In the mathematical theory of probability, the Hsu–Robbins–Erdős theorem states that if
X
1
,
…
,
X
n
is a sequence of i.i.d. random variables with zero mean and finite variance and
S
n
=
X
1
+
⋯
+
X
n
,
then
∑
n
⩾
1
P
(

S
n

>
ε
n
)
<
∞
for every
ε
>
0
.
The result was proved by PaoLu Hsu and Herbert Robbins in 1947.
This is an interesting strengthening of the classical strong law of large numbers in the direction of the Borel–Cantelli lemma. The idea of such a result is probably due to Robbins, but the method of proof is vintage Hsu. Hsu and Robbins further conjectured in that the condition of finiteness of the variance of
X
is also a necessary condition for
∑
n
⩾
1
P
(

S
n

>
ε
n
)
<
∞
to hold. Two years later, the famed mathematician Paul Erdős proved the conjecture.
Since then, many authors extended this result in several directions.