Notes on large deviations for branching processes indexed by a Poisson process

Lithuanian Mathematical Journal - Consider a continuous-time process {ZNt}, where {Zn} is a Galton–Watson process with offspring mean m, and {Nt} is a Poisson process independent of {Zn}. It...     

Multiplicity distributions in a two-component branching process

Probability (multiplicity) distributions and those densities (KNO scaling functions) are investigated in a two-component (charged and neutral) branching process. It is shown that the two-component KNO scaling functions depend effectively on one variable in two typical cases. A formula for multiplicity correlation between two components (charged and neutral particles) is formulated. It is applied to the analysis of experimental data.     

An estimator for parameters of a nonlinear nonnegative multidimensional AR(1) process

Let e _t=(e _t1,...e _tp) be a p-dimensional nonnegative strict white noise with finite second moments. Let h _ij(x) be nondecreasing functions from [0,) onto [0,) such that h _ij(x) x for i, j = 1,...,p. Let U = (u _ij) be a p×p matrix with nonnegative elements having all its roots inside the unit circle. Define a process X t=(X _t1,...,X _tp) by for

Large deviations for posterior distributions on the parameter of a multivariate text{ AR}(p) process

We prove the large deviation principle for the posterior distributions on the (unknown) parameter of a multivariate autoregressive process with i.i.d. Normal innovations. As a particular case, we recover a previous result for univariate first-order autoregressive processes. We also show that the rate function can be expressed in terms of the divergence between two spectral densities.     

Limiting distributions of time averages of additive functionals defined on a semi-Markov process

We consider an additive functional _ton a continuous time semi-Markov process with an arbitrary state space. We study the existence of the limiting distribution of _t/t without the condition of the finiteness of the mean occupation time of a fixed state.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 42, No. 6, pp. 843–847, June, 1990.     

Asymptotic inference for AR(1) panel data

A general asymptotic theory is given for the panel data AR(1) model with time series independent in different cross sections. The theory covers the cases of stationary process, local to unity process, unit root process, mildly integrated, mildly explosive and explosive processes. It is assumed that the cross-sectional dimension and time-series dimension are respectively N and T. The results in this paper illustrate that whichever the process is, with an appropriate regularization, the least squares estimator of the autoregressive coefficient converges in distribution to a normal distribution with rate at least O(N-1/3). Since the variance is the key to characterize the normal distribution, it is important to discuss the variance of the least squares estimator. We will show that when the autoregressive coefficient ρ satisfies |ρ| 1, the variance declines at the rate O((NT)-1), while the rate changes to O(N~(-1) T~(-2)) when ρ = 1 and O(N~(-1)ρ~(-2 T+4)) when |ρ| 1. ρ = 1 is the critical point where the convergence rate changes radically. The transition process is studied by assuming ρ depending on T and going to 1. An interesting phenomenon discovered in this paper is that, in the explosive case, the least squares estimator of the autoregressive coefficient has a standard normal limiting distribution in the panel data case while it may not has a limiting distribution in the univariate time series case.     

Aggregation of random-coefficient AR(1) process with infinite variance and common innovations

We discuss aggregation of random-coefficient AR(1) processes X _i,t = a _i X _i,t?1 + ε _t , i = 1,…,N, with i.i.d. coefficients a _i ∈ (?1, 1) and common i.i.d. innovations {ε _t } belonging to the domain of attraction of an α-stable law (0 < α ≤ 2). Particular attention is given to the case of slope coefficient having probability density growing regularly to infinity at points a = 1 and a = ?1. We obtain conditions under which the limit aggregate \( {\bar X_t} = {\lim_{N \to \infty }}{N^{ - 1}}\sum\nolimits_{i = 1}^N {{X_{i,t}}} \) exists and exhibits long memory in a certain sense. In particularly, we show that suitably normalized partial sums of the \( {\bar X_t} \)’s tend to a fractional α-stable motion and that \( \left\{ {{{\bar X}_t}} \right\} \) satisfies the long-range dependence (sample Allen variance) property of Heyde and Yang. We also extend some results of Zaffaroni from the finite variance case to the infinite variance case.     

Some limit theorems for the empirical process indexed by functions

Summary Let,n1, be a sequence of classes of real-valued measurable functions defined on a probability space (S,,P). Under weak metric entropy conditions on,n1, and under growth conditions on we show that there are non-zero numerical constantsC ₁ andC ₂ such that where (n) is a non-decreasing function ofn related to the metric entropy of. A few applications of this general result are considered: we obtain a.s. rates of uniform convergence for the empirical process indexed by intervals as well as a.s. rates of uniform convergence for the empirical characteristic function over expanding intervals.Portions of this article were presented during the conference on Mathematical Stochastics (February 19–25, 1984) at Oberwolfach, West Germany     

Immortal particle for a catalytic branching process

We study the existence and asymptotic properties of a conservative branching particle system driven by a diffusion with smooth coefficients for which birth and death are triggered by contact with a set. Sufficient conditions for the process to be non-explosive are given. In the Brownian motions case the domain of evolution can be non-smooth, including Lipschitz, with integrable Martin kernel. The results are valid for an arbitrary number of particles and non-uniform redistribution after branching. Additionally, with probability one, it is shown that only one ancestry line survives. In special cases, the evolution of the surviving particle is studied and for a two particle system on a half line we derive explicitly the transition function of a chain representing the position at successive branching times.     

