Let
{
Z
n
:
n
=
0
,
1
,
2
,
…
}
\{ {Z_n}:n = 0,1,2, \ldots \}
be a Galton-Watson branching process with offspring p.g.f.
f
(
s
)
=
Σ
0
∞
p
j
s
j
f(s) = \Sigma _0^\infty {p_j}{s^j}
. Assume (i)
1
>
m
=
f
′
(
1
−
)
=
Σ
1
∞
j
p
j
>
∞
1 > m = f’(1 - ) = \Sigma _1^\infty j{p_j} > \infty
, (ii)
Σ
1
∞
j
2
p
j
>
∞
\Sigma _1^\infty {j^2}{p_j} > \infty
and (iii)
γ
0
=
f
′
(
q
)
>
0
{\gamma _0} = f’(q) > 0
, where
q
q
is the extinction probability of the process. Let
w
(
x
)
w(x)
denote the density function of
W
W
, the almost sure limit of
Z
n
m
−
n
{Z_n}{m^{ - n}}
with
Z
0
=
1
,
w
(
i
)
(
x
)
{Z_0} = 1,{w^{(i)}}(x)
the
i
i
-fold convolution of
w
(
x
)
,
P
n
(
i
,
j
)
=
P
(
Z
n
=
j
|
Z
0
=
i
)
,
δ
0
=
(
log
γ
0
−
1
)
(
log
m
)
−
1
w(x),{P_n}(i,j) = P({Z_n} = j|{Z_0} = i),{\delta _0} = (\log \gamma _0^{ - 1}){(\log m)^{ - 1}}
and
β
0
=
m
δ
0
/
(
3
+
δ
0
)
{\beta _0} = {m^{{\delta _0}/(3 + {\delta _0})}}
. Then for any
0
>
β
>
β
0
0 > \beta > {\beta _0}
and
i
i
we can find a constant
C
=
C
(
i
,
β
)
C = C(i,\beta )
such that
\[
|
m
n
P
n
(
i
,
j
)
−
w
(
i
)
(
m
−
n
j
)
|
≦
C
[
β
0
−
n
(
m
−
n
j
)
−
1
+
β
−
n
]
|{m^n}{P_n}(i,j) - {w^{(i)}}({m^{ - n}}j)| \leqq C[\beta _0^{ - n}{({m^{ - n}}j)^{ - 1}} + {\beta ^{ - n}}]
\]
for all
j
≧
1
j \geqq 1
. Applications to the boundary theory of the associated space time process are also discussed.