Some Limit Theorems for the Critical Galton–Watson Branching Processes

We consider the critical Galton–Watson processes starting from a random number of particles and determine the effect of the mean value of initial state on the asymptotic state of the process. For processes starting from large numbers of particles and satisfying condition (S), we prove the limit theorem similar to the result obtained by W. Feller. We also prove the theorem under the condition W(n) > 0 for the critical processes satisfying the conditions (S) and (M).


Introduction
Suppose that {⇠(k, j), k, j 2 N} is a sequence of independent identically distributed random variables taking nonnegative integer values.Assume that the random variable ⇠(1, 1) has the following distribution: with a generating function and p 0 + p 1 6 = 1.Consider a process W (k), k ≥ 0, defined by the following recurrence relation: where ⌘ is a random variable, which takes positive integer values and is independent on the sequence of random variables {⇠(k, j), k, j 2 N}.
In the present paper, we consider only critical processes.We denote the Galton-Watson process generated by the ith particle in the initial state by W i (n), n = 0, 1, . . . .Clearly, W i (n), n = 0, 1, . . ., i ≥ 1, form independent and identically distributed Galton-Watson branching processes.It is known [1] that W (n) can be represented as follows: (1. 2) The independence of random variables ⌘ and ⇠(i, j), i ≥ 1, j ≥ 1, implies the independence of W i (n) and the random variable ⌘.Let P (n) be the probability of degeneration of the process {W (k), k ≥ 0} at the nth step, i.e., P (n) = P (W (n) = 0).By R(n) we denote the probability of continuation of the process W 1 (n) in the nth step, i.e., R(n In what follows, we need the following notation: The case where the process {W (k), k ≥ 0} starts with one particle (⌘ = 1) was studied by numerous authors.Thus, in 1938, Kolmogorov [2] obtained the following famous result for the probability of continuation R(n) of the critical Galton-Watson process: In 1947, Yaglom [3] studied the conditional distribution of the variable W (n) (given W (n) > 0) and obtained the following result: where it was required that F 000 (1) < 1.The presented results (1.3), (1.4) were later obtained by Spitzer, Kesten, and Ney [4] under the condition F 00 (1) < 1.In [5], Zolotarev established similar results for branching processes with continuous parameters.In 1968, Slack [6] considered the case of where L(x) is a slowly varying function in a neighborhood of zero, and obtained the following result: This result yields the result obtained by Yaglom (1.4) for ↵ ⌘ 1 and F 00 (1) < 1.It should be noted that, in the case considered by Slack, the equality F 00 (1) = 1 can be satisfied.
In [8], K. Mitov, G. Mitov, and N. M. Yanev considered the critical case (F 0 (1) = 1) in which the second factorial moment is finite, i.e., F 00 (1) = σ 2 < 1, and the generating function of the number of particles in the initial state satisfies the condition where L 0 (x) is a function slowly varying at infinity.They obtained the following results: ) With the help of Tauber's theorem, it is not difficult to show that the condition (M) implies that the average number of particles in the initial state is infinite.However, it follows from (1.7) that, in this case, the critical Galton-Watson process also degenerates with probability 1.
In 2007, Nagaev and Wachtel [9] considered the case where ↵ = 0 in the condition (S), i.e., and obtained the following results: where Thus, an analog of the Yaglom theorem is formulated for all critical processes satisfying the condition (S) for ↵ 2 [0, 1].It should be noted that, for ↵ = 0, not the distribution of the process itself but the distribution of the process obtained after substitution converges to an exponential distribution.
All results presented above were obtained for distributions satisfying the condition W (n) > 0.
In 1951, Feller [7] studied the critical Galton-Watson process starting with a large number of particles and satisfying the condition F 00 (1) = σ 2 < 1, i.e., he considered the case where the equality holds for process (1.1), where x is a parameter and obtained the following result without the condition W (n) > 0: In the present paper, we consider the critical Galton-Watson processes starting from a random number of particles and determine the effect of the mean value of the initial state on the asymptotic state of the process.We prove the limit theorem that generalizes Feller's result for the processes starting from a large number of particles and satisfying the condition (S).Actually, we prove the limit theorem for the critical processes W (n) satisfying the conditions (S) and (M) under the condition W (n) > 0.

