Let
V
t
{V_t}
be a regenerative process whose successive generations are not necessarily identically distributed and let A be a measurable set in the range of
V
t
{V_t}
. Let
μ
n
{\mu _n}
be the mean length of the nth generation and
α
n
{\alpha _n}
be the mean time
V
t
{V_t}
is in A during the nth generation. We give conditions ensuring
lim
t
→
∞
prob
{
V
t
∈
A
}
=
α
/
μ
{\lim _{t \to \infty }}\,\operatorname {prob} \{ \,{V_t}\, \in \,A\,\} \, = \,\alpha /\mu
where
lim
n
→
∞
(
1
/
n
)
Σ
j
=
1
n
μ
j
=
μ
\lim \limits _{n \to \infty } (1/n)\Sigma _{j = 1}^n\,{\mu _j}\, = \mu
and
lim
n
→
∞
(
1
/
n
)
Σ
j
=
1
n
α
j
=
α
\lim \limits _{n \to \infty } (1/n)\Sigma _{j = 1}^n\,{\alpha _j}\, = \,\alpha
.