1 Introduction
The basic version of the Galton–Watson process (GW-process) was conceived as a stochastic model of the population growth or extinction of a single species of individuals [3, 7]. The GW-process
{
Z
n
}
n
≥
0
unfolds in the discrete time setting, with
Z
n
standing for the population size at the generation n under the assumption that each individual is replaced by a random number of offspring. It is assumed that the offspring numbers are independent random variables having the same distribution
{
p
(
j
)
}
j
≥
0
.
By allowing the offspring number distribution
{
p
n
(
j
)
}
j
≥
0
to depend on the generation number n, we arrive at the GW-process in a varying environment [4].
This more flexible model is fully described by a sequence of probability generating functions
f
n
(
s
)
=
∑
j
≥
0
p
n
(
j
)
s
j
,
0
≤
s
≤
1
,
n
≥
1
.
Introduce the composition of generating functions
F
n
(
s
)
=
f
1
∘
…
∘
f
n
(
s
)
,
0
≤
s
≤
1
,
n
≥
1
.
Given that the GW-process starts at time zero with a single individual,
we get
E
(
s
Z
n
)
=
F
n
(
s
)
,
P
(
Z
n
=
0
)
=
F
n
(
0
)
.
The state 0 of the GW-process is absorbing and the extinction probability for the modeled population is determined by
q
=
lim
F
n
(
0
)
(here and throughout, all limits are taken as
n
→
∞
, unless otherwise specified).
In the case of proper reproduction laws with
f
n
(
1
)
=
1
for all
n
≥
1
, we get
E
(
Z
n
)
=
F
n
′
(
1
)
=
f
1
′
(
1
)
⋯
f
n
′
(
1
)
,
E
(
Z
n
|
Z
n
>
0
)
=
F
n
′
(
1
)
1
-
F
n
(
0
)
.
In [5], the usual ternary classification of the GW-processes into supercritical, critical, and subcritical processes [1], was adapted to the framework of the varying environment. Given
0
<
f
n
′
(
1
)
<
∞
for all n, it was shown that under a regularity condition (A) in [5], it makes sense to
distinguish among four classes of the GW-processes in a varying environment: supercritical, asymptotically degenerate, critical, and subcritical processes. In a more recent paper [10] devoted to the Markov theta-branching processes in a varying environment, the quaternary classification of [5] was further refined into a quinary classification, which can be adapted to the discrete time setting as follows:
supercritical case:
q
<
1
and
lim
E
(
Z
n
)
=
∞
,
asymptotically degenerate case:
q
<
1
and
lim inf
E
(
Z
n
)
<
∞
,
critical case:
q
=
1
and
lim
E
(
Z
n
|
Z
n
>
0
)
=
∞
,
strictly subcritical case:
q
=
1
and a finite
lim
E
(
Z
n
|
Z
n
>
0
)
exists,
loosely subcritical case:
q
=
1
and
lim
E
(
Z
n
|
Z
n
>
0
)
does not exist.
Our paper is build upon the properties of a special parametric family of generating functions [9] leading to what will be called here the Galton–Watson theta-processes or
GW
θ
-processes. The remarkable property of the
GW
θ
-processes in a varying environment is that the generating functions
F
n
(
s
)
have explicit expressions presented in Section 2. An important feature of the
GW
θ
-processes is that they allow for defective reproduction laws. If the generating function
f
i
(
s
)
is defective, in that
f
i
(
1
)
<
1
, then
F
n
(
1
)
<
1
for all
n
≥
i
. In the defective case [6, 11], a single individual, with probability
1
-
f
i
(
1
)
may force the entire GW-process to visit to an ancillary absorbing state Δ by the observation time n with probability
P
(
Z
n
=
Δ
)
=
1
-
F
n
(
1
)
.
In Sections 3 and 4, we state ten limit theorems for the
GW
θ
-processes in a varying environment. These results are illuminated in Section 5 by ten examples describing different growth and extinction patterns under environmental variation.
The proofs are collected in Section 6.
2 Proper and Defective Reproduction Laws
Definition 1.
Consider a sequence
(
θ
,
r
,
a
n
,
c
n
)
n
≥
1
satisfying one of the following sets of conditions:
θ
∈
(
0
,
1
]
,
r
=
1
, and for
n
≥
1
,
0
<
a
n
<
∞
,
c
n
>
0
,
c
n
≥
1
-
a
n
,
θ
∈
(
0
,
1
]
,
r
>
1
, and for
n
≥
1
,
0
<
a
n
<
1
,
(
1
-
a
n
)
r
-
θ
≤
c
n
≤
(
1
-
a
n
)
(
r
-
1
)
-
θ
,
θ
∈
(
-
1
,
0
)
,
r
=
1
, and for
n
≥
1
,
0
<
a
n
<
1
,
0
<
c
n
≤
1
-
a
n
,
θ
∈
(
-
1
,
0
)
,
r
>
1
, and for
n
≥
1
,
0
<
a
n
<
1
,
(
1
-
a
n
)
(
r
-
1
)
-
θ
≤
c
n
≤
(
1
-
a
n
)
r
-
θ
,
θ
=
0
,
r
=
1
, and for
n
≥
1
,
0
<
a
n
<
1
,
0
≤
c
n
<
1
,
θ
=
0
,
r
>
1
, and for
n
≥
1
,
0
<
a
n
<
1
,
0
≤
c
n
≤
1
.
A
GW
θ
-process with parameters
(
θ
,
r
,
a
n
,
c
n
)
n
≥
1
is a GW-process in a varying environment characterized by a sequence of probability generating functions
(
f
n
(
s
)
)
n
≥
1
defined by
(2.1)
f
n
(
s
)
=
r
-
(
a
n
(
r
-
s
)
-
θ
+
c
n
)
-
1
θ
,
0
≤
s
<
r
,
f
n
(
r
)
=
r
,
for
θ
≠
0
, and for
θ
=
0
, defined by
(2.2)
f
n
(
s
)
=
r
-
(
r
-
c
n
)
1
-
a
n
(
r
-
s
)
a
n
,
0
≤
s
≤
r
.
Definition 1 is motivated by [9, Definitions 14.1 and 14.2], which also mentions a trivial case of
θ
=
-
1
not included here. Observe that in the setting of varying environment, the key parameters
θ
∈
(
-
1
,
1
]
and
r
≥
1
stay constant over time, while the parameters
(
a
n
,
c
n
)
may vary. The case
θ
=
r
=
1
is the well studied case of the linear-fractional reproduction law.
This section contains two key lemmas.
Lemma 1 gives the explicit expressions for the generating functions
F
n
(
s
)
in terms of positive constants
A
n
,
C
n
,
D
n
=
D
n
(
r
)
defined by
A
0
=
1
,
A
n
=
∏
i
=
1
n
a
i
,
C
n
=
∑
i
=
1
n
A
i
-
1
c
i
,
D
n
=
∏
i
=
1
n
(
r
-
c
i
)
A
i
-
1
-
A
i
.
Lemmas 2 presents the asymptotic properties of the constants
A
n
,
C
n
,
D
n
leading to the limit theorems stated in Sections 3 and 4.
Lemma 1.
Consider a
GW
θ
-process with parameters
(
θ
,
r
,
a
n
,
c
n
)
.
