Renewed Limit Theorems for the discrete-time Branching Process and its Conditioned Limiting Law interpretation

Our principal aim is to observe the Markov discrete-time process of population growth with long-living trajectory. First we study asymptotical decay of generating function of Galton-Watson process for all cases as the Basic Lemma. Afterwards we get a Differential analogue of the Basic Lemma. This Lemma plays main role in our discussions throughout the paper. Hereupon we improve and supplement classical results concerning Galton-Watson process. Further we investigate properties of the population process so called Q-process. In particular we obtain a joint limit law of Q-process and its total state. And also we prove the analogue of Law of large numbers and the Central limit theorem for total state of Q-process. Keywords: Branching process; transition function; Q-process; invariant measures; ergodic chain; total states; joint distribution; limit theorem.


Introduction
The Galton-Watson branching process (GWP) is a famous classical model for population growth. Although this process is well-investigated but it seems to be wholesome to deeper discuss and improve some famed facts from classical theory of GWP. In first half part of the paper, Sections 2 and 3, we will develop discrete-time analogues of Theorems from the paper of the author [5]. These results we will exploit in subsequent sections to discuss properties of so-called Q-process as GWP with infinite-living trajectory.
Let a random function Z n denotes the successive population size in the GWP at the moment n ∈ N 0 , where; N 0 = {0} ∪ N and N = {1, 2, . . .}. The state sequence {Z n , n ∈ N 0 } can be expressed in the form of Z n+1 = ξ n1 + ξ n2 + · · · + ξ nZ n , where ξ nk , n, k ∈ N 0 , are independent variables with general offspring law p k := P {ξ 11 = k}. They are interpreted as a number of descendants of k-th individual in n-th generation. Owing to our assumption {Z n , n ∈ N 0 } is a homogeneous Markov chain with state space S ⊂ N 0 and transition functions P i j := P Z n+1 = j Z n = i = ∑ k 1 + ··· +k i = j p k 1 · p k 2 · · · p k i , (1.1) for any i, j ∈ S , where p j = P 1 j and ∑ j∈S p j = 1. And on the contrary, any chain satisfying to property (1.1) represents GWP with the evolution law {p k , k ∈ S }. Thus, our GWP is completely defined by setting the distribution {p k }; see [1, pp.1-2], [9, p.19]. From now on we will assume that p k = 1 and p 0 > 0, p 0 + p 1 < 1.
A probability generating function (GF) and its iterations is important analytical tool in researching of properties of GWP. Let F(s) = ∑ k∈S p k s k , for 0 ≤ s < 1.
Obviously that A := Eξ 11 = F ′ (s ↑ 1) denotes the mean per capita number of offspring provided the series ∑ k∈S kp k is finite. Owing to homogeneous Markovian nature transition functions P i j (n) := P i Z n = j = P Z n+r = j Z r = i , for any r ∈ N 0 satisfy to the Kolmogorov-Chapman equation P i j (n + 1) = ∑ k∈S P ik (n)P k j , for i, j ∈ S .
Hence E i s Z n := ∑ j∈S P i j (n)s j = F n (s) where GF F n (s) = E 1 s Z n is n-fold functional iteration of F(s); see [3, pp.16-17]. Throughout this paper we write E and P instead of E 1 and P 1 respectively. It follows from (1.2) that EZ n = A n . The GWP is classified as sub-critical, critical and supercritical, if A < 1, A = 1 and A > 1, accordingly.
The event {Z n = 0} is a simple absorbing state for any GWP. The limit q = lim n→∞ P 10 (n) denotes the process starting from one individual eventually will be lost and called the extinction probability of GWP. It is the least non-negative root of F(q) = q ≤ 1 and that q = 1 if the process is non-supercritical. Moreover the convergence lim n→∞ F n (s) = q holds uniformly for 0 ≤ s ≤ r < 1. An assertion describing decrease speed of the function R n (s) := q − F n (s), due to its importance, is called the Basic Lemma (in fact this name is usually used for the critical situation).
In Section 2 we follow on intentions of papers [7] and [5] and prove an assertion about asymptote of the function R ′ n (s) as Differential Analogue of Basic Lemma. This simple assertion (and its corollaries, Theorem 1 and 2) will lays on the basis of our reasoning in Section 3.
