Renewed limit theorems for noncritical Galton-Watson branching systems (Q6592159)

This paper discusses the Galton-Watson stochastic branching system. The authors deal only with the noncritical case. Their goal is to improve the recent results on explicitly calculated famous constant in the theory of subcritical Galton-Watson branching systems (Kolmogorov (1938)). Those results include the rate of convergence to the Kolmogorov constant, and the speed of approximation rate in classical limit theorems for Galton-Watson branching systems.\N\NWhen \(Z(n)\) is a population size at the moment \(n \in \mathbb{N}_0\) \(:=\) \(\mathbb{N} \cup \{ 0 \}\) in the Galton-Watson branching (GWB) system with branching rates \(\{ p_k, k \in \mathbb{N}_0 \}\), each individual in the system lives a unit length life time and then gives \(k \in \mathbb{N}_0\) descendants with probability \(p_k\). This is a reducible, homogeneous-discrete-time Markov chain with a state space \(\mathcal{S}_0\) \(=\) \(\mathcal{S} \cup \{ 0 \}\) and \(\mathcal{S} \subset \mathbb{N}\). In this paper the authors assume that \(p_0 > 0\), \(p_0 + p_1 > 0\), \(p_0 + p_1 < 1\) and \(m :=\) \(\sum_{ k \in \mathcal{S} } k \cdot p_k\) \(< \infty\). They are interested in the subcritical and supercritical cases, which are assigned the values \(m < 1\) and \(m > 1\), respectively. Denote \(q\) an extinction probability of the system initiated by the single founder individual. \(f(s)\) \(=\) \(\sum_{ k \in \mathcal{S}_0 } p_k \cdot s^k\) is the offspring generating function (GF). Let \(P_i \{ * \}\) \(:=\) \(\mathbb{P} \{ * \vert Z(0) = i \}\), and put into consideration \(n\)-step transition probabilities \(P_{ij}(n)\) \(=\) \(P_i \{ Z(n) = j \}\). \(f_n(s)\) is the GF of probabilities \(p_k(n)\) \(*=\) \(P_{1 k} (n)\), i.e., \N\[f_n(s) = \sum_{ k \in \mathcal{S}_0 } p_k(n) s^k\tag{1}\N\]\Nis an \(n\)-fold iteration of \(f(s)\). In order to state the main results, it is necessary to introduce the following condition [K]. We assume:\N\NThe Kolmogorov Condition [K]\N\[\Nm \not= 1 \qquad \text{and} \qquad f''(1-) < \infty \qquad \text{for} \quad m < 1. \tag{2}\N\]\NAs additional notation, \N\[\N\mathcal{K}_q = \mathcal{A}_q (0) = \frac{q}{ 1 + q \cdot \gamma_q} \qquad \text{with} \quad \gamma_q := \frac{ f''(q)}{ 2 \beta ( 1 - \beta)}, \qquad \beta = f'(q),\tag{3}\N\]\Nwhere \(\mathcal{K}_q\) is the Kolmogorov constant. Here are the main results.\N\NTheorem 1. The following asymptotic relations hold:\N\N(a) For the system survival probability, \N\[\N\frac{ Q(n)}{ \mathcal{K}_q \beta^n} = 1 - B_2 \mathcal{K}_q \beta^n ( 1 + \rho(n)), \tag{4}\N\]\Nwhere \(Q(n) := R_n(0) = \mathbb{P} \{ n < \mathcal{H} < \infty \}\), with \(R_n(s) = q - f_n(s)\), \N\[\N\mathcal{H} := \min \{ n \in \mathbb{N}: \quad Z(n) = 0 \}.