If
θ
≠
0
, then
F
n
(
s
)
=
r
-
(
A
n
(
r
-
s
)
-
θ
+
C
n
)
-
1
θ
,
0
≤
s
<
r
,
F
n
(
r
)
=
r
,
n
≥
1
,
and if
θ
=
0
, then
F
n
(
s
)
=
r
-
(
r
-
s
)
A
n
D
n
,
0
≤
s
≤
r
,
n
≥
1
.
Here:
for
θ
∈
(
0
,
1
]
,
r
=
1
,
0
<
A
n
<
∞
,
C
n
>
0
,
C
n
≥
1
-
A
n
,
F
n
(
1
)
=
1
,
F
n
′
(
1
)
=
A
n
-
1
θ
,
n
≥
1
,
for
θ
∈
(
0
,
1
]
,
r
>
1
,
0
<
A
n
<
1
,
(
1
-
A
n
)
r
-
θ
≤
C
n
≤
(
1
-
A
n
)
(
r
-
1
)
-
θ
,
F
n
(
1
)
≤
1
,
n
≥
1
,
with
F
n
(
1
)
=
1
if and only if
c
k
=
(
1
-
a
k
)
(
r
-
1
)
-
θ
,
1
≤
k
≤
n
, implying
F
n
′
(
1
)
=
A
n
,
for
θ
∈
(
-
1
,
0
)
,
r
=
1
,
0
<
A
n
<
∞
,
0
<
C
n
≤
1
-
A
n
,
F
n
(
1
)
=
1
-
C
n
-
1
θ
,
n
≥
1
,
for
θ
∈
(
-
1
,
0
)
,
r
>
1
,
0
<
A
n
<
1
,
(
1
-
A
n
)
(
r
-
1
)
-
θ
≤
C
n
≤
(
1
-
A
n
)
r
-
θ
,
F
n
(
1
)
≤
1
,
n
≥
1
,
with
F
n
(
1
)
=
1
if and only if
c
k
=
(
1
-
a
k
)
(
r
-
1
)
-
θ
,
1
≤
k
≤
n
, implying
F
n
′
(
1
)
=
A
n
,
for
θ
=
0
,
r
=
1
,
0
<
A
n
<
1
,
0
<
D
n
≤
1
,
F
n
(
1
)
=
1
,
F
n
′
(
1
)
=
∞
,
n
≥
1
,
for
θ
=
0
,
r
>
1
,
0
<
A
n
<
1
,
(
r
-
1
)
1
-
A
n
≤
D
n
≤
r
1
-
A
n
,
F
n
(
1
)
≤
1
,
n
≥
1
,
with
F
n
(
1
)
=
1
if and only if
c
k
=
1
,
1
≤
k
≤
n
, implying
F
n
′
(
1
)
=
A
n
.
Lemma 2.
Denote the limits
A
=
lim
A
n
,
C
=
lim
C
n
,
D
=
lim
D
n
, whenever they exist, whether finite or infinite.
If
θ
∈
(
0
,
1
]
,
r
=
1
, then
C
∈
[
1
,
∞
]
, and if
C
<
∞
, then
A
∈
[
0
,
∞
]
.
If
θ
∈
(
0
,
1
]
,
r
>
1
, then
A
∈
[
0
,
1
)
and
(
1
-
A
)
r
-
θ
≤
C
≤
(
1
-
A
)
(
r
-
1
)
-
θ
.
If
θ
∈
(
-
1
,
0
)
,
r
=
1
, then
A
∈
[
0
,
1
)
and
0
<
C
≤
1
-
A
.
If
θ
∈
(
-
1
,
0
)
,
r
>
1
, then
A
∈
[
0
,
1
)
and
(
1
-
A
)
(
r
-
1
)
-
θ
≤
C
≤
(
1
-
A
)
r
-
θ
.
If
θ
=
0
,
r
=
1
, then
A
∈
[
0
,
1
)
and
D
=
∏
n
≥
1
(
1
-
c
n
)
A
n
-
1
-
A
n
with
D
∈
[
0
,
1
]
.
If
θ
=
0
,
r
>
1
, then
A
∈
[
0
,
1
)
and
D
=
∏
n
≥
1
(
r
-
c
n
)
A
n
-
1
-
A
n
with
(
r
-
1
)
1
-
A
≤
D
≤
r
1
-
A
.
3 Limit Theorems for the Proper
GW
θ
-Processes
Theorems 1–5 deal with the
GW
θ
-process in the case
θ
∈
(
0
,
1
]
,
r
=
1
, when
by Lemma 1,
E
(
Z
n
)
=
A
n
-
1
θ
,
P
(
Z
n
>
0
)
=
(
A
n
+
C
n
)
-
1
θ
.
Putting
B
n
=
C
n
A
n
, we obtain
E
(
Z
n
|
Z
n
>
0
)
=
(
1
+
B
n
)
1
θ
.
These five theorems fully cover the five regimes of reproduction in a varying environment and could be summarized as follows. Let
θ
∈
(
0
,
1
]
,
r
=
1
,
given
C
<
∞
, the
GW
θ
-process is
supercritical if
A
n
→
0
, see Theorem 1,
asymptotically degenerate if
A
n
→
A
∈
(
0
,
∞
)
, see Theorem 2,
strictly subcritical if
A
n
→
∞
, see Theorem 4,
given
C
=
∞
, the
GW
θ
-process is
critical if
B
n
→
∞
, see Theorem 3,
strictly subcritical if
B
n
→
B
∈
[
0
,
∞
)
, see Theorem 4,
loosely subcritical if the
lim
B
n
does not exist, see Theorem 5.
This section also includes Theorem 6 addressing the proper case
θ
=
0
,
r
=
1
. Notice that Theorem 6 deals with the case of infinite mean values, when the above mentioned quinary classification does not apply.
Theorem 1.
Let
θ
∈
(
0
,
1
]
,
r
=
1
, and
C
<
∞
. If
A
n
→
0
, then
q
=
1
-
C
-
1
θ
and
A
n
1
θ
Z
n
almost surely converges to a random variable W such that
E
(
e
-
λ
W
)
=
1
-
(
λ
-
θ
+
C
)
-
1
θ
,
λ
≥
0
.
Theorem 2.
Let
θ
∈
(
0
,
1
]
,
r
=
1
, and
C
<
∞
. If
A
n
→
A
∈
(
0
,
∞
)
, then
q
=
1
-
(
A
+
C
)
-
1
θ
,
E
(
Z
n
)
→
A
-
1
θ
,
and
Z
n
almost surely converges to a random variable
Z
∞
such that
E
(
Z
∞
)
=
A
-
1
θ
,
E
(
s
Z
∞
)
=
1
-
(
A
(
1
-
s
)
-
θ
+
C
)
-
1
θ
,
0
≤
s
≤
1
.
Theorem 3.
Let
θ
∈
(
0
,
1
]
,
r
=
1
, and
C
=
∞
. If
B
n
→
∞
, then
q
=
1
,
P
(
Z
n
>
0
)
∼
C
n
-
1
θ
,
E
(
Z
n
|
Z
n
>
0
)
∼
B
n
1
θ
,
and with
λ
n
=
λ
B
n
-
1
θ
,
E
(
e
-
λ
n
Z
n
|
Z
n
>
0
)
→
1
-
(
1
+
λ
-
θ
)
-
1
θ
,
λ
≥
0
.
Theorem 4.