We start the Section 3 with recalling the Lemma 3 proved in [1, p.15]. Until the Theorem 6 we study ergodic property of transition functions P i j (n) , having carried out the comparative analysis of known results. We discuss a role of µ j = lim n→∞ P 1 j (n) P 11 (n) qua the invariant measures and seek an analytical form of GF M (s) = ∑ j∈S µ j s j and also we discuss R-classification of GWP. Further consider the variable H denoting an extinction time of GWP, that is H = min {n : Z n = 0}. An asymptote of P {H = n} has been studied in [12] and [20]. The event {n < H < ∞} represents a condition of {Z n = 0} at the moment n and {Z n+k = 0} for some k ∈ N. By the extinction theorem P i {H < ∞} = q i . Therefore in non-supercritical case Hence, Z n → 0 with probability one, so in these cases the process will eventually die out. We also consider a conditional distribution in the section. The classical limit theorems state that if q > 0 then under certain moment assumptions the limit P i j (n) := P H (n) i Z n = j exists always; see [1, p.16].
In particular, Seneta [19] has proved that if A = 1 then the set ν j := lim n→∞ P 1 j (n) represents a probability distribution and, limiting GF V (s) = ∑ j∈S ν j s j satisfies to Schroeder equation where β = F ′ (q). The equation (1.3) determines an invariant property of numbers ν j with respect to the transition functions P 1 j (n) and, the set ν j is called Rinvariant measure with parameter R = β −1 ; see [17]. In the critical case we know the Yaglom theorem about a convergence of conditional distribution of 2Z n F ′′ (1)n given that {H > n} to the standard exponential law. In the end of the Section we investigate an ergodic property of probabilities P i j (n) and we refine above mentioned result of Seneta, having explicit form of V (s).
More interesting phenomenon arises if we observe the limit of P k → ∞ and fixed n ∈ N. In Section 4 we observe the conditioned limit lim k→∞ P H (n+k) i Z n = j which represents an honest probability measures Q = Q i j (n) and defines homogeneous Markov chain called the Q-process. Let W n be the state at the moment n ∈ N in Q-Process. Then W 0 d = Z 0 and P i W n = j = Q i j (n). The Q-process was considered first by Lamperti and Ney [15]; see, also [1, pp.56-60]. Some properties of it were discussed by Pakes [17], [18], and in [6], [8]. The considerable part of the paper of Klebaner, Rösler and Sagitov [13] is devoted to discussion of this process from the viewpoint of branching transformation called the Lamperti-Ney transformation. Continuous-time analogue of Q-process was considered by the author [7]. Section 5 is devoted to classification properties of Markov chain W n , n ∈ N . Unlike of GWP the Q-process is classified on two types depending on value of pos- is an invariant measure for Q-process. The section studies properties of the invariant measure.
Sections 6 and 7 are devoted to examine of structure and long-time behaviors of the total state S n = ∑ n−1 k=0 W k in Q-process until time n. First we consider the joint distribution of the cumulative process W n , S n . As a result of calculation we will know that in case of β < 1 the variables W n and S n appear asymptotically not dependent. But in the case β = 1 we state that under certain conditions the normalized cumulative process W n EW n ; S n ES n weakly converges to the two-dimensional random vector having a finite distribution. Comparing results of old researches we note that in case of β = 1 the properties of S n essentially differ from properties of the total progeny of simple GWP. In this connection we refer the reader to [2], [10] and [11] in which an interpretation and properties of total progeny of GWP in various contexts was investigated. In case of β < 1, in accordance with the asymptotic independence property of W n and S n we seek a limiting law of S n separately. So in Section 7 we state and prove an analogue of Law of Large Numbers and the Central Limit Theorem for S n .

