Let
{
X
k
:
k
=
1
,
2
,
…
}
\{ {X_k}:k = 1,2, \ldots \}
be a sequence of random variables with zero mean and finite variance
σ
k
2
\sigma _k^2
. We say that
{
X
k
}
\{ {X_k}\}
is blockwise
m
m
-dependent if for each
p
p
large enough the following is true: if we remove
m
m
or more consecutive
X
X
’s from the dyadic block
{
X
2
p
−
1
+
1
,
…
,
X
2
p
}
\{ {X_{{2^{p - 1}} + 1}}, \ldots ,{X_{{2^p}}}\}
, then the two remaining portions are independent. We say that
{
X
k
}
\{ {X_k}\}
is blockwise quasiorthogonal if for each
p
p
, the expectations
E
(
X
k
X
l
)
E({X_k}{X_l})
are small in a certain sense again within the dyadic block
{
X
2
p
−
1
+
1
,
…
,
X
2
p
}
\{ {X_{{2^{p - 1}} + 1}}, \ldots ,{X_{{2^p}}}\}
. Blockwise independence and blockwise orthogonality are particular cases of the above notions, respectively. We study the a.s. behavior of the series
∑
k
=
1
∞
X
k
\sum \nolimits _{k = 1}^\infty {{X_k}}
and that of the first arithmetic means
(
1
/
n
)
∑
k
=
1
n
X
k
(1/n)\sum \nolimits _{k = 1}^n {{X_k}}
. It turns out that the classical strong limit theorems, with one exception, remain valid in this more general setting, too.