the previous relations (mean square convergence implies convergence in probability, which in turn implies convergence in distribution).

Resnick, S. I. (1999) A probability path, Birkhauser.

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