Distribution of the branching-process population among generations

1971 ◽  
Vol 8 (04) ◽  
pp. 655-667 ◽  
Author(s):  
M. L. Samuels
Keyword(s):  
Normal Form ◽  
Density Function ◽  
Branching Process ◽  
Random Variables ◽  
Normal Density ◽  
Asymptotically Linear ◽  
Random Functions ◽  
Normal Density Function ◽  
Age Dependent ◽  
The Mean

Summary In a standard age-dependent branching process, let Rn (t) denote the proportion of the population belonging to the nth generation at time t. It is shown that in the supercritical case the distribution {Rn (t); n = 0, 1, …} has asymptotically, for large t, a (non-random) normal form, and that the mean ΣnRn (t) is asymptotically linear in t. Further, it is found that, for large n, Rn (t) has the shape of a normal density function (of t). Two other random functions are also considered: (a) the proportion of the nth generation which is alive at time t, and (b) the proportion of the nth generation which has been born by time t. These functions are also found to have asymptotically a normal form, but with parameters different from those relevant for {Rn (t)}. For the critical and subcritical processes, analogous results hold with the random variables replaced by their expectations.

Distribution of the branching-process population among generations

1971 ◽  
Vol 8 (4) ◽  
pp. 655-667 ◽  
Author(s):  
M. L. Samuels
Keyword(s):  
Normal Form ◽  
Density Function ◽  
Branching Process ◽  
Random Variables ◽  
Normal Density ◽  
Asymptotically Linear ◽  
Random Functions ◽  
Normal Density Function ◽  
Age Dependent ◽  
The Mean

SummaryIn a standard age-dependent branching process, let Rn(t) denote the proportion of the population belonging to the nth generation at time t. It is shown that in the supercritical case the distribution {Rn(t); n = 0, 1, …} has asymptotically, for large t, a (non-random) normal form, and that the mean ΣnRn(t) is asymptotically linear in t. Further, it is found that, for large n, Rn(t) has the shape of a normal density function (of t).Two other random functions are also considered: (a) the proportion of the nth generation which is alive at time t, and (b) the proportion of the nth generation which has been born by time t. These functions are also found to have asymptotically a normal form, but with parameters different from those relevant for {Rn(t)}.For the critical and subcritical processes, analogous results hold with the random variables replaced by their expectations.

A note on the subcritical generalized age-dependent branching process

1976 ◽  
Vol 13 (4) ◽  
pp. 798-803 ◽  
Author(s):  
R. A. Doney
Keyword(s):  
Branching Process ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Malthusian Parameter ◽  
Age Dependent ◽  
The Mean ◽  
Mean Function ◽  
Necessary And Sufficient

For a subcritical Bellman-Harris process for which the Malthusian parameter α exists and the mean function M(t)∼ aeat as t → ∞, a necessary and sufficient condition for e–at (1 –F(s, t)) to have a non-zero limit is known. The corresponding condition is given for the generalized branching process.

scholarly journals Point-Symmetric Multivariate Density Function and Its Decomposition

2014 ◽  
Vol 2014 ◽  
pp. 1-6
Author(s):  
Kiyotaka Iki ◽  
Sadao Tomizawa
Keyword(s):  
Density Function ◽  
Point Symmetry ◽  
Normal Density ◽  
Multivariate Normal ◽  
Normal Density Function ◽  
Multivariate Density

For aT-variate density function, the present paper defines the point-symmetry, quasi-point-symmetry of orderk(<T), and the marginal point-symmetry of orderkand gives the theorem that the density function isT-variate point-symmetric if and only if it is quasi-point-symmetric and marginal point-symmetric of orderk. The theorem is illustrated for the multivariate normal density function.

