Infinite sums of i.i.d. random variables discounted by a multiplicative random walk are called perpetuities and have been studied by many authors. The present paper provides a log-type moment result for such random variables under minimal conditions which is then utilized for the study of related moments of a.s. limits of certain martingales associated with the supercritical branching random walk. The connection, first observed by the second author in [Iksanov, A.M. (2004). Elementary fixed points of the BRW smoothing transforms with infinite number of summands. Stoch. Proc. Appl. 114, 27-50.], arises upon consideration of a size-biased version of the branching random walk originally introduced by Lyons in [Lyons, R.(1997). A simple path to Biggins' martingale convergence for branching random walk. In Athreya, K.B., Jagers, P. (eds.). Classical and Modern Branching Processes, IMA Volumes in Mathematics and its Applications, vol. 84, Springer, Berlin, 217-221.]. We also provide a necessary and sufficient condition for uniform integrability of these martingales in the most general situation which particularly means that the classical (LlogL)-condition is not always needed.
Lévy-type perpetuities being the a.s. limits of particular generalized Ornstein-Uhlenbeck processes are a natural continuous-time generalization of discrete-time perpetuities. These are random variables of the form $S:=\int_{[0,\infty)}e^{-X_{s-}}{\mathrm{d}}Z_s$, where $(X,Z)$ is a two-dimensional Lévy process, and $Z$ is a drift-free Lévy process of bounded variation. We prove an ultimate criterion for the finiteness of power moments of $S$. This result and the previously known assertion due to Erickson and Maller (2005) concerning the a.s. finiteness of $S$ are then used to derive ultimate …
Let $W_n, n\in\mn_{0}$ be an intrinsic martingale with almost sure limit $W$ in a supercritical branching random walk. We provide criteria for the $L_p$-convergence of the series $\sum_{n\ge 0} e^{an}(W-W_n)$ for $p>1$ and $a>0$. The result may be viewed as a statement about the exponential rate of convergence of $\me |W-W_n|^p$ to zero.
Let $(W_n(θ))_{n\in\mathbb N_0}$ be the Biggins martingale associated with a supercritical branching random walk and denote by $W_\infty(θ)$ its limit. Assuming essentially that the martingale $(W_n(2θ))_{n\in\mathbb N_0}$ is uniformly integrable and that $\text{Var} W_1(θ)$ is finite, we prove a functional central limit theorem for the tail process $(W_\infty(θ) - W_{n+r}(θ))_{r\in\mathbb N_0}$ and a law of the iterated logarithm for $W_\infty(θ)-W_n(θ)$, as $n\to\infty$.
We provide sufficient conditions which ensure that the intrinsic martingale in the supercritical branching random walk converges exponentially fast to its limit. The case of Galton-Watson processes is particularly included so that our results can be seen as a generalization of a result given in the classical treatise by Asmussen and Hering. As an auxiliary tool, we prove ultimate versions of two results concerning the exponential renewal measures which may be interesting on its own and which correct, generalize and simplify some earlier works.
In his, by now, classical work from 1981, Nerman made extensive use of a crucial martingale $(W_t)_{t \geq 0}$ to prove convergence in probability, in mean and almost surely, of supercritical general branching processes (a.k.a. Crump-Mode-Jagers branching processes) counted with a general characteristic. The martingale terminal value $W$ figures in the limits of his results. We investigate the rate at which the martingale, now called Nerman's martingale, converges to its limit $W$. More precisely, assuming the existence of a Malthusian parameter $α> 0$ and $W_0\in L^2$, we prove a functional c…
We consider the additive martingale $W_t(λ)$ and the derivative martingale $\partial W_t(λ)$ for one-dimensional supercritical super-Brownian motions with general branching mechanism. In the critical case $λ=λ_0$, we prove that $\sqrt{t}W_t(λ_0)$ converges in probability to a positive limit, which is a constant multiple of the almost sure limit $\partial W_\infty(λ_0)$ of the derivative martingale $\partial W_t(λ_0)$. We also prove that, on the survival event, $\limsup_{t\to\infty}\sqrt{t}W_t(λ_0)=\infty$ almost surely.
In this paper we use the spine decomposition and martingale change of measure to establish a Kesten-Stigum $L\log L$ theorem for branching Hunt processes. This result is a generalization of the results in Asmussen-Hering (1976) and Hering (1978) for branching diffusions.
We prove necessary and sufficient conditions for the $L^p$-convergence, $p>1$, of the Biggins martingale with complex parameter in the supercritical branching random walk. The results and their proofs are much more involved (especially in the case $p\in (1,2)$) than those for the Biggins martingale with real parameter. Our conditions are ultimate in the case $p\geq 2$ only.
