Let
X
,
X
1
,
X
2
,
…
X,{X_1},{X_2}, \ldots
be i.i.d. random variables. It is shown that
E
|
X
|
log
+
log
+
|
X
|
>
∞
E\left | X \right |{\log ^ + }{\log ^ + }\left | X \right | > \infty
is a sufficient condition for Riemann
R
1
{R_1}
-summability of
{
X
n
}
\left \{ {{X_n}} \right \}
to
E
X
EX
. Counterexamples are provided which indicate that the strongest possible necessary condition of moment type is
E
|
X
|
>
∞
E\left | X \right | > \infty
. However under weak regularity conditions on the tails of the distribution of
X
X
the sufficient condition is also shown to be necessary.