Basic Lemma and its Differential analogue
In this section we observe an asymptotic property of the function R n (s) := q − F n (s) and its derivative. In the critical situation an asymptotic explicit expansion of this function is known from the classical literature which is given in the formula (2.10) below.
Let A = 1. First we consider s ∈ [0; q). The mean value theorem gives where ξ n (s) = q − θ R n (s), 0 < θ < 1. We see that ξ n (s) < q. Since the GF and its derivatives are monotonically non-decreasing then consecutive application of (2.1) leads R n (s) < qβ n . Collecting last finding and seeing that β < 1 we write following inequalities: In (2.2) the top index means derivative of a corresponding order. Considering together representation (2.1) and inequalities (2.2) we take relations In turn, by Taylor formula and the iteration for F(s) we have expansion where and throughout this section ξ n (s) is such for which are satisfied relations (2.2). Assertions (2.2)-(2.4) yield: Repeated application of (2.5) leads us to the following: Taking limit as n → ∞ from here we have estimation where We see that last two series converge. Designating we rewrite the relation (2.6) as following: Clearly that So there is a positive δ = δ (s) such that ∆ 1 ≤ δ ≤ ∆ 2 and the limit in (2.7) is equal to Having spent similar reasoning for s ∈ [q; 1) as before, we will be convinced that the limit lim n→∞ β n R n (s) = A (s) holds for all s ∈ [0; 1).
So we can formulate the following Basic Lemma.
where the function A (s) is defined in (2.8); The following lemma is discrete-time analogue of Lemma 2 from [5].
Proof. Concerning the first part of the lemma we have equality Let at first s ∈ [0; q). As the function R n (s) monotonously decreases by s, then its derivative R ′ n (s) < 0 and, hence R ′ n+1 (s) R ′ n (s) > 0. Therefore, taking the logarithm and after, summarizing along n, we transform the equality (2.13) to the form of where L n (s) = 1 − F ′′ ξ n (s) β R n (s).

Using elementary inequalities
for L k (s) (a relevance of the use is easily be checked), we write On the other hand as R n (s) < q · β n , then F n (s) > q · 1 − β n and hence Using this relation in (2.14) we obtain

Hence in our designations
Considering together the estimations (2.18) and (2.19) we conclude The function β n R ′ n (s) is continuous and monotone by s for each n ∈ N 0 . Inequalities (2.20) entail that the functions ln −β n R ′ n (s) converge uniformly for 0 ≤ s ≤ z < q as n → ∞. From here we get (2.11) for 0 ≤ s < q. By similar reasoning we will be convinced that convergence (2.11) is fair for s ∈ [q; 1) and ergo for all values of s, such that 0 ≤ s < 1.
Let's prove now the formula (2.12). The Taylor expansion and iteration of F(s) produce In the left-side part of (2.21) we apply the mean value Theorem and have where s < c(s) < F(s). If we use a derivative's monotonicity property of any GF, a functional iteration of F(s) entails From here, using iteration again we have It follows from relations (2.22), (2.23) and the fact F n (s) ↑ 1, that Designatingh(s) the mid-part of last inequalities leads us to the representation (2.12). Lemma 2 is proved. [5]; see also [7]. Really, it can check up that in the conditions of the Lemma 1,

Remark 1. The function A (s) plays the same role, as the akin function in the Basic Lemma for the continuous-time Markov branching process established in
and also it is asymptotically satisfied to the Schroeder equation: for all 0 ≤ s < 1.
Now due to the Lemma 2 we can calculate the probability of return to an initial state Z 0 = 1 in time n. So since F ′ n (0) = P 11 (n), putting s = 0 in (2.11) and (2.12) we directly obtain the following two local limit theorems.
where the function K (s) is defined in (2.11).

An Ergodic behavior of Transition Functions {P i j (n)} and Invariant Measures
We devote this section to ergodicity property of transition functions P i j (n) . Herewith we will essentially use the Lemma 2 with combining the following ratio limit property (RLP) [1].

