Local Limit Theorems in the Theory of Branching Random Processes

Let $\mu _t $ be a continuous branching process and let us consider the function \[ f(x) = \sum\limits_{k = 0}^\infty {p_{k^{x^k } } } , \] where \[p_k = \mathop {\lim }\limits_{t \to 0} \frac{{{\bf P}\{ \mu _t = k\} }} {t}(k \ne 1) \quad {\text{and}} \quad p_1 = \mathop {\lim }\limits_{t \to 0} \frac{{1 - {\bf P}\{ \mu _t = 1\} }} {t}.\] Denote: \[ a = f'(1),\quad b = f''(1),\quad c = f'''(1),\quad d = f''''(1). \] The following theorems are true 1. Let$a = 0,b,c,d$be finite, then when$t \to \infty $and$0 < c_1 \leqq z_{n,t} \leqq c_2 $\[ \frac{{bt}}{2}P_n * (t) = e^{ - z_{n,t} } + O\left(t^{ - \frac{1}{2}} \sqrt {\log t} \right), \]where\[ P_n * (t) = \frac{{{\bf P}\{ \mu _t = n\} }}{{1 - {\bf P}\{ \mu _t = 0\} }}\quad \textit{and} \quad z_{n,t} = \frac{{2n}}{{bt}}. \] 2. Let$a > 0$and$b < \infty $. If$t \to \infty $and$0 < c_1 \leqq z_{n,t} \leqq c_2 $, then\[ \frac{{e^{at} }}{{1 - \lambda }}P_n * (t) = \left\{ \begin{gathered} hs(z_{n,t} ) + o(1) \quad {\textit{for}} \quad n \equiv 1(\bmod h) \hfill \\ \qquad\ 0\qquad {\textit{for}} \quad n \not\equiv 1(\bmod h), \hfill \\ \end{gathered} \right.\]where a) $\lambda $is the smallest non-negative real root of the equation$f(x) = 0$; b) $z_{n,t} = ne^{ - at} (1 - \lambda )$; c) his the greatest common divisor for twin differences of indicesnfor which$p_n \ne 0$; d) $s(y)$is the density of the distribution\[ S(y) = \mathop {\lim }\limits_{t \to \infty } {{\bf P}}\left\{ {\frac{{\mu _t (1 - \lambda )}}{{e^{at} }} < y \mid\mu _t > 0} \right\}. \]

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