Main Results
Suppose that a given critical Galton-Watson process is defined by relation (1.1).The following theorem shows the influence of the average number of particles in the initial state on the asymptotics of survival probability of the process.

Theorem 2.2. If the condition (S) holds, then
In the case ⌘ = 1, the equality A = 1 holds and, in this case, Theorem 2.2 turns into the Slack theorem.
The following theorem determines the asymptotic distribution of the critical Galton-Watson process, which initially has many particles, on the average, and the law of particle multiplication satisfies the condition (S).
Theorem 2.3.If the condition (S) is satisfied and, for the initial state W (0), the condition Theorem 2.4.If the conditions (M) and (S) are satisfied, then In the case where F 00 (1) < 1, Theorem 2.4 implies the result obtained by Mitov, Mitov, and Yanev.If we formally set ✓ = 1 and ↵ = 1 in the last Laplace substitution, then we get the Laplace substitution (1 + λ) −1 of the exponential distribution.

Proofs of the Main Results
Proof of Theorem 2.1.It is not difficult to see that It is clear that, according to (3.1), Since h 00 (1) < 1,by virtue of the Taylor formula, where ✓ s is such that s  ✓ s  1.Since h is a generating function, both this function and its derivatives monotonically increase.Therefore, Further, replacing s in (3.3) with F n (0) and taking into account (3.2), we obtain This yields Thus, taking into account the fact that F n (0) ! 1 as n ! 1 and the relations , and (1.5), we arrive at the following relation: Theorem 2.1 is proved.
Proof of Theorem 2.2.It is clear that, according to the total probability formula, we have whence it follows that The asymptotics of P (W (n) = 0) in the last relation is known according to Theorem 2.1.We now determine the asymptotics of E(exp{−λ(1 − H n (0))W (n)}).In view of the fact that the variables W i (n) are independent, identically distributed, and independent of the random variable ⌘, by using relation (1.2), we obtain: According to the total probability formula, we have Further, applying Theorem 2.1 and relation (1.5), we get By using this inequality, the relation valid for the critical process, and (3.10), we obtain Further, by virtue of (3.11) and relations (1.5) and (1.6), it follows from equality (3.9) that By virtue of (3.8) and (3.12), we get Further, according to the asymptotic relations, we find Proof of Theorem 2.3.In view of the independence and identical distributions of the variables W i (n), by using (1.2), we obtain Further, we determine the asymptotic behavior of By virtue of the total probability formula, we obtain If we use results (1.5) and (1.6) in the last equation, then we get It follows from (3.15) and (3.16) that the following equality is true: .
If we now pass to the limit in the last equation as n ! 1, then we get the statement of Theorem 2.3.

Proof of Theorem 2.4. We have
According to the notation, this yields Thus, .17) In the last relation, if we replace s with e −λ(1−Fn(0)) , where λ > 0, then we get According to the condition (M) for the function h, we obtain By virtue of (1.6), we find In this case, according to (3.20), for any number " > 0, there exists a number N such that, for any n ≥ N, In this case, by virtue of Lemma 1 in [6], we get as n ! 1.The statement of Theorem 2.4 follows from the last relation, (3.18), (3.19), and (3.20).
Proof of Theorem 2.5.We first prove the following lemma.
Lemma 3.1.Let ⇠ n , n = 1, 2, . . ., be a sequence of nonnegative random variables and let V (x) be a continuous, increasing, and slowly varying function.Also let G(x) be the function inverse to V (x).If there exist a continuous function '(x) and a sequence of numbers a n > 0 such that a n ! 1 as n ! 1 for all x > 0 and then, for all x > 0, Proof.The proof follows the same scheme as the proof of Lemma 1 in [9].Let " > 0 be an arbitrary fixed number.It is not difficult to show that Hence, passing to the limit as " !0, in view of the continuity of (x), we get the assertion of the lemma.We now prove the theorem.By virtue of (M) and (3.17), we obtain (3.27) We set where G(x) is the function inverse to V (x) and Then, as shown in [9], (3.28)By Lemma 1 in [6], we obtain Hence, it follows from Lemma 3.1 and (3.21) that Theorem 2.5 is proved.