Lemma 3 (see [1, p.15]).
If p 1 = 0, then for all i, j ∈ S the RLP holds: we see that a GF analogue of assertion (3.1) is n (s) and M (s) = ∑ j∈S µ j s j . The properties of numbers µ j are of some interest within our purpose. In view of their non-negativity the limiting GF M (s) is monotonously not decreasing by s. And according to the assertion (3.2) in studying of behavior of P i j (n) P 11 (n) is enough to consider function M n (s).
It has been proved in [1, pp.12-14] the sequence µ j satisfies to equation whenever A (s) and K (s) are functions in (2.9) and (2.11) respectively;

6)
and Proof. The convergence property of GF M (s) was proved in [1, p.13].
In our designations we write In case A = 1 it follows from (2.9) that and, considering (2.24) implies Combining (3.7) and (3.8) we obtain M (s) in form of (3.5).
Taking limit from here we find the limiting GF in the form of (3.6). The proof is completed.

Remark 2.
The theorem above is an enhanced form of Theorem 2 from [1, p.13] in sense that in our case we get the information on analytical form of limiting GF M (s).
The following assertions follow from the theorem proved above.
Now from the Lemma 3 and Theorems 1 and 2 we get complete account about asymptotic behaviors of transition functions P i j (n). Following theorems are fair. (1)) , as n → ∞.
2B is finite then for transition functions the following asymptotic representation holds: Further we will discuss the role of the set µ j as invariant measures concerning transition probabilities P i j (n) . An invariant (or stationary) measure of the GWP is a set of nonnegative numbers µ * j satisfying to equation If ∑ j∈S µ * j < ∞ (or without loss of generality ∑ j∈S µ * j = 1) then it is called as invariant distribution. As P 00 (n) = 1 then according to (3.13) µ * 0 = 0 for any invariant measure µ * j . If P 10 (n) = 0 then condition (3.13) becomes µ * j = ∑ j k=1 µ * k P k j (n). If P 10 (n) > 0 then P i0 (n) > 0 and hence µ * j > 0. In virtue of Theorem 4 in non-critical situation the transition functions P i j (n) exponentially decrease to zero as n → ∞. Following a classification of the continuoustime Markov process we characterize this decrease by a "decay parameter" We classify the non-critical Markov chain {Z n , n ∈ N 0 } as R-transient if ∑ n∈N e Rn P ii (n) < ∞ and R-recurrent otherwise. This chain is called as R-positive if lim n→∞ e Rn P ii (n) > 0, and R -null if last limit is equal to zero. Now assertion (3.11) and Theorem 4 yield the following statement. In critical situation the set µ j directly enters to a role of invariant measure for the GWP. Indeed, in this case β = 1 and according to (3.3) the following invariant equation holds: µ j = ∑ k∈S µ k P k j , for all j ∈ S , and owing to (3.12) ∑ j∈S µ j = ∞ . to be the appropriate GF. As it has been noticed in the introduction section that if q > 0, then the limit ν j := lim n→∞ P 1 j (n) always exists. In case of A = 1 the set ν j represents a probability distribution. And limiting GF V (s) = ∑ j∈S ν j s j satisfies to Schroeder's equation (1.3) for 0 ≤ s ≤ 1. But if A = 1 then ν j ≡ 0; see [19] and [1, p.16]. In forthcoming two theorems we observe the limit of P i j (n) as n → ∞ for any i, j ∈ S . Unlike aforementioned results of Seneta we get the explicit expressions for the appropriate GF.

14)
where the function A (s) is defined in (2.8).
Proof. We write In turn P i Z n = j, n < H < ∞ = P n < H < ∞ Z n = j · P i j (n).
Since the vanishing probability of j particles is equal to q j then from last form we receive that P i Z n = j, n < H < ∞ = q j · P i j (n) (3.16) Using relation (3.16) implies Now it follows from (3.15)-(3.17) and Lemma 3 that as n → ∞. It can be verified the limit distribution ν j defines the GF V (s) = M (qs) M (q). Applying here equality (3.5) we get to (3.14).

Remark 4. The mean of distribution measure
, as n → ∞ and, the limit distribution ν j has the finite mean V ′ (s ↑ 1) = q A (0).

