Heavy tailed branching process with immigration (original) (raw)

ASYMPTOTIC BEHAVIOR OF THE MEASURE VALUED BRANCHING PROCESS WITH IMMIGRATION

1993

The measure-valued branching process with immigration is defined as Y t = X t + I t , t ≥ 0, where X t satisfies the branching property and I t with I 0 = 0 is independent of X t . This formulation leads to the model of . We prove a large number law for Y t . Equilibrium distributions and spatial transformations are also studied.

Limit Theorems for Branching Processes with Immigration in a Random Environment

2020

We investigate subcritical Galton-Watson branching processes with immigration in a random environment. Using Goldie's implicit renewal theory we show that under general Cramer condition the stationary distribution has a power law tail. We determine the tail process of the stationary Markov chain, prove point process convergence, and convergence of the partial sums. The original motivation comes from Kesten, Kozlov and Spitzer seminal 1975 paper, which connects a random walk in a random environment model to a special Galton-Watson process with immigration in a random environment. We obtain new results even in this very special setting.

Limit theorems for time averages of continuous-state branching processes with immigration

Cornell University - arXiv, 2022

In this work we investigate limit theorems for the time-averaged process 1 t t 0 X x s ds t≥0 where X x is a subcritical continuous-state branching process with immigration starting in x ≥ 0. Under a second moment condition on the branching and immigration measures we first prove the law of large numbers in L 2 and afterward establish the central limit theorem. Assuming additionally that the big jumps of the branching and immigration measures have finite exponential moments of some order, we prove in our main result the large deviation principle and provide a semi-explicit expression for the good rate function in terms of the branching and immigration mechanisms. Our methods are deeply based on a detailed study of the corresponding generalized Riccati equation and related exponential moments of the time-averaged process.

Branching processes with immigration in atypical random environment

Extremes

Motivated by a seminal paper of Kesten et al. (Ann. Probab., 3(1), 1–31, 1975) we consider a branching process with a conditional geometric offspring distribution with i.i.d. random environmental parameters An, n ≥ 1 and with one immigrant in each generation. In contrast to above mentioned paper we assume that the environment is long-tailed, that is that the distribution F of xin:=log((1−An)/An)\xi _{n}:=\log ((1-A_{n})/A_{n})xin:=log((1−An)/An) ξ n : = log ( ( 1 − A n ) / A n ) is long-tailed. We prove that although the offspring distribution is light-tailed, the environment itself can produce extremely heavy tails of the distribution of the population size in the n th generation which becomes even heavier with increase of n. More precisely, we prove that, for all n, the distribution tail mathbbP(Zngem)\mathbb {P}(Z_{n} \ge m)mathbbP(Zngem) ℙ ( Z n ≥ m ) of the n th population size Zn is asymptotically equivalent to noverlineF(logm)n\overline F(\log m)noverlineF(logm) n F ¯ ( log m ) as m grows. In this way we generalise Bhattacharya and Palmowski (Stat. Probab. Lett., 15...

Limit theorems for normalized nearly critical branching processes with immigration

Publicationes Mathematicae Debrecen, 2008

Functional limit theorems are proved for a sequence of Galton-Watson processes with immigration, where the offspring mean tends to its critical value 1 under weak conditions for the variances of offspring and immigration processes. In the limit theorems the norming factors depend on these variances, respectively.

Fluctuation limit of branching processes with immigration and estimation of the means

Advances in Applied Probability, 2005

A sequence of Galton-Watson branching processes with immigration is investigated, when the offspring mean tends to its critical value one and the offspring variance tends to zero. It is shown that the fluctuation limit is an Ornstein-Uhlenbeck type process. As a consequence, in contrast to the case where the offspring variance tends to a positive limit, the conditional least squares estimator of the offspring mean turns out to be asymptotically normal. The norming factor is n 3/2 , in contrast to the subcritical case where it is n 1/2 , and in contrast to the nearly critical case with positive limiting offspring variance, where it is n.

Limit Theorems for Exceedances of Sequence of Branching Processes

The Bulletin of the Malaysian Mathematical Society Series 2

A problem of the first exceedance of a given level by the family of independent branching processes with and without immigration is considered. Using limit theorems for large deviations for processes with and without immigration limit theorems for the index of the first process exceeding some fixed or increasing levels in critical, subcritical and supercritical cases are proved. Asymptotic formulas for the expectation of the index are also obtained.