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LONG-TERM BEHAVIOR OF SURVIVAL CHANCES IN SUBCRITICAL GALTON-
WATSON PROCESSES WITH DISCRETE TIME
Misliddin Murtazaev
TMCI-Technology,
Management, Communication Institute in Tashkent.
Senior lecturer of the department.
+998914572979.
misliddin1991@mail.ru
https://doi.org/10.5281/zenodo.14845513
Abstract
. In this paper, Kolmogorov's famous theorem is generalized to the subcritical
case, known from the discrete-time subcritical Galton-Watson branching processes theory. In this
theorem, the Kolmogorov constant for the subcritical state is defined, and its exact form is
determined. The results highlighted several important classical results in the subcritical random
branching processes theory. In particular, using our results, local limit theorems are derived for
subcritical random branching theories.
Keywords
: Generating function, Kolmogorov constant, extinction probability, survival
probability.
Introduction
In this paper, we consider the Galton-Watson (G-W) process. In this case, the age
distribution of the particles is given by a unit-step function. The process evolves discretely in time
as follows. Each particle, independently of the number of particles in the system and its past, will
leave
𝑘 ∈ 𝑁
0
offspring with probability
𝑝
𝑘
> 0
at the end of its life, where
𝑁
0
= {0} ∪ 𝑁
and
𝑁
denotes the set of natural numbers. Let the number of particles at time
𝑛
be denoted by
𝑍(𝑛)
, and
we observe that for this sequence, the following equation holds:
𝑍(𝑛 + 1) = 𝜉
𝑛1
+ 𝜉
𝑛2
+ ⋯ 𝜉
𝑛𝑍(𝑛)
,
where
𝜉
𝑛𝑘
are independent random variables with the common distribution
𝑃{𝜉
11
= 𝑘} =
𝑝
𝑘
, which are interpreted as the number of offspring of the
𝑘
-th particle in the
𝑛
-th generation; [1,
pp. 1-2], [10, p. 19]. To exclude trivial cases, we impose the conditions
𝑝
𝑘
≠ 1,
and
𝑝
0
+ 𝑝
1
< 1
.
The sequence of offspring
{𝑍(𝑛), 𝑛 ∈ 𝑁
0
}
of the defined process generates a discrete state space,
a branching, and a discrete-time homogeneous Markov chain. Its state space is
𝑆
0
= {0} ∪ 𝑆
,
{0}
is an absorbing state, and
𝑆 ⊂ 𝑁
forms the set of connected significant states.
The transition probabilities of this chain for all
𝑖, 𝑗 ∈ 𝑆
0
and
𝑛 ∈ 𝑁
0
are given by:
𝑃
𝑖𝑗
≔ 𝑃{𝑍(𝑛 + 1) = 𝑗|𝑍(𝑛) = 𝑖} =
∑
𝑝
𝑘1
∙ 𝑝
𝑘2
𝑘
1
+𝑘
2
+⋯+𝑘
𝑖
=𝑗
∙ … ∙ 𝑝
𝑘𝑖
(1)
which satisfies the branching property. Conversely, any Markov chain
{𝑝
𝑘
, 𝑘 ∈ 𝑁
0
}
that
satisfies property (1) is a Galton-Watson (G-W) process with a reproductive law. From the above,
it follows that the distribution
{𝑝
𝑘
}
of the law completely determines the G-W process. Now, let
us consider the transition probability from an arbitrary state
𝑖 ∈ 𝑆
to a state
𝑗 ∈ 𝑆
0
in
𝑛 ∈ 𝑁
steps:
𝑃
𝑖𝑗
(𝑛) ≔ 𝑃
𝑖
{𝑍(𝑛) = 𝑗}: = 𝑃{𝑍(𝑛 + 𝑘) = 𝑗|𝑍(𝑘) = 𝑖},
which implies that this probability is independent of the index
𝑘 ∈ 𝑁
0
. For the generating
function, according to the Kolmogorov-Chepmen equation, the following relation holds:
𝐸
𝑖
𝑠
𝑍(𝑛)
≔ ∑
𝑃
𝑖𝑗
(𝑛)𝑠
𝑗
= [𝐹(𝑛; 𝑠)]
𝑖
,
𝑗∈𝑆
0
(2)
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where
𝐹(𝑛; 𝑠) = ∑
𝑝
𝑗
(𝑛)𝑠
𝑗
𝑗∈𝑆
0
is
the
generating
function
with
𝑝
𝑗
(𝑛) ≔
𝑃
1𝑗
(𝑛)
representing the transition probabilities. This function, in turn, is expressed as the
𝑛
-fold
iteration of the generating function:
𝑓(𝑠) ≔ ∑ 𝑝
𝑗
𝑗∈𝑆
0
𝑠
𝑗
which satisfies
𝐹(𝑡; 𝑠) = 𝑓(𝐹(𝑡 − 1; 𝑠))
with
𝐹(1; 𝑠) = 𝑓(𝑠)
, and is given by the
recurrence:
𝐹(𝑛 + 1; 𝑠) = 𝑓(𝐹(𝑛; 𝑠)).