Remark 5.
It is a curious fact that in last theorem we managed to be saved of undefined variable p 1 ∈ [p 1 ; 1].
Now define the stochastic process Z n with the transition matrix P i j (n) . It is easy to be convinced that Z n represents a discrete-time Markov chain. According to last theorems the properties of its trajectory lose independence on initial state with growth the numbers of generations.
In non-critical case, according to the Theorem 7, for GWP Z n there is (up to multiplicative constant) unique set of nonnegative numbers ν j which are not all zero and ∑ j∈S ν j = 1. Moreover as M (qs) = M (q) · V (s) then using the formula (3.4) we can establish the following invariant equation: where V (s) = ∑ j∈S ν j s j and F(s) = F(qs) q.
So we have the following Theorem 9. Let A = 1 and F ′′ (q) < ∞. Then where transition functions P i j (n) have an ergodic property and their limits ν j = lim n→∞ P i j (n) present |ln β |-invariant distribution for the Markov chain Z n .
In critical situation we have the following assertion which directly implies from Theorem 8 and taking into account the continuity theorem for GF. So we have an honest probability measure Q = Q i j (n) . The stochastic process {W n , n ∈ N 0 } defined by this measure is called the Q-process. By definition that the Q-process can be considered as GWP with a non-degenerating trajectory in remote future, that is it conditioned on event {H = ∞}. Harris [4] has established that if A = 1 and 2B := F ′′ (1) < ∞ the distribution of Z n Bn conditioned on {H = ∞} has the limiting Erlang's law. Thus the Q-process {W n , n ∈ N 0 } represents a homogeneous Markov chain with initial state W 0 d = Z 0 and general state space which will henceforth denoted as E ⊂ N. The variable W n denote the state size of this chain in instant n with the transition matrix Q i j (n) = P i W n+k = j = jq j−i iβ n P i j (n), for all i, j ∈ E , (4.1) and for any n, k ∈ N . Put into consideration a GF As F n (s) → q owing to (4.2) and (4.3), Q i j (n) Q 1 j (n) → 1, at infinite growth of the number of generations. Using (4.2) and iterating F(s) produce a following functional relation: where F(s) = F(qs) q and Y (s) := Y 1 (s). We see that Q-process is completely defined by GF Y (s) = s F ′ (qs) β and, its evolution is regulated by the positive parameter β . In fact, if the first moment α := Y ′ (1) is finite then differentiating of (4.3) in s = 1 gives where γ := (α − 1) (1 − β ) and α = 1 + F ′′ (1) β > 1.

Classification and ergodic behavior of states of Q-processes
The formula (4.5) shows that if β < 1, then and, provided that β = 1 The Q-Process has the following properties: (I) if β < 1, then it is positive-recurrent; (II) if β = 1, then it is transient.
In the transient case W n → ∞ with probability 1; see [1, p.59]. Let's consider first the positive-recurrent case. In this case according to (2.11), (4.2), (4.3) the limit π(s) := lim n→∞ Y (i) n (s) exists provided that α < ∞. Then owing to (4.4) we make sure that GF π(s) = ∑ j∈E π j s j satisfies to invariant equation π(s)·F(qs) q = Y (s) · π F(qs) q . Applying this equation reduces to where F n (s) = F n (qs) q. A transition function analogue of (5.1) is form of π j = ∑ i∈E π i Q i j (n). Taking limit in (5.1) as n → ∞ it follows that π F n (s) ∼ F n (s) and it in turn entails ∑ j∈E π j = 1 since F n (s) → 1. So in this case the set π j , j ∈ E represents an invariant distribution. Differentiation (5.1) and taking into account (4.5) we easily compute that π ′ (1) = ∑ j∈E jπ j = 1 + γ, (5.2) where as before γ : If we remember the form of function A (s) the last condition becomes For the function δ = δ (s) all cases are disregarded except for the unique case σ = 0 for the following simple reason. All functions having a form of (1 − s) −σ monotonically increase to infinity as s ↑ 1 when 0 < σ < 1 and this fact contradicts the boundedness of function δ = δ (s). In the case σ < 0 cannot be occurred (5.3) since the limit in the left-hand part is equal to zero while γ = 0. In unique case σ = 0 the limit is constant and in view of (5.3) We proved the following theorem.
where π(s) is probability GF having a form of .
The set π j , j ∈ E coefficients in power series expansion of π(s) = ∑ j∈E π j s j are invariant distribution for the Q-process.
In transient case the following theorem hold.
with Y (s) ≤ sh(s) ≤ s. Nonnegative numbers µ j , j ∈ E satisfy to invariant equation Proof. The convergence (5.5) immediately follows as a result of combination of (2.12), (4.2) and (4.3). Taking limit in (4.4) reduces to equation µ(s)F n (s) = Y n (s)µ (F n (s)) which equivalent to (5.6) in the context of transition probabilities. On the other hand it follows from (5.5) that µ (F n (s)) ∼ n 2 F n (s) as n → ∞. Hence ∑ j∈E µ j = ∞ .
Another invariant measure for Q-process are numbers which don't depend on i ∈ E . In fact a similar way as in GWP (see Lemma 3) case it is easy to see that this limit exists. Owing to Kolmogorov-Chapman equation Last equality and (5.11), taking into account that Q i1 (n + 1) Q i1 (n) → 1 gives us an invariant relation υ j = ∑ i∈E υ i Q i j (1).

