The local limit theorem and some related aspects of super-critical branching processes

Abstract:

Let $\{ {Z_n}:n = 0,1,2, \ldots \}$ be a Galton-Watson branching process with offspring p.g.f. $f(s) = \Sigma _0^\infty {p_j}{s^j}$. Assume (i) $1 < m = f’(1 - ) = \Sigma _1^\infty j{p_j} < \infty$, (ii) $\Sigma _1^\infty {j^2}{p_j} < \infty$ and (iii) ${\gamma _0} = f’(q) > 0$, where $q$ is the extinction probability of the process. Let $w(x)$ denote the density function of $W$, the almost sure limit of ${Z_n}{m^{ - n}}$ with ${Z_0} = 1,{w^{(i)}}(x)$ the $i$-fold convolution of $w(x),{P_n}(i,j) = P({Z_n} = j|{Z_0} = i),{\delta _0} = (\log \gamma _0^{ - 1}){(\log m)^{ - 1}}$ and ${\beta _0} = {m^{{\delta _0}/(3 + {\delta _0})}}$. Then for any $0 < \beta < {\beta _0}$ and $i$ we can find a constant $C = C(i,\beta )$ such that \[ |{m^n}{P_n}(i,j) - {w^{(i)}}({m^{ - n}}j)| \leqq C[\beta _0^{ - n}{({m^{ - n}}j)^{ - 1}} + {\beta ^{ - n}}]\] for all $j \geqq 1$. Applications to the boundary theory of the associated space time process are also discussed.