The event
{𝑍(𝑛) = 0}
represents the extinction of the Galton-Watson process at time
𝑛
.
The limit of the probability of the event is
𝑞 = 𝑙𝑖𝑚
𝑛→∞
𝑃{𝑍(𝑛) = 0}
, which is called the
extinction probability of a process that starts with a single particle. This probability is equal to the
smallest positive solution for
𝑠 ∈ [0,1]
of the equation
𝑓(𝑠) = 𝑠
. Furthermore, for any
𝑟 < 1
, the
relation
𝑙𝑖𝑚
𝑛→∞
𝐹(𝑛; 𝑠) = 𝑞
holds uniformly for all
𝑠 ∈ [0,1]
. Let the series
𝑚 ≔ ∑
𝑗𝑝
𝑗
𝑗∈𝑆
converge. Then,
𝑚 = 𝑓′(1−)
, and this value is equal to the expected number of offspring left by
a single particle. According to the relation (2),
𝐸𝑍(𝑛) = 𝑚
𝑛
. Based on this formula, the Galton-
Watson process is classified as subcritical, critical, or supercritical if
𝑚 < 1, 𝑚 = 1
, or
𝑚 > 1
,
respectively. If
𝑚 ≤ 1
, then
𝑞 = 1
, and for
𝑚 > 1
,
𝑞 < 1
is valid.
We recall the result proven by A. Kolmogorov [3] for the probability of continuation
𝑄(𝑛) = 𝑃{𝑍(𝑛) > 0}
of a subcritical branching Galton-Watson process
{𝑍(𝑛), 𝑛 ∈ 𝑁}
.
Kolmogorov's Theorem [3].
For a subcritical branching Galton-Watson process, if the
condition
𝑓
′′
(1 −) < ∞
holds, then the following asymptotic relation is valid
:
𝑄(𝑛) = 𝐾𝑚
𝑛
(1 + 𝑜(1)), 𝑛 → ∞,
(3)
where
𝐾
is a constant, known as the Kolmogorov constant, which depends on the
numerical parameters of the reproductive law
{𝑝
𝑘
, 𝑘 ∈ 𝑆
0
}
According to the formula (3), the probability of survival of the offspring of a single particle
tends to zero. On the other hand, using the total probability formula, it is straightforward to
calculate that
𝐸[𝑍(𝑛)|𝑍(𝑛) > 0] = 1/𝑄(𝑛).
According to our conditions, the denominator of this
relation is positive for all fixed
𝑛 ∈ 𝑁
. Thus, there exists a class of positive trajectories such that,
for the subcritical case as
𝑛 → ∞
,
𝑚
𝑛
𝑄(𝑛)
=
𝐸𝑍(𝑛)
𝑃{𝑍(𝑛)>0}
= 𝐸[𝑍(𝑛)|𝑍(𝑛) > 0] ≈ 1/𝐾
(4)
meaning that the expected number of offspring of a single particle converges to
1/𝐾
as
time progresses along positive trajectories.
According to the relation (4) above, the physical meaning of the constant
𝐾
is that it
represents the equivalence coefficient between the expected value of the population size at the
current time,
𝐸𝑍(𝑛) = 𝑚
𝑛
, and the survival probability of the population,
𝑄(𝑛) = 𝑃{𝑍(𝑛) > 0}
.