Let
θ
∈
(
0
,
1
]
and
r
=
1
. If
A
n
→
∞
and
B
n
→
B
∈
[
0
,
∞
)
, then
q
=
1
,
P
(
Z
n
>
0
)
∼
(
1
+
B
)
-
1
θ
A
n
-
1
θ
,
E
(
Z
n
|
Z
n
>
0
)
→
(
1
+
B
)
1
θ
,
and
E
(
s
Z
n
|
Z
n
>
0
)
→
1
-
(
(
1
+
B
)
(
1
-
s
)
-
θ
+
B
+
B
2
)
-
1
θ
,
0
≤
s
≤
1
.
Theorem 5.
Let
θ
∈
(
0
,
1
]
,
r
=
1
, and assume that
lim
B
n
does not exist. Then
q
=
1
and letting
B
k
n
→
B
∈
[
0
,
∞
]
along a subsequence
k
n
→
∞
, we get:
if
B
=
∞
, then
P
(
Z
k
n
>
0
)
∼
C
k
n
-
1
θ
,
E
(
Z
k
n
|
Z
k
n
>
0
)
∼
B
k
n
1
θ
,
and with
λ
n
=
λ
B
n
-
1
θ
,
E
(
e
-
λ
k
n
Z
k
n
|
Z
k
n
>
0
)
→
1
-
(
1
+
λ
-
θ
)
-
1
θ
,
λ
≥
0
,
if
B
∈
[
0
,
∞
)
, then
A
k
n
→
∞
,
P
(
Z
k
n
>
0
)
∼
(
1
+
B
)
-
1
θ
A
k
n
-
1
θ
,
E
(
Z
k
n
|
Z
k
n
>
0
)
→
(
1
+
B
)
1
θ
,
and
E
(
s
Z
k
n
|
Z
k
n
>
0
)
→
1
-
(
(
1
+
B
)
(
1
-
s
)
-
θ
+
B
+
B
2
)
-
1
θ
,
0
≤
s
≤
1
.
Theorem 6.
Suppose
θ
=
0
and
r
=
1
. Then
P
(
Z
n
>
0
)
=
D
n
, so that
q
=
1
-
D
, with D given by Lemma 2 (e).
Furthermore:
if
A
=
0
and
D
=
0
, then
q
=
1
and
P
(
A
n
ln
Z
n
≤
x
|
Z
n
>
0
)
→
1
-
e
-
x
,
x
≥
0
,
if
A
=
0
and
D
>
0
, then
q
<
1
and
P
(
A
n
ln
Z
n
≤
x
)
→
1
-
e
-
x
D
,
x
≥
0
,
if
A
∈
(
0
,
1
)
and
D
=
0
, then
q
=
1
and
E
(
s
Z
n
|
Z
n
>
0
)
→
1
-
(
1
-
s
)
A
,
0
≤
s
≤
1
,
if
A
∈
(
0
,
1
)
and
D
>
0
, then
q
<
1
and
Z
n
almost surely converges to a random variable
Z
∞
such that
E
(
Z
∞
)
=
∞
,
E
(
s
Z
∞
)
=
1
-
(
1
-
s
)
A
D
,
0
≤
s
≤
1
.
Remarks
We make the following observations.
It is a straightforward exercise to check that the above mentioned regularity condition (A) in [5] is valid for the
GW
θ
-process in the case
θ
∈
(
0
,
1
]
,
r
=
1
.
The limiting distribution obtained in Theorem 3 coincides with that of [12] obtained for the critical GW-processes in a constant environment with a possibly infinite variance for the offspring number.
Statement (ii) of Theorem 6 is of the Darling–Seneta-type limit theorem obtained in [2] for GW-processes with infinite mean.
Part (iv) of Theorem 6 presents the pattern of limit behavior similar to the asymptotically degenerate regime in the case of infinite mean values. The conditions of Theorem 6 (iv) hold if and only if
(3.1)
∑
n
≥
1
(
1
-
a
n
)
<
∞
and
(3.2)
∑
n
≥
1
(
1
-
a
n
)
ln
1
1
-
c
n
<
∞
.
4 Limit Theorems for the Defective
GW
θ
-Process
In the defective case, there are two kinds of absorption times:
τ
0
the absorption time of the
GW
θ
-process at 0,
τ
Δ
the absorption time of the
GW
θ
-process at the state Δ.
Let
τ
=
min
(
τ
0
,
τ
Δ
)
be the absorption time of the
GW
θ
-process either at 0 or at the state Δ. Let us
recall that
q
=
P
(
τ
0
<
∞
)
and denote
q
Δ
=
P
(
τ
Δ
<
∞
)
,
Q
=
P
(
τ
<
∞
)
=
q
+
q
Δ
.
Clearly,
P
(
τ
≤
n
)
=
P
(
τ
0
≤
n
)
+
P
(
τ
Δ
≤
n
)
=
F
n
(
0
)
+
1
-
F
n
(
1
)
,
implying
P
(
τ
>
n
)
=
F
n
(
1
)
-
F
n
(
0
)
.
Furthermore,
E
(
Z
n
;
τ
Δ
>
n
)
=
F
n
′
(
1
)
,
E
(
s
Z
n
;
τ
Δ
>
n
)
=
F
n
(
s
)
,
0
≤
s
≤
1
,
so that
E
(
Z
n
|
τ
>
n
)
=
F
n
′
(
1
)
F
n
(
1
)
-
F
n
(
0
)
,
E
(
s
Z
n
|
τ
>
n
)
=
F
n
(
s
)
-
F
n
(
0
)
F
n
(
1
)
-
F
n
(
0
)
,
0
≤
s
≤
1
.
Theorems 7–10 present the transparent asymptotical results on these absorption probabilities and the limit behavior of the
GW
θ
-process in the four defective cases.
Corollaries of Theorems 7–9 deal with the proper sub-cases, where
τ
=
τ
0
. All three corollaries describe a strictly subcritical case, when
A
=
0
, and an asymptotically degenerate case, when
A
∈
(
0
,
1
)
.
Theorem 7.
Consider the case
θ
∈
(
0
,
1
]
,
r
>
1
. Then
q
=
r
-
(
A
r
-
θ
+
C
)
-
1
θ
,
q
Δ
=
1
-
r
+
(
A
(
r
-
1
)
-
θ
+
C
)
-
1
θ
,
where
A
∈
[
0
,
1
)
and
(
1
-
A
)
r
-
θ
≤
C
≤
(
1
-
A
)
(
r
-
1
)
-
θ
.
If
A
=
0
, then
q
=
1
-
q
Δ
=
r
-
C
-
1
θ
∈
[
0
,
1
]
,
so that
Q
=
1
.
Furthermore,
A
n
-
1
P
(
τ
>
n
)
→
(
(
r
-
1
)
-
θ
-
r
-
θ
)
θ
-
1
C
-
1
θ
-
1
,
E
(
Z
n
|
τ
>
n
)
→
(
r
-
1
)
-
θ
-
1
(
r
-
1
)
-
θ
-
r
-
θ
,
E
(
s
Z
n
|
τ
>
n
)
→
(
r
-
s
)
-
θ
-
r
-
θ
(
r
-
1
)
-
θ
-
r
-
θ
,
0
≤
s
≤
1
.
If
A
∈
(
0
,
1
)
, then
Q
∈
[
0
,
1
)
,
E
(
Z
n
;
τ
Δ
>
n
)
→
A
(
A
+
C
(
r
-
1
)
θ
)
-
1
θ
-
1
,
and
Z
n
almost surely converges to a random variable
Z
∞
taking values in the set
{
Δ
,
0
,
1
,
2
,
…
}
, with
P
(
Z
∞
=
Δ
)
=
1
-
r
+
(
A
(
r
-
1
)
-
θ
+
C
)
-
1
θ
,
E
(
s
Z
∞
;
Z
∞
≠
Δ
)
=
r
-
(
A
(
r
-
s
)
-
θ
+
C
)
-
1
θ
,
0
≤
s
≤
1
.