Biggins [Uniform convergence of martingales in the branching random walk. {\em Ann. Probab.}, 20(1):137--151, 1992] proved local uniform convergence of additive martingales in $d$-dimensional supercritical branching random walks at complex parameters $λ$ from an open set $Λ\subseteq \mathbb{C}^d$. We investigate the martingales corresponding to parameters from the boundary $\partial Λ$ of $Λ$. The boundary can be decomposed into several parts. There may be a part of the boundary, on which the martingales do not exist, on other parts it exists, but diverges or vanishes in the limit. In the rema…
We provide sufficient conditions for polynomial rate of convergence in the weak law of large numbers for supercritical general indecomposable multi-type branching processes. The main result is derived by investigating the embedded single-type process composed of all individuals having the same type as the ancestor. As an important intermediate step, we determine the (exact) polynomial rate of convergence of Nerman's martingale in continuous time to its limit. The techniques used also allow us to give streamlined proofs of the weak and strong laws of large numbers and ratio convergence for the …
Let $(W_n(θ))_{n \in \mathbb{N}_0}$ be Biggins' martingale associated with a supercritical branching random walk, and let $W(θ)$ be its almost sure limit. Under a natural condition for the offspring point process in the branching random walk, we show that if the law of $W_1(θ)$ belongs to the domain of normal attraction of an $α$-stable distribution for some $α\in (1,2)$, then, as $n\to\infty$, there is weak convergence of the tail process $(W(θ) - W_{n-k}(θ))_{k \in \mathbb{N}_0}$, properly normalized, to a random scale multiple of a stationary autoregressive process of order one with $α$-sta…
We consider a supercritical branching process $Z_n$ in a stationary and ergodic random environment $ξ=(ξ_n)_{n\ge0}$. Due to the martingale convergence theorem, it is known that the normalized population size $W_n=Z_n/ (\mathbb E (Z_n|ξ))$ converges almost surely to a random variable $W$. We prove that if $W$ is not concentrated at $0$ or $1$ then for almost every environment $ξ$ the law of $W$ conditioned on the environment $ξ$ is absolutely continuous with a possible atom at $0$. The result generalizes considerably the main result of \cite{kaplan:1974}, and of course it covers the well-known…
We study the limits of the additive and derivative martingales of one-dimensional branching Brownian motion in a periodic environment. Then we prove the existence of pulsating travelling wave solutions of the corresponding F-KPP equation in the supercritical and critical cases by representing the solutions probabilistically in terms of the limits of the additive and derivative martingales. We also prove that there is no pulsating travelling wave solution in the subcritical case. Our main tools are the spine decomposition and martingale change of measures.
The long-term behavior of a supercritical branching random walk can be described and analyzed with the help of Biggins' martingales, parametrized by real or complex numbers. The study of these martingales with complex parameters is a rather recent topic. Assuming that certain sufficient conditions for the convergence of the martingales to non-degenerate limits hold, we investigate the fluctuations of the martingales around their limits. We discover three different regimes. First, we show that for parameters with small absolute values, the fluctuations are Gaussian and the limit laws are scale …
We work under the A\"ıdékon-Chen conditions which ensure that the derivative martingale in a supercritical branching random walk on the line converges almost surely to a nondegenerate nonnegative random variable that we denote by $Z$. It is shown that $\mathbb{E} Z\mathbf{1}_{\{Z\le x\}}=\log x+o(\log x)$ as $x\to\infty$. Also, we provide necessary and sufficient conditions under which $\mathbb{E} Z\mathbf{1}_{\{Z\le x\}}=\log x+{\rm const}+o(1)$ as $x\to\infty$. This more precise asymptotics is a key tool for proving distributional limit theorems which quantify the rate of convergence of the …
Evans (1992) described the semi-group of a superprocess with quadratic branching mechanism under a martingale change of measure in terms of the semi-group of an immortal particle and the semigroup of the superprocess prior to the change of measure. This result, commonly referred to as the spine decomposition, alludes to a pathwise decomposition in which independent copies of the original process `immigrate' along the path of the immortal particle. For branching particle diffusions the analogue of this decomposition has already been demonstrated in the pathwise sense, see for example Hardy and …
Let $\mm_n, n=0,1,...$ be the supercritical branching random walk, in which the number of direct descendants of one individual may be infinite with positive probability. Assume that the standard martingale $W_n$ related to $\mm_n$ is regular, and $W$ is a limit random variable. Let $a(x)$ be a nonnegative function which regularly varies at infinity, with exponent greater than -1. The paper presents sufficient conditions of the almost sure convergence of the series $\sum_{n=1}^{\infty}a(n)(W-W_n)$. Also we establish a criterion of finiteness of $\me W\log^+ W a(\log^+W)$ and $\me \log^+|\zi| a(…
