A note on the multitype Markov branching process

We consider a supercritical, p-dimensional Markov branching process (MBP). Based on the finite and the infinite lines of descent of particles of this p-dimensional MBP, we construct an associated 2p-dimensional process. We show that such a process is a 2p-dimensional, supercritical MBP. This 2p-dimensional process retains the branching property when conditioned on the sets of extinction and non-extinction. Asymptotic results and central limit theorems for the associated process and the original process are established by using the results of Gadag and Rajarshi (1987).

Download Full-text

Non-central limit theorems for functionals of random fields on hypersurfaces

ESAIM Probability and Statistics ◽

10.1051/ps/2020006 ◽

2020 ◽

Vol 24 ◽

pp. 315-340

Author(s):

Andriy Olenko ◽

Volodymyr Vaskovych

Keyword(s):

Rate Of Convergence ◽

Central Limit ◽

Random Fields ◽

Limit Theorems ◽

Gaussian Random Fields ◽

Central Limit Theorems ◽

Integral Functionals ◽

Asymptotic Results ◽

Non Linear ◽

Linear Integral

This paper derives non-central asymptotic results for non-linear integral functionals of homogeneous isotropic Gaussian random fields defined on hypersurfaces in ℝd. We obtain the rate of convergence for these functionals. The results extend recent findings for solid figures. We apply the obtained results to the case of sojourn measures and demonstrate different limit situations.

Download Full-text

Central limit theorems for logarithmic averages

Studia Scientiarum Mathematicarum Hungarica ◽

10.1556/sscmath.38.2001.1-4.6 ◽

2001 ◽

Vol 38 (1-4) ◽

pp. 79-96

Author(s):

I. Berkes ◽

L. Horváth ◽

X. Chen

Keyword(s):

Wiener Process ◽

Central Limit ◽

Limit Theorems ◽

Random Variables ◽

Central Limit Theorems ◽

Partial Sums ◽

Asymptotic Results

We prove central limit theorems and related asymptotic results for where W is a Wiener process and Sk are partial sums of i.i.d. random variables with mean 0 and variance 1. The integrability and smoothness conditions made on f are optimal in a number of important cases.

Download Full-text

On the properties of processes associated with a markov branching process

Journal of Applied Probability ◽

10.1017/s0021900200042388 ◽

1991 ◽

Vol 28 (03) ◽

pp. 520-528

Author(s):

V. G. Gadag ◽

R. P. Gupta

Keyword(s):

Generating Function ◽

Finite Time ◽

Branching Process ◽

Asymptotic Properties ◽

Probability Generating Function ◽

Asymptotic Distributions ◽

Time Interval ◽

Original Process ◽

Markov Branching Process ◽

Branching Property

Consider a time-homogeneous Markov branching process. We construct reduced processes, based on whether the length of line of descent of particles of this process are (a) greater than or (b) at most equal to, τ units of time, for some fixed τ ≧ 0. We show that in both cases the reduced processes retain the branching property, but the latter does not retain the time homogeneity. We investigate finite-time and asymptotic properties of the reduced processes. Based on a realization of the original process and a realization of a reduced process, observed continuously over a time interval [0, T] for T > 0, we propose estimators for the different parameters involved, including qτ , the probability that the original process becomes extinct before τ units of time, and f (j)(qτ ), the jth derivative of the offspring probability generating function f(s) at q τ when q τ is known. We study the properties of these estimators and derive their asymptotic distributions, under the assumption that the original process is supercritical.

Download Full-text

On the properties of processes associated with a markov branching process

Journal of Applied Probability ◽

10.2307/3214488 ◽

1991 ◽

Vol 28 (3) ◽

pp. 520-528

Author(s):

V. G. Gadag ◽

R. P. Gupta

Keyword(s):

Generating Function ◽

Finite Time ◽

Branching Process ◽

Asymptotic Properties ◽

Probability Generating Function ◽

Asymptotic Distributions ◽

Time Interval ◽

Original Process ◽

Markov Branching Process ◽

Branching Property

Consider a time-homogeneous Markov branching process. We construct reduced processes, based on whether the length of line of descent of particles of this process are (a) greater than or (b) at most equal to, τ units of time, for some fixed τ ≧ 0. We show that in both cases the reduced processes retain the branching property, but the latter does not retain the time homogeneity. We investigate finite-time and asymptotic properties of the reduced processes. Based on a realization of the original process and a realization of a reduced process, observed continuously over a time interval [0, T] for T > 0, we propose estimators for the different parameters involved, including qτ, the probability that the original process becomes extinct before τ units of time, and f(j)(qτ), the jth derivative of the offspring probability generating function f(s) at qτ when qτ is known. We study the properties of these estimators and derive their asymptotic distributions, under the assumption that the original process is supercritical.

Download Full-text

Multitype branching processes based on exact progeny lengths of particles in a Galton-Watson branching process

Journal of Applied Probability ◽

10.1017/s0021900200041747 ◽

1989 ◽

Vol 26 (01) ◽

pp. 1-8

Author(s):

V. G. Gadag ◽

M. B. Rajarshi

Keyword(s):

Branching Process ◽

Branching Processes ◽

White Male ◽

Asymptotic Results ◽

Male Population ◽

Multitype Branching Process ◽

Original Process ◽

The Usa ◽

Multitype Branching Processes ◽

Lines Of Descent

In Gadag and Rajarshi (1987), we studied a bivariate (multitype) branching process based on infinite and finite lines of descent, of particles of a supercritical one-dimensional (multitype) Galton-Watson branching process (GWBP). In this paper, we discuss a few more meaningful and interesting univariate and multitype branching processes, based on exact progeny lengths of particles in a GWBP. Our constructions relax the assumption of supercriticality made in Gadag and Rajarshi (1987). We investigate some finite-time and asymptotic results of these processes in some details and relate them to the original process. These results are then used to propose new and better estimates of the offspring mean. An illustration based on the branching process of the white male population of the USA is also given. We believe that our work offers a rather finer understanding of the branching property.

Download Full-text

Multitype branching processes based on exact progeny lengths of particles in a Galton-Watson branching process

Journal of Applied Probability ◽

10.2307/3214311 ◽

1989 ◽

Vol 26 (1) ◽

pp. 1-8 ◽

Cited By ~ 1

Author(s):

V. G. Gadag ◽

M. B. Rajarshi

Keyword(s):

Branching Process ◽

Branching Processes ◽

White Male ◽

Asymptotic Results ◽

Male Population ◽

Multitype Branching Process ◽

Original Process ◽

The Usa ◽

Multitype Branching Processes ◽

Lines Of Descent

In Gadag and Rajarshi (1987), we studied a bivariate (multitype) branching process based on infinite and finite lines of descent, of particles of a supercritical one-dimensional (multitype) Galton-Watson branching process (GWBP). In this paper, we discuss a few more meaningful and interesting univariate and multitype branching processes, based on exact progeny lengths of particles in a GWBP. Our constructions relax the assumption of supercriticality made in Gadag and Rajarshi (1987). We investigate some finite-time and asymptotic results of these processes in some details and relate them to the original process. These results are then used to propose new and better estimates of the offspring mean. An illustration based on the branching process of the white male population of the USA is also given. We believe that our work offers a rather finer understanding of the branching property.

Download Full-text