(5.12)
In GF context the equality (5.12) is equivalent to Schroeder type functional equation where F n (s) = F n (qs) q and U (s) = ∑ j∈E υ j s j with υ 1 = 1.
Hence, considering (5.11), we generalize the statement (5.7): for all i, j ∈ E . By similar way for β = 1 it is discovered that where Q 1 is defined in (5.8).
Providing that Y ′′ (1) < ∞ it can be estimated the convergence speed in Theorem 12. It is proved in [16] and K(s) is some bounded function depending on form of F(s). Since the finiteness of C is equivalent to condition Y ′′ (1) < ∞ then from combination of relations (2.12), (4.2), (4.3) and (5.13) we receive the following theorem for the case β = 1.

Joint distribution law of Q-process and its total state
Consider the Q-process {W n , n ∈ N 0 } with structural parameter β = F ′ (q). Let's define a random variable S n = W 0 +W 1 + · · · +W n−1 , a total state in Q-process until time n. Let J n (s; x) = ∑ j∈E ∑ l∈N P W n = j, S n = l s j x l be the joint GF of W n and S n on a set of

Lemma 4.
For all (s; x) ∈ K and any n ∈ N a recursive equation Proof. Let's consider the cumulative process W n , S n which is evidently a bivariate Markov chain with transition functions P W n+1 = j, S n+1 = l W n = i, S n = k = P i W 1 = j, S 1 = l δ l,i+k , where δ i j is the Kronecker's delta function. Hence we have Using this result and the formula of composite probabilities, we discover that The formula (4.2.) is used in last step. The last equation reduces to (6.1). Now by means of relation (6.1) we can take an explicit expression for GF J n (s; x). In fact, sequentially having applied it, taking into account(4.4) and, after some transformations we have where the sequence of functions {H k (s; x)} is defined for (s; x) ∈ K by following recurrence relations: then provided that α := Y ′ (1) it follows from 6.2) and (6.3) that where as before γ := (α − 1) (1 − β ).