Corollary.
Consider the case
θ
∈
(
0
,
1
]
,
r
>
1
assuming
(4.1)
c
n
=
(
1
-
a
n
)
(
r
-
1
)
-
θ
,
n
≥
1
,
so that
C
=
(
1
-
A
)
(
r
-
1
)
-
θ
implying
q
Δ
=
0
.
If
A
=
0
, then
q
=
1
with
A
n
-
1
P
(
Z
n
>
0
)
→
(
(
r
-
1
)
-
θ
-
r
-
θ
)
θ
-
1
(
r
-
1
)
θ
+
1
.
Furthermore,
E
(
Z
n
|
Z
n
>
0
)
→
(
r
-
1
)
-
θ
-
1
θ
(
(
r
-
1
)
-
θ
-
r
-
θ
)
,
E
(
s
Z
n
|
Z
n
>
0
)
→
(
r
-
s
)
-
θ
-
r
-
θ
(
r
-
1
)
-
θ
-
r
-
θ
,
0
≤
s
≤
1
.
If
A
∈
(
0
,
1
)
, then
q
=
1
-
r
+
(
A
r
-
θ
+
C
)
-
1
θ
,
E
(
Z
n
)
→
A
,
so that
q
∈
(
0
,
1
)
,
and
Z
n
almost surely converges to a random variable
Z
∞
such that
E
(
Z
∞
)
=
A
,
E
(
s
Z
∞
)
=
r
-
(
A
(
r
-
s
)
-
θ
+
(
1
-
A
)
(
r
-
1
)
-
θ
)
-
1
θ
,
0
≤
s
≤
1
.
Theorem 8.
Consider the case
θ
∈
(
-
1
,
0
)
,
r
>
1
and put
α
=
-
1
θ
, so that
α
>
1
. Then
q
=
r
-
(
A
r
1
α
+
C
)
α
,
q
Δ
=
1
-
r
+
(
A
(
r
-
1
)
1
α
+
C
)
α
,
where
A
∈
[
0
,
1
)
and
(
1
-
A
)
(
r
-
1
)
1
α
≤
C
≤
(
1
-
A
)
r
1
α
.
If
A
=
0
, then
q
=
1
-
q
Δ
=
r
-
C
α
∈
[
0
,
1
]
,
so that
Q
=
1
.
Furthermore,
A
n
-
1
P
(
τ
>
n
)
→
α
C
α
-
1
(
r
1
α
-
(
r
-
1
)
1
α
)
,
E
(
Z
n
|
τ
>
n
)
→
(
r
-
1
)
1
α
-
1
r
1
α
-
(
r
-
1
)
1
α
,
E
(
s
Z
n
|
τ
>
n
)
→
r
1
α
-
(
r
-
s
)
1
α
r
1
α
-
(
r
-
1
)
1
α
,
0
≤
s
≤
1
.
If
A
∈
(
0
,
1
)
, then
Q
∈
[
0
,
1
)
,
E
(
Z
n
;
τ
Δ
>
n
)
→
A
(
A
+
C
(
r
-
1
)
-
1
α
)
α
-
1
,
and
Z
n
almost surely converges to a random variable
Z
∞
taking values in the set
{
Δ
,
0
,
1
,
2
,
…
}
, with
P
(
Z
∞
=
Δ
)
=
1
-
r
+
(
A
(
r
-
1
)
1
α
+
C
)
α
,
E
(
s
Z
∞
;
Z
∞
≠
Δ
)
=
r
-
(
A
(
r
-
s
)
1
α
+
C
)
α
,
0
≤
s
≤
1
.
Corollary.
Consider the case
θ
∈
(
-
1
,
0
)
,
r
>
1
assuming (4.1), so that
C
=
(
1
-
A
)
(
r
-
1
)
1
α
implying
q
Δ
=
0
.
If
A
=
0
, then
q
=
1
with
A
n
-
1
P
(
Z
n
>
0
)
→
α
(
r
-
1
)
1
-
1
α
(
r
1
α
-
(
r
-
1
)
1
α
)
.
Furthermore,
E
(
Z
n
|
Z
n
>
0
)
→
(
r
-
1
)
-
θ
-
1
θ
(
(
r
-
1
)
-
θ
-
r
-
θ
)
,
E
(
s
Z
n
|
Z
n
>
0
)
→
(
r
-
s
)
-
θ
-
r
-
θ
(
r
-
1
)
-
θ
-
r
-
θ
,
0
≤
s
≤
1
.
If
A
∈
(
0
,
1
)
, then
q
=
1
-
r
+
(
A
r
1
α
+
(
1
-
A
)
(
r
-
1
)
1
α
)
α
,
E
(
Z
n
)
→
A
,
so that
q
∈
(
0
,
1
)
, and
Z
n
almost surely converges to a random variable
Z
∞
such that
E
(
Z
∞
)
=
A
,
E
(
s
Z
∞
)
=
r
-
(
A
(
r
-
s
)
1
α
+
(
1
-
A
)
(
r
-
1
)
1
α
)
α
,
0
≤
s
≤
1
.
Theorem 9.
Consider the case
θ
=
0
,
r
>
1
implying
q
=
r
-
r
A
D
,
q
Δ
=
1
-
r
+
(
r
-
1
)
A
D
,
Q
=
1
-
(
r
A
-
(
r
-
1
)
A
)
D
,
where D is given by Lemma 2 (f).
If
A
=
0
, then
Q
=
1
,
and
P
(
τ
>
n
)
∼
(
ln
r
-
ln
(
r
-
1
)
)
A
n
D
n
.
Moreover,
E
(
Z
n
|
τ
>
n
)
→
(
r
-
1
)
-
1
ln
r
-
ln
(
r
-
1
)
,
P
(
s
Z
n
|
τ
>
n
)
→
ln
r
-
ln
(
r
-
s
)
ln
r
-
ln
(
r
-
1
)
,
0
≤
s
≤
1
.
If
A
∈
(
0
,
1
)
, then
Q
<
1
,
(
r
-
1
)
1
-
A
≤
D
≤
r
1
-
A
,
E
(
Z
n
;
τ
Δ
>
n
)
→
A
(
r
-
1
)
A
-
1
D
,
and
Z
n
almost surely converges to a random variable
Z
∞
taking values in the set
{
Δ
,
0
,
1
,
2
,
…
}
, with
P
(
Z
∞
=
Δ
)
=
1
-
r
+
(
r
-
1
)
A
D
,
E
(
s
Z
∞
;
Z
∞
≠
Δ
)
=
r
-
(
r
-
s
)
A
D
,
0
≤
s
≤
1
.
Corollary.
Given
θ
=
0
,
r
>
1
, assume
c
n
≡
1
. Then
D
=
(
r
-
1
)
1
-
A
implying
q
Δ
=
0
.
If
A
=
0
, then
q
=
1
,
and
P
(
Z
n
>
0
)
∼
(
ln
r
-
ln
(
r
-
1
)
)
A
n
D
n
.
Moreover,
E
(
Z
n
|
Z
n
>
0
)
→
(
r
-
1
)
-
1
ln
r
-
ln
(
r
-
1
)
,
P
(
s
Z
n
|
Z
n
>
0
)
→
ln
r
-
ln
(
r
-
s
)
ln
r
-
ln
(
r
-
1
)
,
0
≤
s
≤
1
.
