(Equivalent sequences of r.v.) The sequences of r.v. are "equivalent" iff

(Property of equivalent sequences of r.v.) If are equivalent then converges a.s. Furthermore if then

It follows from the definition ( Equivalent sequences of r.v. ) and the proposition ( Borel-Cantelli lemma, part 1 ) that We perform the equivalent transformation of the above statement as follows (see the section ( Operations on sets and logical statements )): According to the section ( Operations on sets and logical statements ) we recover the meaning of the above as follows: Such set has the probability 1. Therefore, starting from some all the terms in the series are zero everywhere except a set of measure 0.

(Law of large number for iid r.v. with finite mean) Let be a family of iid r.v. with a finite mean. Then

Let be the common distribution function. We have We introduce the variables The are iid. We have According to the proposition ( Estimate of mean by probability series ), and by the proposition ( Property of equivalent sequences of r.v. ) it suffices to show that But such conclusion follows from the proposition ( Simple law of large numbers ) because the have the same finite second moment.

(Law of large numbers for independent r.v.). Let be a sequence of independent r.v. with d.f. . Let is a sequence, , . Suppose

1. as

2. as .

Then

The idea of the proof is similar to the previous proposition. Introduce the r.v. Then (1) is equivalent to and (2) is equivalent to Also,