Generalizing classical studies of power series with sequences of independent random variables as coefficients, we study series of the forms
\[
g
x
,
ϕ
(
z
)
=
∑
n
=
0
∞
ϕ
(
T
n
x
)
z
n
and
f
x
,
ϕ
(
z
)
=
∑
n
=
1
∞
ϕ
(
x
)
ϕ
(
T
x
)
⋯
ϕ
(
T
n
−
1
x
)
z
n
,
{g_{x,\phi }}(z) = \sum \limits _{n = 0}^\infty {\phi ({T^n}x){z^n}\quad {\text {and}}\quad {f_{x,\phi }}(z) = \sum \limits _{n = 1}^\infty {\phi (x)\phi (Tx) \cdots \phi ({T^{n - 1}}x){z^n},} }
\]
where
T
T
is an ergodic measure-preserving transformation on a probability space
(
X
,
B
,
μ
)
(X,\mathcal {B},\mu )
and
ϕ
\phi
is a measurable complex-valued function which is a.e. nonzero. When
f
x
,
ϕ
{f_{x,\phi }}
is entire, its order of growth at infinity measures the speed of divergence of the ergodic averages of
log
|
ϕ
|
\log |\phi |
. We give examples to show that any order is possible for any
T
T
and that different orders are possible for fixed
ϕ
\phi
. For fixed
T
T
, the set of
ϕ
\phi
which produce infinite order is residual in the subset of
L
1
(
X
)
{L^1}(X)
consisting of those
ϕ
\phi
which are a.e. nonzero and produce entire
f
x
,
ϕ
{f_{x,\phi }}
. As in a theorem of Pólya for gap series, if
f
x
,
ϕ
{f_{x,\phi }}
is entire and has finite order, then it assumes every value infinitely many times. The functions
ϕ
∈
L
1
(
X
)
\phi \in {L^1}(X)
for which
g
x
,
ϕ
{g_{x,\phi }}
is rational a.e. are exactly the finite sums of eigenfunctions of
T
T
; their poles are all simple and are the inverses of the corresponding eigenvalues. By combining this result with a skew product construction, we can also characterize when
f
x
,
ϕ
{f_{x,\phi }}
is rational, provided that
ϕ
\phi
takes one of several particular forms.