Let F be a nonarithmetic probability distribution on
(
0
,
∞
)
(0,\infty )
and suppose
1
−
F
(
t
)
1 - F(t)
is regularly varying at
∞
\infty
with exponent
α
,
0
>
α
≦
1
\alpha ,0 > \alpha \leqq 1
. Let
U
(
t
)
=
Σ
F
n
∗
(
t
)
U(t) = \Sigma {F^{{n^ \ast }}}(t)
be the renewal function. In this paper we first derive various asymptotic expressions for the quantity
U
(
t
+
h
)
−
U
(
t
)
U(t + h) - U(t)
as
t
→
∞
,
h
>
0
t \to \infty ,h > 0
fixed. Next we derive asymptotic relations for the convolution
U
∗
z
(
t
)
,
t
→
∞
{U^ \ast }z(t),t \to \infty
, for a large class of integrable functions z. All of these asymptotic relations are expressed in terms of the truncated mean function
m
(
t
)
=
∫
0
t
[
1
−
F
(
x
)
]
d
x
m(t) = \smallint _0^t[1 - F(x)]dx
, t large, and appear as the natural extension of the classical strong renewal theorem for distributions with finite mean. Finally in the last sections of the paper we apply the special case
α
=
1
\alpha = 1
to derive some limit theorems for the distributions of certain waiting times associated with a renewal process.