If
A
∈
(
0
,
1
)
, then
q
=
r
-
r
A
(
r
-
1
)
1
-
A
,
E
(
Z
n
)
→
A
,
so that
q
∈
(
0
,
1
)
, and
Z
n
almost surely converges to a proper random variable
Z
∞
, such that
E
(
Z
∞
)
=
A
,
E
(
s
Z
∞
)
=
r
-
(
r
-
s
)
A
(
r
-
1
)
1
-
A
,
0
≤
s
≤
1
.
Theorem 10.
In the case
θ
∈
(
-
1
,
0
)
,
r
=
1
, put
α
=
-
1
θ
, so that
α
>
1
. Then
q
=
1
-
(
A
+
C
)
α
,
q
Δ
=
C
α
,
Q
=
1
-
(
A
+
C
)
α
+
C
α
,
where
A
∈
[
0
,
1
)
and
0
<
C
≤
1
-
A
.
If
A
=
0
, then
q
=
1
-
q
Δ
=
1
-
C
α
,
Q
=
1
,
and
A
n
-
1
P
(
τ
>
n
)
→
α
C
α
-
1
.
Moreover,
E
(
s
Z
n
|
τ
>
n
)
→
1
-
(
1
-
s
)
1
α
,
0
≤
s
≤
1
.
If
A
∈
(
0
,
1
)
, then
Q
<
1
,
E
(
Z
n
;
τ
Δ
>
n
)
=
∞
,
and
Z
n
almost surely converges to a random variable
Z
∞
taking values in the set
{
Δ
,
0
,
1
,
2
,
…
}
, with
P
(
Z
∞
=
Δ
)
=
C
α
,
E
(
s
Z
∞
;
Z
∞
≠
Δ
)
=
1
-
(
A
(
1
-
s
)
1
α
+
C
)
α
,
0
≤
s
≤
1
.
Remarks
We make the following observations.
Theorem 7 (ii) should be compared to the more general [6, Theorem 1], which allows the limit
Z
∞
to take the value
∞
with a positive probability.
The convergence results for the conditional expectation should be compared to the statements of [6, Theorems 3 and 4].
The conditional convergence in distribution stated in Theorem 7 (i) should be compared to [11, Theorem 2a (
k
=
0
)] in the more general setting under the assumption of constant environment.
5 Examples
The following ten examples illustrate each of the ten theorems of this paper. Observe that given
(5.1)
c
n
=
(
1
-
a
n
)
σ
,
n
≥
1
,
for some suitable positive constant σ, we get
C
n
=
(
1
-
A
n
)
σ
,
n
≥
1
. Similarly, if
(5.2)
c
n
=
(
a
n
-
1
)
σ
,
n
≥
1
,
for some suitable positive constant σ, then
C
n
=
(
A
n
-
1
)
σ
,
n
≥
1
.
Example 1.
Suppose
θ
∈
(
0
,
1
]
,
r
=
1
, and
(5.3)
a
n
=
n
n
+
1
,
A
n
=
1
n
+
1
,
n
≥
1
.
If (5.1) holds for some
σ
≥
1
, then
by Theorem 1,
q
=
1
-
σ
-
1
θ
,
n
-
1
θ
E
(
Z
n
)
→
1
,
and
n
-
1
θ
Z
n
→
W
almost surely, with
E
(
e
-
λ
W
)
=
1
-
(
λ
-
θ
+
σ
)
-
1
θ
,
λ
≥
0
.
Example 2.
Suppose
θ
∈
(
0
,
1
]
,
r
=
1
, and
(5.4)
a
n
=
n
(
n
+
3
)
(
n
+
1
)
(
n
+
2
)
,
A
n
=
n
+
3
3
(
n
+
1
)
,
n
≥
1
.
If (5.1) holds for some
σ
≥
1
, then
by Theorem 2,
q
=
1
-
(
3
1
+
2
σ
)
1
θ
,
E
(
Z
n
)
→
3
1
θ
,
and
Z
n
→
Z
∞
almost surely, with
E
(
Z
∞
)
=
3
1
θ
,
E
(
s
Z
∞
)
=
1
-
3
1
θ
(
2
σ
+
(
1
-
s
)
-
θ
)
-
1
θ
,
0
≤
s
≤
1
.
Example 3.
Suppose
θ
∈
(
0
,
1
]
and
r
=
1
. Let
a
1
=
1
2
,
a
2
n
=
4
,
a
2
n
+
1
=
1
4
,
c
2
n
-
1
=
1
,
c
2
n
=
2
,
A
2
n
-
1
=
1
2
,
A
2
n
=
2
,
n
≥
1
.
Then
C
=
∞
and
B
n
→
∞
implying the conditions of Theorem 3. Observe that for this example,
lim
A
n
does not exist.
Example 4.
Suppose
θ
∈
(
0
,
1
]
and
r
=
1
. Recall that Theorem 4 is the only one among Theorems 1–5 which may hold both with
C
<
∞
and
C
=
∞
. For this reason, we present two examples (1) and (2) for each of these two situations:
Let
a
n
=
n
+
1
n
,
c
n
=
1
n
2
(
n
+
1
)
,
n
≥
1
,
implying
A
n
=
n
+
1
,
C
n
=
n
n
+
1
,
B
n
=
n
(
n
+
1
)
2
,
n
≥
1
.
In this case, according to Theorem 4,
P
(
Z
n
>
0
)
∼
n
-
1
θ
,
E
(
Z
n
|
Z
n
>
0
)
→
1
,
and
E
(
s
Z
n
|
Z
n
>
0
)
→
s
,
0
≤
s
≤
1
.
Let
a
n
=
n
+
1
n
,
A
n
=
n
+
1
,
n
≥
1
,
and (5.2) hold for some
σ
>
0
. Then
C
n
=
σ
n
,
B
n
=
σ
n
n
+
1
,
n
≥
1
.
In this case, according to Theorem 4,
P
(
Z
n
>
0
)
∼
(
1
+
σ
)
-
1
θ
n
-
1
θ
,
E
(
Z
n
|
Z
n
>
0
)
→
(
1
+
σ
)
1
θ
,
and
E
(
s
Z
n
|
Z
n
>
0
)
→
1
-
(
(
1
+
σ
)
(
1
-
s
)
-
θ
+
σ
+
σ
2
)
-
1
θ
,
0
≤
s
≤
1
.
Example 5.
Suppose
θ
∈
(
0
,
1
]
and
r
=
1
.
Let
a
n
=
{
n
for
n
=
2
k
-
1
,
k
≥
1
,
1
n
-
1
for
n
=
2
k
,
k
≥
1
,
1
otherwise
,
A
n
=
{
n
for
n
=
2
k
-
1
,
k
≥
1
,
1
otherwise
.
Taking
c
n
=
{
1
for
n
=
2
k
,
k
>
1
,
1
n
2
otherwise
,
we get
C
n
=
∑
k
:
2
≤
2
k
≤
n
(
2
k
-
1
-
2
-
2
k
)
+
∑
k
=
1
n
k
-
2
,
n
≥
1
,
implying
C
k
n
∼
2
n
+
1
,
provided
2
n
-
1
≤
k
n
<
2
n
+
1
-
1
.
