For certain conservative, ergodic, infinite measure preserving transformations
T
T
we identify increasing functions
A
A
, for which
\[
lim sup
n
→
∞
1
A
(
n
)
∑
k
=
1
n
f
∘
T
k
=
∫
X
f
d
μ
a
.e
.
\limsup \limits _{n \to \infty } \frac {1} {{A(n)}}\sum \limits _{k = 1}^n {f \circ } {T^k} = \int _X {fd\mu } \quad {\text {a}}{\text {.e}}{\text {.}}
\]
holds for any nonnegative integrable function
f
f
. In particular the results apply to some Markov shifts and number-theoretic transformations, and include the other law of the iterated logarithm.