\tag{5}\N\]\N(b) For the expected value of \(\mathbb{E} Z(n)\) on the positive trajectories, \N\[\N\mathbb{E} [ Z(n) \vert \quad n < \mathcal{H} < \infty ] = \frac{q}{\mathcal{K}_q } \left( 1 + B_2 \mathcal{K}_q \beta^n ( 1 + \rho(n) ) \right), \tag{6}\N\]\Nwhere \(B_2 = f''(q) / ( 2 \beta)\) and \(\rho(n) \to 0\) (as \(n \to \infty\)).\N\NTheorem 2. Under the condition \(f''(1-) < \infty\), \(m < 1\), the following estimate for the remainder term \(\rho(n)\) in (4) and (6) is true: \(\rho(n) = O( \beta^n)\) (as \(n \to \infty\)). \N\NTheorem 3. The following asymptotic relations hold: \N\[\N\frac{ p_1(n)}{ \beta^n} = \frac{1}{ q^2} \mathcal{K}_q^2 \left( 1+ 2 \gamma_q \cdot \mathcal{K}_q \beta^n ( 1 + o(1)) \right), \qquad \text{as} \quad n \to \infty.\tag{7}\N\]\NTheorem 4. Let \(\mathcal{P}(s) := \sum_{ j \in \mathcal{S} } \pi_j s^j\) be the limit function, which satisfies the Abel functional equation \(\mathcal{P} ( f(s) )\) \(=\) \(\beta \mathcal{P} (s)\) \(+\) \(\mathcal{P} p_0)\). Let \(p_1 > 0\). Then \(\mathcal{P} ( s)\) converges for all \(s \in U_q[0,1)\), and it has an explicit form: \N\[\N\mathcal{P} (s) = \frac{ q^2}{ \mathcal{K}_q^2 } ( \mathcal{K}_q - \mathcal{A}_q(s) ).\tag{8}\N\]\NFurthermore, \(\mathcal{P}_n (s) = \mathcal{P} (s) ( 1 + O( \beta^n ) )\) and \N\[\N\frac{ p_j(n)}{ p_1(n) } = \pi_j ( 1 + O( \beta^n)), \qquad \text{for all} \quad j \in \mathcal{S}, \tag{9}\N\]\Nas \(n \to \infty\), where \(U_q[0, 1)\) \(=\) \(\{ [0, q) \cup (q,1) \}\), and \(\mathcal{S} \subset \mathbb{N}\) is the class of possible essential communication states. \N\NTheorem 5. Let \(p_1 > 0\). Then for all \(i,j \in \mathcal{S}\) the limit \N\[\N\lim_{ n \to \infty} P_i^{ \mathcal{H}(n)} \{ Z(n) = j \} = \nu_j\tag{10}\N\]\Nexists. If condition [K] holds, then \(\mathcal{V} (s)\) \(:=\) \(\sum_{ j \in \mathcal{S} } \nu_j s^j\) has the following form: \N\[\N\mathcal{V} (s) = \frac{s}{ 1 + ( 1 -s ) q \gamma_q }.\tag{11}\N\]\NFurthermore, \N\[\N\mathcal{V}_n^{(i)}(s) = \mathcal{V} (s) ( 1 + O ( \beta^n)), \qquad \text{as} \quad n \to \infty,\tag{12} \N\]\Nwhere we define \(P_i^{ \mathcal{H}(n)} \{ * \}\) \(:=\) \(P_i \{ * \vert n < \mathcal{H} < \infty \}\), and \N\[\N\mathcal{V}_n^{(i)} (s) := \sum_{ j \in \mathcal{S} } P_i^{ \mathcal{H} (n) } \{ Z(n) = j \} s^j.\tag{13} \N\]\NFor other related works, see, e.g., [\textit{V. M. Zolotarev}, Teor. Veroyatn. Primen. 2, 256--266 (1957; Zbl 0089.34202)] about more exact statements of several threorems in the theory of branching processes, and [\textit{A. A. Imomov} and \textit{M. S. Murtazaev}, J. Appl. Probab. 61, No. 3, 927--941 (2024; Zbl 07925100)] about the Kolmogorov constant explicit form in the theory of Discrete-time stochastic branching systems.