Proof
We introduce the
notation
To investigate the convergence in
,
we
calculate
The
are uncorrelated, hence, the second term is
zero:
Since the second moments have common
bound
we conclude
that
Hence,
It follows by the proposition
(
Convergence in Lp and in
probability 1
) that
also

It remains to prove the a.s. convergence. According to the formula
(
Chebyshev
inequality
)
Hence, if we restrict our attention to the subindexing
then
Therefore, according to the proposition
(
Borel-Cantelli lemma, part
1
),
Then, according to the proposition
(
IO criteria for AS
convergence
),
It remains to consider the middle terms
for every
.
We introduce the notation
:
for
.
We estimate for every
s.t.
:

Hence, it suffices to prove that
We
calculate
The
are uncorrelated, hence the cross terms
vanish:
Therefore, according to the formula
(
Chebyshev inequality
),
It follows, according to the proposition
(
Borel-Cantelli lemma, part
1
),
Then, according to the proposition
(
IO criteria for AS
convergence
),