Thus, by Theorem 5, for
k
n
=
2
n
,
λ
n
=
λ
(
2
n
)
-
1
θ
,
P
(
Z
k
n
>
0
)
∼
(
2
k
n
)
-
1
θ
,
E
(
e
-
λ
k
n
Z
k
n
|
Z
k
n
>
0
)
→
1
-
(
1
+
λ
-
θ
)
-
1
θ
,
λ
≥
0
,
and on the other hand, for
k
n
=
2
n
-
1
,
P
(
Z
k
n
>
0
)
∼
(
3
k
n
)
-
1
θ
,
E
(
s
Z
k
n
|
Z
k
n
>
0
)
→
1
-
(
3
(
1
-
s
)
-
θ
+
6
)
-
1
θ
,
0
≤
s
≤
1
.
Example 6.
Suppose
θ
=
0
,
r
=
1
, and assume
c
n
=
1
-
e
-
n
σ
,
-
∞
<
σ
<
∞
,
n
≥
1
,
yielding
D
n
=
exp
(
-
∑
i
=
1
n
i
σ
(
A
i
-
1
-
A
i
)
)
,
n
≥
1
.
Notice that (5.3) implies
A
=
0
and
D
n
=
exp
(
-
∑
i
=
1
n
i
σ
-
1
i
+
1
)
,
n
≥
1
,
on the other hand,
(5.4) implies
A
=
1
3
and
D
n
=
exp
(
-
∑
i
=
1
n
2
i
σ
-
1
3
(
i
+
1
)
)
,
n
≥
1
.
If (5.3) holds and
σ
≥
1
, then
A
n
∼
n
-
1
,
D
n
=
exp
(
-
∑
i
=
1
n
i
σ
-
1
i
+
1
)
→
0
,
so that the conditions of Theorem 6 (i) are satisfied.
If (5.3) holds and
σ
<
1
, then
A
n
∼
n
-
1
,
D
=
exp
(
-
∑
i
=
1
∞
i
σ
-
1
i
+
1
)
,
so that the conditions of Theorem 6 (ii) are satisfied.
If (5.4) holds and
σ
≥
1
, then
A
=
1
3
,
D
n
=
exp
(
-
∑
i
=
1
n
2
i
σ
-
1
3
(
i
+
1
)
)
→
0
,
so that the conditions of Theorem 6 (iii) are satisfied.
If (5.4) holds and
σ
<
1
, then
A
=
1
3
,
D
=
exp
(
-
∑
i
=
1
∞
2
i
σ
-
1
3
(
i
+
1
)
)
,
so that the conditions of Theorem 6 (iv) are satisfied.
Example 7.
Suppose
θ
∈
(
0
,
1
]
,
r
>
1
assuming (5.1) with
r
-
θ
≤
σ
≤
(
r
-
1
)
-
θ
.
If (5.3),
then the conditions of Theorem 7 (i) hold with
A
n
∼
n
-
1
and
C
=
σ
.
If (5.4),
then
the conditions of Theorem 7 (ii) hold with
A
=
1
3
and
C
=
2
σ
3
.
Example 8.
Suppose
θ
∈
(
-
1
,
0
)
,
r
>
1
assuming (5.1) with
r
-
1
≤
σ
α
≤
r
,
where
α
=
-
1
θ
.
If (5.3),
then the conditions of Theorem 8 (i) hold with
A
n
∼
n
-
1
and
C
=
σ
.
If (5.4),
then
the conditions of Theorem 8 (ii) hold with
A
=
1
3
and
C
=
2
σ
3
.
Example 9.
Suppose
θ
=
0
and
r
>
1
and assume
c
n
=
σ
,
0
≤
σ
≤
1
,
n
≥
1
,
which implies
D
n
=
(
r
-
σ
)
1
-
A
n
,
n
≥
1
.
If (5.3), then by Theorem 9 (i), we get in particular,
P
(
τ
>
n
)
∼
γ
n
-
1
,
γ
=
(
r
-
σ
)
ln
r
r
-
1
.
If (5.3), then by Theorem 9 (ii), we get in particular,
q
=
r
-
r
1
3
(
r
-
σ
)
2
3
,
q
Δ
=
1
-
r
+
(
r
-
1
)
1
3
(
r
-
σ
)
2
3
,
Q
=
1
-
(
r
1
3
-
(
r
-
1
)
1
3
)
(
r
-
σ
)
2
3
.
Example 10.
Suppose
θ
∈
(
-
1
,
0
)
,
r
=
1
. Put
α
=
-
1
θ
and assume (5.1) with
0
<
σ
≤
1
.
If (5.3),
then by Theorem 10 (i), we get in particular,
q
Δ
=
σ
α
and
P
(
τ
>
n
)
∼
α
σ
α
-
1
n
-
1
.
If (5.4), then by Theorem 10 (ii), we get in particular,
Q
=
1
-
(
1
3
+
2
σ
3
)
α
+
2
σ
3
α
.
6 Proofs
In this section we sketch the proofs of lemmas and theorems of this paper.
The corollaries to Theorems 7–9 are easily obtained from the corresponding theorems.
Proof of Lemma 1
Relations (2.1) and (2.2) imply respectively
(
r
-
f
k
∘
f
k
+
1
(
s
)
)
-
θ
=
a
k
(
r
-
f
k
+
1
(
s
)
)
-
θ
+
c
k
=
a
k
a
k
+
1
(
r
-
s
)
-
θ
+
c
k
+
a
k
c
k
+
1
,
and
r
-
f
k
∘
f
k
+
1
(
s
)
=
(
r
-
c
k
)
1
-
a
k
(
r
-
f
k
+
1
(
s
)
)
a
k
=
(
r
-
c
k
)
1
-
a
k
(
r
-
c
k
+
1
)
(
1
-
a
k
+
1
)
a
k
(
r
-
s
)
a
k
a
k
+
1
,
entailing the main claims of Lemma 1. Parts (a)–(f) follow from the respective restrictions (a)–(f) on
(
a
n
,
c
n
)
stated in Definition 1.
Proof of Lemma 2
(a) In the case
θ
∈
(
0
,
1
]
,
r
=
1
, the claim follows from the existence of
lim
C
n
and
lim
(
A
n
+
C
n
)
, which in turn, follows from monotonicity of the two sequences. To see that
A
n
+
C
n
≤
A
n
+
1
+
C
n
+
1
, it suffices to observe that
A
n
-
A
n
+
1
=
A
n
(
1
-
a
n
+
1
)
≤
A
n
c
n
+
1
=
C
n
+
1
-
C
n
.
The second part of Lemma 2 is a direct implication of the definition of
C
n
.
(b)–(f) The rest of the stated results follows immediately from the restrictions (b)–(f) imposed on
(
a
n
,
c
n
)
in Definition 1.
Church–Lindvall Condition for the
GW
θ
-Process
In [8] it was shown for the GW-processes in a varying environment that the almost surely convergence
Z
n
→
a.s.
Z
∞
holds with
P
(
0
<
Z
∞
<
∞
)
>
0
if and only if the following condition holds:
(6.1)
∑
n
≥
1
(
1
-
p
n
(
1
)
)
<
∞
.
Relation (6.1) is equivalent to
(6.2)
∏
n
≥
n
0
p
n
(
1
)
>
0
for some
n
0
≥
1
.
For the
GW
θ
-process, the equality
p
n
(
1
)
=
f
n
′
(
0
)
implies
(6.3)
p
n
(
1
)
=
a
n
(
a
n
+
c
n
r
θ
)
-
1
θ
-
1
for
θ
≠
0
, and for
θ
=
0
,
(6.4)
p
n
(
1
)
=
a
n
(
1
-
c
n
r
-
1
)
1
-
a
n
.
