Given an aperiodic dynamical system
(
X
,
T
,
μ
)
(X,T,\mu )
then there is an
f
∈
L
2
(
μ
)
f \in {L^2}(\mu )
with
∫
f
d
μ
=
0
\smallint fd\mu = 0
satisfying the Central Limit Theorem, i.e. if
S
m
f
=
f
+
f
∘
T
+
⋯
+
f
∘
T
m
−
1
{S_m}f = f + f \circ T + \cdots + f \circ {T^{m - 1}}
and
σ
m
=
‖
S
m
f
‖
2
{\sigma _m} = {\left \| {{S_m}f} \right \|_2}
then
\[
μ
{
x
|
S
m
f
(
x
)
σ
m
>
u
}
→
(
2
π
)
−
1
/
2
∫
−
∞
u
exp
[
−
υ
2
2
]
d
υ
.
\mu \left \{ {x|\frac {{{S_m}f(x)}}{{{\sigma _m}}} > u} \right \} \to {(2\pi )^{ - 1/2}}\int _{ - \infty }^u {{\text {exp}}} \left [ {\frac {{ - {\upsilon ^2}}}{2}} \right ]d\upsilon .
\]
The analogous result also holds for flows.