A Unit Root Test for an AR(1) Process with AR Errors by Using Random Weighted Bootstrap

A great deal of economic problems are related to detecting the stability of time series data,where the main interest is in the unit root test.In this paper,we consider the unit root testing problem with errors being long-memory processes with the LARCH structure.A new test statistic is developed by using the random weighted bootstrap method.It turns out that the proposed statistic has a chisquared distribution asymptotically regardless of the process being stationary or nonst at ionary,and with ...     

Trimmed stable AR(1) processes

In this paper we investigate the distribution of trimmed sums of dependent observations with heavy tails. We consider the case of autoregressive processes of order one with independent innovations in the domain of attraction of a stable law. We show if the $d$ largest (in magnitude) terms are removed from the sample, then the sum of the remaining elements satisfies a functional central limit theorem with random centering provided $d=d\left(n\right)\ge n^{\gamma }$ (for some $\gamma >0$) and $d\left(n\right)/n\to 0$. This result is used to get asymptotics for the widely used CUSUM process in case of dependent heavy tailed observations.     

Limiting process for integral functionals of a wiener process on a cylinder

We prove that integral functionals, whose integrands are bounded functions of a Wiener process on a cylinder, weakly converge to the processw ₁(τ(t)), τ(t) = β₁ t + (β₂ − β₁)mes {s:w ₂(s)≥0,s<t}, wherew ₁(t andw ₂(t) are independent one-dimensional Wiener processes, β₁ and β₂ are nonrandom values, and β₂≥β₁≥0. Kiev University, Kiev. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 46, No. 6, pp. 765–768, June, 1994.     

Asymptotic inference for AR(1) processes with (nonnormal) stable errors

This is the first of several papers in which we consider problems related to the asymptotic distribution of the least squares estimate of the parameter γ in theAR(1) model $$X_k = \gamma X_{k - 1} + \varepsilon _k , k = 1,...,n,$$ where ε_k are independent identically distributed (i.i.d.) random variables in the domain of attraction of a stable law. In §1 we give a summary in the case ε_k is in the domain of attraction of the normal distribution. In §2 we consider errors in the domain of attraction of a (nonnormal) stable distribution. In §3 we prove a result in the case of the completely asymmetric stable distribution with α=β=1.     

Limiting distributions of time averages for processes with a semi-Markov interference

We obtain the limiting distribution of the time averages of processes with a semi-Markov interference without assuming the finiteness of the moments of the interference time.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 42, No. 2, pp. 281–284, February, 1990.     

Jumps in binomial AR(1) processes

We consider the binomial AR(1) model for serially dependent processes of binomial counts. After a review of its definition and known properties, we investigate marginal and serial properties of jumps in such processes. Based on these results, we propose the jumps control chart for monitoring a binomial AR(1) process. We show how to evaluate the performance of this control chart and give design recommendations.     

Inference in binomial AR(1) models

This paper studies inference methods for stationary time series with binomial distributions. Such series describe, for example, the number of rainy days in consecutive weeks. First, we formulate the renewal sequence version of the model that seemingly generates a new class of stationary binomial series. The model is shown to obey an AR(1) recursion in cases where the renewal lifetime has a constant hazard rate past lag one. Explicit asymptotic variances of the parameter estimators in the AR(1) case are derived from conditional least squares methods; likelihood techniques are also considered.     

Extinction probability for a critical general branching process

A general branching process begins with a single individual born at time t=0. At random ages during its random lifespan L it gives birth to offspring, N(t) being the number born in the age interval [0,t]. Each offspring behaves as a probabilistically independent copy of the initial individual. Let Z(t) be the population at time t, and let N=N(∞). Theorem: If a general branching process is critical, i. e E{N}=1, and if $\text{σ}{}^2\text{=E {N(N?1)}<∞, 0<a≡}\text{∫}\text{0}\text{∞}\text{tdE{N(t)}}$,and as t → ∞ both $\text{t}{}^2\text{(1?}\text{E}\text{{N(t)})→0}$ and $\text{t}{}^2\text{P}\text{[L>t]→0}$, then $\text{tP[Z(t)>0]→}\text{2a}\text{σ}{}^2$ as t→∞.     

Parametric estimation for a simple branching diffusion process

A simple branching diffusion process is given as an elementary model of spatial evolution. A parametric estimation theory is presented for this model. As side results, a spatial central limit theorem and spatial strong law of large numbers are also obtained.     

Discrimination of converging hypotheses for a branching process

For a discrete-time homogeneous branching process starting with m particles, we consider the problem of discrimination of converging hypotheses for m using a statistic which is equal to the total number of particles in the process during its entire evolution.Translated from Statisticheskie Metody, pp. 63–69, 1982.