Lemma 3.
In the case
θ
∈
(
0
,
1
]
and
r
=
1
, relation (6.1) holds if and only if
(6.5)
A
n
→
A
∈
(
0
,
∞
)
and
(6.6)
∑
n
≥
1
c
n
<
∞
.
Proof.
In view of (6.3), we have
∏
i
=
1
n
p
i
(
1
)
=
A
n
G
n
-
1
θ
-
1
,
G
n
:=
∏
i
=
1
n
(
a
i
+
c
i
)
.
Since
a
n
+
c
n
≥
1
, we have
lim
G
n
=
G
∈
[
1
,
∞
]
.
If
G
=
∞
, then (6.2) is not valid, implying that (6.1) is equivalent to (6.5) plus
G
<
∞
. It remains to verify that under (6.5), the inequality
G
<
∞
is equivalent to (6.6). Suppose (6.5) holds, and observe that in this case,
G
<
∞
is equivalent to
∏
n
≥
1
(
1
+
c
n
a
n
)
<
∞
,
which is true if and only if
∑
n
≥
1
c
n
a
n
<
∞
.
Since under (6.5),
a
n
→
1
, the latter condition is equivalent to (6.6).
∎
Lemma 4.
In the case
θ
=
0
and
r
=
1
, relation (6.1) holds if and only if
A
∈
(
0
,
1
)
and
D
∈
(
0
,
1
)
.
Proof.
In view of (6.4), we have
∏
n
≥
1
p
n
(
1
)
=
A
∏
n
≥
1
(
1
-
c
n
)
1
-
a
n
.
It remains to observe that given
A
∈
(
0
,
1
)
the relation
D
∈
(
0
,
1
)
is equivalent to
∏
n
≥
1
(
1
-
c
n
)
1
-
a
n
>
0
.
∎
Lemma 5.
Assume that
θ
≠
0
and
r
>
1
, and consider
{
Z
~
n
}
, a GW-process in a varying environment with the proper probability generating functions
f
~
n
(
s
)
=
f
n
(
s
)
f
n
(
1
)
=
r
-
(
a
n
(
r
-
s
)
-
θ
+
c
n
)
-
1
θ
r
-
(
a
n
(
r
-
1
)
-
θ
+
c
n
)
-
1
θ
.
Relation (3.1) implies
∑
n
=
1
∞
(
1
-
p
~
n
(
1
)
)
<
∞
.
Proof.
Assume
θ
∈
(
0
,
1
]
and
r
>
1
together with (3.1). Then
A
n
→
A
∈
(
0
,
1
)
,
a
n
→
1
, and
c
n
→
0
.
We have
p
~
n
(
1
)
=
f
~
n
′
(
0
)
=
a
n
h
n
-
1
θ
-
1
k
n
-
1
,
where
h
n
=
a
n
+
c
n
r
θ
,
k
n
=
r
-
(
a
n
(
r
-
1
)
-
θ
+
c
n
)
-
1
θ
are such that
h
n
≥
1
and
k
n
∈
(
0
,
1
]
.
The statement follows from the representation
∏
n
≥
1
p
n
(
1
)
=
A
H
-
1
θ
-
1
K
-
1
,
where
H
=
∏
n
≥
1
h
n
and
K
=
∏
n
≥
1
k
n
.
It is easy to show that (3.1) and
(
1
-
a
n
)
r
-
θ
≤
c
n
≤
(
1
-
a
n
)
(
r
-
1
)
-
θ
yield
∑
n
≥
1
(
h
n
-
1
)
≤
r
θ
∑
n
≥
1
c
n
≤
r
θ
(
r
-
1
)
-
θ
∑
n
≥
1
(
1
-
a
n
)
<
∞
,
implying
H
∈
[
1
,
∞
)
. On the other hand,
K
∈
(
0
,
1
]
, since
∑
n
≥
1
(
1
-
k
n
)
<
∞
,
which follows from
1
-
k
n
≤
(
r
-
1
)
(
a
n
+
c
n
(
r
-
1
)
θ
)
-
1
θ
-
1
)
≤
r
(
1
-
(
a
n
+
c
n
(
r
-
1
)
θ
)
1
θ
)
≤
r
θ
-
1
(
1
-
a
n
)
.
In the other case, when (3.1) holds together with
θ
∈
(
-
1
,
0
)
and
r
>
1
, the lemma is proven similarly.
∎
Proof of Theorems 1–5
The proofs of these theorems are done using the usual for these kind of results arguments applied to the explicit expressions available for
F
n
(
s
)
. In particular, the following standard formula is a starting point for computing the conditional limit distributions:
(6.7)
E
(
s
Z
n
|
Z
n
>
0
)
=
E
(
s
Z
n
)
-
P
(
Z
n
=
0
)
P
(
Z
n
>
0
)
=
1
-
1
-
F
n
(
s
)
1
-
F
n
(
0
)
.
Thus in the case
θ
∈
(
0
,
1
]
and
r
>
1
, Lemma 1 and (6.7) imply
E
(
s
Z
n
|
Z
n
>
0
)
=
1
-
(
(
1
-
s
)
-
θ
+
B
n
)
-
1
θ
(
1
+
B
n
)
-
1
θ
→
1
-
(
(
1
-
s
)
-
θ
+
B
)
-
1
θ
(
1
+
B
)
-
1
θ
,
proving the main statement of Theorem 4. The almost sure convergence stated in Theorem 2 follows from Lemma 3 and the earlier cited criterium of [8].
Proof of Theorem 6
Suppose
θ
=
0
,
r
=
1
, in which case
A
∈
[
0
,
1
)
and
D
∈
[
0
,
1
]
.
(i) Suppose
A
=
D
=
0
. In this case
q
=
1
-
D
=
1
, and by (6.7) and Lemma 1,
E
(
s
Z
n
|
Z
n
>
0
)
=
1
-
(
1
-
s
)
A
n
.
Putting here
s
n
=
exp
(
-
λ
e
-
x
A
n
)
, we get as
n
→
∞
,
E
(
s
n
Z
n
|
Z
n
>
0
)
=
1
-
(
1
-
exp
(
-
λ
e
-
x
A
n
)
)
A
n
=
1
-
exp
(
A
n
ln
(
λ
e
-
x
A
n
(
1
+
o
(
1
)
)
)
→
1
-
e
-
x
.
This implies a convergence in distribution
(
Z
n
e
-
x
A
n
|
Z
n
>
0
)
→
d
W
(
x
)
,
where the limit
W
(
x
)
has a degenerate distribution with
P
(
W
(
x
)
≤
w
)
=
(
1
-
e
-
x
)
1
{
0
≤
w
<
∞
}
.
In other words,
P
(
Z
n
≤
w
e
x
A
n
|
Z
n
>
0
)
→
(
1
-
e
-
x
)
1
{
0
≤
w
<
∞
}
.
After taking the logarithm of
Z
n
, we arrive at the statement of Theorem 6 (i).
(ii) Statement (ii) follows from Lemma 1 and relation (6.7) in a similar way as statement (i).
(iii) If
A
∈
(
0
,
1
)
and
D
=
0
, then
q
=
1
and by relation (6.7) and Lemma 1,
E
(
s
Z
n
|
Z
n
>
0
)
=
1
-
(
1
-
s
)
A
n
→
1
-
(
1
-
s
)
A
.