Suppose that $X=\{X_t, t\ge 0; \mathbb{P}_μ\}$ is a supercritical superprocess in a locally compact separable metric space $E$. Let $ϕ_0$ be a positive eigenfunction corresponding to the first eigenvalue $λ_0$ of the generator of the mean semigroup of $X$. Then $M_t:=e^{-λ_0t}\langleϕ_0, X_t\rangle$ is a positive martingale. Let $M_\infty$ be the limit of $M_t$. It is known (see, J. Appl. Probab. 46 (2009), 479--496) that $M_\infty$ is non-degenerate iff the $L\log L$ condition is satisfied. In this paper we are mainly interested in the case when the $L\log L$ condition is not satisfied. We pr…
Suppose that $X=\{X_t, t\ge 0\}$ is a supercritical superprocess on a locally compact separable metric space $(E, m)$. Suppose that the spatial motion of $X$ is a Hunt process satisfying certain conditions and that the branching mechanism is of the form $$ ψ(x,λ)=-a(x)λ+b(x)λ^2+\int_{(0,+\infty)}(e^{-λy}-1+λy)n(x,dy), \quad x\in E, \quadλ> 0, $$ where $a\in \mathcal{B}_b(E)$, $b\in \mathcal{B}_b^+(E)$ and $n$ is a kernel from $E$ to $(0,\infty)$ satisfying $$ \sup_{x\in E}\int_0^\infty y^2 n(x,dy)<\infty. $$ Put $T_tf(x)=\mathbb{P}_{δ_x}< f,X_t>$. Let $λ_0>0$ be the largest eigenvalue of the g…
The branching random walk (BRW) smoothing transform $T$ is defined as $T:\text{distr}(U_{1})\mapsto \text{distr} (\sum_{i=1}^{L}X_{i}U_{i})$, where given realizations $\{X_{i}\}_{i=1}^{L}$ of a point process, $U_{1},U_{2},...$ are conditionally independent identically distributed random variables, and $0\leq \text{Prob}\{L=\infty \}\leq 1$. Given $α\in (0,1]$, $α$-\emph{elementary} fixed points are fixed points of $T$ whose Laplace-Stieltjes transforms $ϕ$ satisfy $\underset{s\to +0}{\lim}\dfrac{1-ϕ(s)}{s^α}=m$, where $m$ is any given positive number. If $α=1$, these are the fixed points with …
We consider the classical problem of existence, uniqueness and asymptotics of monotone solutions to the travelling wave equation associated to the parabolic semi-group equation of a super-Brownian motion with a general branching mechanism. Whilst we are strongly guided by the probabilistic reasoning of Kyprianou (2004) for branching Brownian motion, the current paper offers a number of new insights. Our analysis incorporates the role of Seneta-Heyde norming which, in the current setting, draws on classical work of Grey (1974). We give a pathwise explanation of Evans' immortal particle picture …
Let $(Z_n)$ be a supercritical branching process in a random environment $ξ$. We study the convergence rates of the martingale $W_n = Z_n/ E[Z_n| ξ]$ to its limit $W$. The following results about the convergence almost sur (a.s.), in law or in probability, are shown. (1) Under a moment condition of order $p\in (1,2)$, $W-W_n = o (e^{-na})$ a.s. for some $a>0$ that we find explicitly; assuming only $EW_1 \log W_1^{α+1} < \infty$ for some $α>0$, we have $W-W_n = o (n^{-α})$ a.s.; similar conclusions hold for a branching process in a varying environment. (2) Under a second moment condition, there…
Consider the supercritical branching random walk on the real line in the boundary case and the associated Gibbs measure $ν_{n,β}$ on the $n^\text{th}$ generation, which is also the polymer measure on a disordered tree with inverse temperature $β$. The convergence of the partition function $W_{n,β}$, after rescaling, towards a nontrivial limit has been proved by A\"ıdékon and Shi in the critical case $β= 1$ and by Madaule when $β>1$. We study here the near-critical case, where $β_n \to 1$, and prove the convergence of $W_{n,β_n}$, after rescaling, towards a constant multiple of the limit of the…
For a subcritical Galton-Watson process $(ζ_n)$, it is well known that under an $X \log X$ condition, the quotient $P(ζ_n > 0)/ Eζ_n$ has a finite positive limit. There is an analogous result for a (one-dimensional) supercritical branching random walk: when $a$ is in the so-called subcritical speed area, the probability of presence around $na$ in the $n$-th generation is asymptotically proportional to the corresponding expectation. In Rouault (1993) this result was stated under a natural $X \log X$ assumption on the offspring point process and a (unnatural) condition on the offspring mean. Her…
Let $W_{\infty}(β)$ be the limit of the Biggins martingale $W_n(β)$ associated to a supercritical branching random walk with mean number of offspring $m$. We prove a functional central limit theorem stating that as $n\to\infty$ the process $$ D_n(u):= m^{\frac 12 n} \left(W_{\infty}\left(\frac{u}{\sqrt n}\right) - W_{n}\left(\frac{u}{\sqrt n}\right) \right) $$ converges weakly, on a suitable space of analytic functions, to a Gaussian random analytic function with random variance. Using this result we prove central limit theorems for the total path length of random trees. In the setting of bina…
Chen [Ann. Appl. Probab. {\bf 11} (2001), 1242--1262] derived exact convergence rates in a central limit theorem and a local limit theorem for a supercritical branching Wiener process.We extend Chen's results to a branching random walk under weaker moment conditions. For the branching Wiener process, our results sharpen Chen's by relaxing the second moment condition used by Chen to a moment condition of the form $ \E X (\ln^+X )^{1+λ}< \infty$. In the rate functions that we find for a branching random walk, we figure out some new terms which didn't appear in Chen's work.The results are establi…