Remark 6. It is known from classical theory that if an evolution law of simple GWP
Z n , n ∈ N 0 is generated by GF F(s) = F(qs) q, then a joint GF of distribution of Z n ,V n , where V n = ∑ n−1 k=0 Z k is the total number of individuals participating until time n, satisfies to the recurrent equation (6.3); see e.g., [14, p.126]. So H n (s; x), (s; x) ∈ K, represents the two-dimensional GF for all n ∈ N and has all properties as E s Z n x V n .
In virtue of the told in Remark 6, in studying of function H k (s; x) we certainly will use properties of GF E s Z n x V n . As well as F ′ (1) = β ≤ 1 and hence the process Z n , n ∈ N 0 is mortal GWP. So there is an integer valued random variable V = lim n→∞ V n -a total number of individuals participating in the process for all time of its evolution. Hence there is a limit and according to (6.3) it satisfied the recurrence relation Provided that the second moment Y ′′ (1) is finite, the following asymptotes for the variances can be found from (6.2) by differentiation: , when β < 1, and as n → ∞. In turn it is matter of computation to verify that Hence letting ρ n denote the correlation coefficient of W n and S n , we have Last statement specifies that in the case β < 1 between the variables W n and S n there is an asymptotic independence property. Contrariwise for the case β = 1 the following "joint theorem" holds, which has been proved in the paper [6].
Theorem 15. Let β = 1 and α = Y ′ (1) < ∞. Then the two-dimensional process W n EW n ; S n ES n weakly converges to the two-dimensional random vector (w; s) having the Laplace transform where chx = e x + e −x 2 and shx = e x − e −x 2.
Supposing λ = 0 in Theorem 15 produces the following limit theorem for S n .
where the limit function F(u) has the Laplace transform Letting θ = 0 from the Theorem 15 we have the following assertion which was proved in the monograph [1, pp.59-60] with applying of the Helly's theorem.
Really, denoting ψ n (λ ) = Ψ n (λ ; 0) we have Here we have used that lim θ ↓0 sh √ θ √ θ = 1. The found Laplace transform corresponds to a distribution of the right-hand side term in (6.6) produced as composition of two exponential laws with an identical density.
7 Asymptotic properties of S n in case of β < 1 In this section we investigate asymptotic properties of distribution of S n in the case β < 1. Consider the GF T n (x) := Ex S n = J n (1; x). Owing to (6.2) it has a form of and F(s) = F(qs) q, h n (x) = Ex V n , V n = ∑ n−1 k=0 Z k . In accordance with (6.3) h n+1 (x) = x F h n (x) . Denoting since |h(x)| ≤ 1 and |h n (s; x)| ≤ 1. Therefore for each n ∈ N and k = 0, 1, . . . , n. Consecutive application of last inequality gives as n → ∞ uniformly for x ∈ K. Further, where the function R n (x) is used, we deal with set K in which this function certainly is not zero. By Taylor expansion and taking into account (7.2), (6.5), we have where |η n (x)| → 0 as n → ∞ uniformly with respect to x ∈ K. Since R n (x) → 0, formula Owing to last equality we transform the formula (7.3) to a form of and, hence where and |ε n (x)| ≤ ε n → 0 as n → ∞ for all x ∈ K. Repeated use of (7.4) leads to the following representation for R n (x): Note that the formula (7.5) was written out in monograph [14, p.130] for the critical case.
The expansions of functions h(x) and u(x) in neighborhood of x = 1 will be useful for our further purpose. Lemma 5. Let β < 1. If b := F ′′ (1) < ∞, then for h(x) = Ex V the following relation holds: Proof. We write down the Taylor expansion as x ↑ 1: In turn by direct differentiation from (6.5) we have which together with (7.7) proves (7.6).
Lemma 9. Let β < 1 and α < ∞. Then the following relation holds: R n e θ u n e θ ∼ as θ → 0 and for each fixed n ∈ N.
Further the following lemma is required.
With the help of the above established lemmas, we state and prove now the analogue of Law of Large Numbers and the Central Limit Theorem for S n . Theorem 16. Let β < 1 and α < ∞. Then Proof. Denoting ψ n (θ ) be the Laplace transform of distribution of S n n it follows from formula (7.1) that ψ n (θ ) = T n (θ n ), where θ n = exp −θ n . The theorem statement is equivalent to that for any fixed θ ∈ R + ψ n (θ ) −→ e −θ (1+γ) , as n → ∞. (7.19) From Lemma 10 follows as n → ∞. The first addendum, owing to (7.10), becomes And the second one, as it is easy to see, has a decrease order of O 1 n 3 . Therefore from (7.20) and (7.21) follows (7.19). The Theorem is proved.
Hence we conclude that ϕ n (θ ) −→ exp − θ 2 2 , as n → ∞, and the theorem statement follows from the continuity theorem for characteristic functions.