(iv) Let
A
>
0
and
D
>
0
. Since
q
=
1
-
D
, similarly to part (iii), we obtain
E
(
s
Z
n
)
→
1
-
(
1
-
s
)
A
D
.
By Lemma 4, the convergence in distribution
Z
n
→
d
Z
∞
can be upgraded to the almost surely convergence
Z
n
→
a.s.
Z
∞
.
Proof of Theorems 7 and 8
In this section we prove only Theorem 7. Theorem 8 is proven similarly.
By Lemma 1,
F
n
(
0
)
=
r
-
(
A
n
r
-
θ
+
C
n
)
-
1
θ
,
F
n
(
1
)
=
r
-
(
A
n
(
r
-
1
)
-
θ
+
C
n
)
-
1
θ
.
It follows that
P
(
τ
>
n
)
=
(
A
n
r
-
θ
+
C
n
)
-
1
θ
-
(
A
n
(
r
-
1
)
-
θ
+
C
n
)
-
1
θ
,
E
(
Z
n
|
τ
>
n
)
=
F
n
′
(
1
)
F
n
(
1
)
-
F
n
(
0
)
=
θ
-
1
A
n
(
A
n
+
C
n
(
r
-
1
)
θ
)
-
1
θ
-
1
(
A
n
r
-
θ
+
C
n
)
-
1
θ
-
(
A
n
(
r
-
1
)
-
θ
+
C
n
)
-
1
θ
,
E
(
s
Z
n
|
τ
>
n
)
=
F
n
(
s
)
-
F
n
(
0
)
F
n
(
1
)
-
F
n
(
0
)
=
(
A
n
r
-
θ
+
C
n
)
-
1
θ
-
(
A
n
(
r
-
s
)
-
θ
+
C
n
)
-
1
θ
(
A
n
r
-
θ
+
C
n
)
-
1
θ
-
(
A
n
(
r
-
1
)
-
θ
+
C
n
)
-
1
θ
.
(i) Assume that
A
=
0
. Then the sequence of positive numbers
V
n
=
A
n
-
1
(
C
-
C
n
)
=
c
n
+
1
+
c
n
+
2
a
n
+
1
+
c
n
+
3
a
n
+
2
a
n
+
1
+
⋯
satisfies
r
-
θ
≤
lim inf
V
n
≤
lim sup
V
n
≤
(
r
-
1
)
-
θ
.
For a given
x
∈
(
0
,
∞
)
, put
W
n
(
x
)
=
A
n
-
1
(
C
-
1
θ
-
(
A
n
x
+
C
n
)
-
1
θ
)
.
Since
W
n
(
x
)
=
A
n
-
1
(
C
-
1
θ
-
(
A
n
(
x
-
V
n
)
+
C
)
-
1
θ
)
=
θ
-
1
C
-
1
θ
-
1
(
x
-
V
n
+
o
(
1
)
)
,
the representation
A
n
-
1
P
(
τ
>
n
)
=
W
n
(
(
r
-
1
)
-
θ
)
-
W
n
(
r
-
θ
)
yields the first asymptotic result stated in part (i) of Theorem 7. The other two asymptotic results follow from the representations
E
(
Z
n
|
τ
>
n
)
=
θ
-
1
(
A
n
+
C
n
(
r
-
1
)
θ
)
-
1
θ
-
1
W
n
(
(
r
-
1
)
-
θ
)
-
W
n
(
r
-
θ
)
,
E
(
s
Z
n
|
τ
>
n
)
=
W
n
(
(
r
-
s
)
-
θ
)
-
W
n
(
r
-
θ
)
W
n
(
(
r
-
1
)
-
θ
)
-
W
n
(
r
-
θ
)
.
(ii) The second claim follows from the equality
E
(
s
Z
n
;
τ
Δ
>
n
)
=
r
-
(
A
n
(
r
-
s
)
-
θ
+
C
n
)
-
1
θ
.
Proof of Theorem 9
If
θ
=
0
and
r
>
1
, then by Lemma 1
P
(
Z
n
=
0
)
=
r
-
r
A
n
D
n
,
P
(
Z
n
≠
Δ
)
=
r
-
(
r
-
1
)
A
n
D
n
.
It follows that
q
=
r
-
r
A
D
,
q
Δ
=
1
-
r
+
(
r
-
1
)
A
D
,
Q
=
1
-
(
r
A
-
(
r
-
1
)
A
)
D
.
(i) If
A
=
0
, then clearly
q
=
r
-
D
,
q
Δ
=
1
-
r
+
D
,
Q
=
1
,
and
P
(
τ
>
n
)
=
(
r
A
n
-
(
r
-
1
)
A
n
)
D
n
∼
(
ln
r
-
ln
(
r
-
1
)
)
A
n
D
n
.
Furthermore,
E
(
Z
n
|
τ
>
n
)
=
F
n
′
(
1
)
F
n
(
1
)
-
F
n
(
0
)
=
A
n
(
r
-
1
)
A
n
-
1
r
A
n
-
(
r
-
1
)
A
n
→
(
r
-
1
)
-
1
ln
r
-
ln
(
r
-
1
)
,
P
(
s
Z
n
|
τ
>
n
)
=
F
n
(
s
)
-
F
n
(
0
)
F
n
(
1
)
-
F
n
(
0
)
=
r
A
n
-
(
r
-
s
)
A
n
r
A
n
-
(
r
-
1
)
A
n
→
ln
r
-
ln
(
r
-
s
)
ln
r
-
ln
(
r
-
1
)
.
(ii) In the case
A
>
0
, the main claim is obtained as
E
(
s
Z
n
;
τ
Δ
>
n
)
=
r
-
(
r
-
s
)
A
n
D
n
→
r
-
(
r
-
s
)
A
D
.
Proof of Theorem 10
If
θ
∈
(
-
1
,
0
)
and
r
=
1
, then by Lemmas 1 and 2,
F
n
(
0
)
=
1
-
(
A
n
+
C
n
)
α
,
F
n
(
1
)
=
1
-
C
n
α
and
q
=
1
-
(
A
+
C
)
α
,
q
Δ
=
C
α
,
where
α
=
-
1
θ
and
0
<
C
≤
1
-
A
.
(i) Suppose
A
=
0
. Then the sequence of positive numbers
V
n
=
A
n
-
1
(
C
-
C
n
)
satisfies
0
≤
lim inf
V
n
≤
lim sup
V
n
≤
1
.
For a given
x
∈
(
0
,
∞
)
, put
W
n
(
x
)
=
A
n
-
1
(
(
A
n
x
+
C
n
)
α
-
C
α
)
.
Since
W
n
(
x
)
=
α
C
α
-
1
(
x
-
V
n
+
o
(
1
)
)
,
the representation
A
n
-
1
P
(
τ
>
n
)
=
W
n
(
1
)
-
W
n
(
0
)
yields the first asymptotic result stated in part (i) of Theorem 10. The other asymptotic result follows from the representation
E
(
s
Z
n
|
τ
>
n
)
=
W
n
(
1
)
-
W
n
(
(
1
-
s
)
1
α
)
W
n
(
1
)
-
W
n
(
0
)
.
(ii)
Claim (ii) is derived as
P
(
s
Z
n
;
τ
>
n
)
=
1
-
(
A
n
(
1
-
s
)
1
α
+
C
n
)
α
→
1
-
(
A
(
1
-
s
)
1
α
+
C
)
α
.
and
Yerakhmet Zhumayev