Later, in 1967, A. Nagayev and I. Badalbayev [7] showed that the asymptotic formula (3)
also holds for subcritical Galton-Watson processes under the condition weaker than Kolmogorov's,
namely:
𝐸𝑍(1)𝑙𝑛
+
𝑍(1) = ∑
𝑝
𝑘
𝑘𝑙𝑛𝑘 < ∞
𝑘∈𝑆
.
(5)
Results And Discussion
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The fact that the exact form of the Kolmogorov constant has not been found is the reason
why many limit theorems for random processes that follow subcritical branching laws remain
incomplete. This also applies to immigration processes.
Now, we consider the function
𝑅(𝑛; 𝑠) ≔ 𝑞 − 𝐹(𝑛; 𝑠).
For a random variable
𝐻
defined as the extinction moment of the process, where
𝐻 ≔
𝑖𝑛𝑓{𝑛: 𝑍(𝑛) = 0}
, the following equation holds for any
𝑛 ∈ 𝑁
0
𝑃{𝑛 < 𝐻 < ∞|𝑍(𝑛) = 𝑘} = 𝑞
𝑘
.
From this equation, using the full probability formula
𝑝
𝑘
(𝑛) = 𝑃{𝑍(𝑛) = 𝑘}
, we get the
following relationships:
𝑄(𝑛) ≔ 𝑃{𝑛 < 𝐻 < ∞} = ∑ 𝑃{𝑛 < 𝐻 < ∞, 𝑍(𝑛) = 𝑘}
𝑘∈𝑆
= ∑ 𝑃{𝑛 < 𝐻 < ∞|𝑍(𝑛) = 𝑘}𝑝
𝑘
𝑘∈𝑆
(𝑛)
= ∑ 𝑝
𝑘
(𝑛)𝑞
𝑘
𝑘∈𝑆
= ∑ 𝑝
𝑘
(𝑛)𝑞
𝑘
𝑘∈𝑆
0
− 𝑝
1
(𝑛)
= 𝐹(𝑛; 𝑞) − 𝐹(𝑛; 0) = 𝑅(𝑛; 0).
(6)
The obtained quantity expresses the probability that the process will continue at time
𝑛 ∈
𝑁
in the subcritical case. For the supercritical case, it corresponds to the probability of continuation
at time
𝑛 ∈ 𝑁
for trajectories that eventually extinguish. At the same time, the above equation
leads us to the idea of extending the asymptotic formula (3) to the critical case.
In the following theorem, we generalize and prove Kolmogorov's theorem for subcritical
branching processes.
Theorem 1
.
For subcritical branching G-W processes, if condition (5) is satisfied, the
following asymptotic relation holds:
𝑄(𝑛) = 𝐾
𝑞
𝛽
𝑛
(1 + 𝑜(1)), 𝑛 → ∞
, (7)
where
𝛽 = 𝑓′(𝑞)
, and
𝐾
𝑞
is a constant, which we refer to as the generalized Kolmogorov
constant. This constant depends on the probability
𝑞
and the value of the converging series
∑
𝑝
𝑘
𝑘∈𝑆
𝑞
𝑘−1
𝑘𝑙𝑛𝑘
.
Theorem 2.
For subcritical branching G-W processes, if the Kolmogorov condition (9) is
satisfied, the following asymptotic relation holds:
𝑄(𝑛) = 𝐾
𝑞
𝛽
𝑛
(1 + 𝑜(1)), 𝑛 → ∞,
(10)
where the generalized Kolmogorov constant is
𝐾
𝑞
=
𝑞
1 + 𝑞𝛾
with
𝛾 = 𝑏
𝑞
/(𝛽 − 𝛽
2
)
and
2𝑏
𝑞
≔ 𝑓
′′
(𝑞)
. In particular, for the subcritical case, the form
of the Kolmogorov constant in formula (3) is
𝐾 =
1
1 + 𝛾
,
where
𝛾 = 𝑏/(𝑚 − 𝑚
2
)
and
2𝑏 ≔ 𝑓
′′
(1 −).
Conclusion.
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In this paper, Kolmogorov's [3] famous theorem has been generalized and proven for
subcritical Galton-Watson branching processes.
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