Given an increasing sequence of positive integers
{
m
n
}
\left \{ {{m_n}} \right \}
, a non-decreasing sequence of positive integers
{
b
n
}
\left \{ {{b_n}} \right \}
, and a measurable, measure-preserving ergodic transformation
τ
\tau
on a probability space
(
Ω
,
F
,
μ
)
\left ( {\Omega ,\mathcal {F},\mu } \right )
, the a.s. convergence of the moving averages
T
n
(
f
)
=
b
n
−
1
∑
k
=
m
n
+
1
m
n
+
b
n
f
(
τ
k
)
{T_n}\left ( f \right ) = b_n^{ - 1}\sum \nolimits _{k = {m_n} + 1}^{{m_n} + {b_n}} {f\left ( {{\tau ^k}} \right )}
is considered, for
f
∈
L
p
(
Ω
)
f \in {L_p}\left ( \Omega \right )
. A counterexample is constructed in the case of polynomial-like
{
m
n
}
\left \{ {{m_n}} \right \}
.