Asymptotic probabilities in a critical age-dependent branching process

1976 ◽  
Vol 13 (3) ◽  
pp. 476-485 ◽  
Author(s):  
Howard J. Weiner
Keyword(s):  
Branching Process ◽  
Comparison Method ◽  
Lifetime Distribution ◽  
Order Moment ◽  
Moment Condition ◽  
Age Dependent ◽  
Total Progeny ◽  
Critical Age ◽  
The Mean ◽  
Critical Process

Let Z(t) denote the number of cells alive at time t in a critical Bellman-Harris age-dependent branching process, that is, where the mean number of offspring per parent is one. A comparison method is used to show for k ≧ 1, and a high-order moment condition on G(t), where G(t) is the cell lifetime distribution, that lim t→∞t2P[Z(t) = k] = ak > 0, where {ak} are constants.The method is also applied to the total progeny in the critical process.

scholarly journals Bounds for the Asymptotic Growth Rate of an Age-Dependent Branching Process

1969 ◽  
Vol 10 (1-2) ◽  
pp. 231-235 ◽  
Author(s):  
P. J. Brockwell
Keyword(s):  
Growth Rate ◽  
Population Size ◽  
Branching Process ◽  
Expected Number ◽  
Lattice Distribution ◽  
Asymptotic Growth ◽  
Age Dependent ◽  
The Mean ◽  
Image Position ◽  
Asymptotic Growth Rate

Let M(t) denote the mean population size at time t (conditional on a single ancestor of age zero at time zero) of a branching process in which the distribution of the lifetime T of an individual is given by Pr {T≦t} =G(t), and in which each individual gives rise (at death) to an expected number A of offspring (1λ A λ ∞). expected number A of offspring (1 < A ∞). Then it is well-known (Harris [1], p. 143) that, provided G(O+)-G(O-) 0 and G is not a lattice distribution, M(t) is given asymptotically by where c is the unique positive value of p satisfying the equation .

The proportions of individuals of different kinds in two-type populations. A branching process problem arising in biology

1969 ◽  
Vol 6 (2) ◽  
pp. 249-260 ◽  
Author(s):  
P. Jagers
Keyword(s):  
Branching Process ◽  
The Other ◽  
Mean Square ◽  
Random Functions ◽  
Age Dependent ◽  
Random Limit

Consider an age-dependent branching process with two types of individuals. Suppose that individuals of one type beget children of both types, whereas those of the other type can only give birth to individuals of their own kind. This paper is a study of the relation between two random functions occurring in such processes starting from an ancestor of the first type, the two functions being the numbers of individuals of the two kinds. Under weak assumptions it is shown that the random proportion of individuals of one type converges as time passes, in mean square as well as almost surely to a non-random limit, easily determined in terms of the reproduction laws and life-length distributions of the process.

scholarly journals Location-scale matching for approximate quasi-order sampling

2019 ◽  
Author(s):  
Ali Ünlü ◽  
Martin Schrepp
Keyword(s):  
Standard Deviation ◽  
Polynomial Regression ◽  
Discrete Distribution ◽  
Linear Functions ◽  
Normal Density ◽  
Representative Sampling ◽  
Normal Density Function ◽  
Inductive Construction ◽  
Quasi Order ◽  
The Mean

Quasi-orders are reflexive and transitive binary relations and have many applications. Examples are the dependencies of mastery among the problems of a psychological test, or methods such as item tree or Boolean analysis that mine for quasi-orders in empirical data. Data mining techniques are typically tested based on simulation studies with unbiased samples of randomly generated quasi-orders. In this paper, we develop techniques for the approximately representative sampling of quasi-orders. Polynomial regression curves are fitted for the mean and standard deviation of quasi-order size as a function of item number. The resulting regression graphs are seen to be quadratic and linear functions, respectively. The extrapolated values for the mean and standard deviation are used to propose two quasi-order sampling techniques. The discrete method matches these location and scale measures with a transformed discrete distribution directly obtained from the sample. The continuous method uses the normal density function with matched expectation and variance. The quasi-orders are constructed according to the biased randomized doubly inductive construction, however they are resampled to become approximately representative following the matched discrete and continuous distributions. In simulations, we investigate the usefulness of these methods. The location-scale matching approach can cope with very large item sets. Close to representative samples of random quasi-orders are constructed for item numbers up to n = 400.

On the normal moments distributions and its characterizations

2017 ◽  
Vol 6 (1-2) ◽  
pp. 16
Author(s):  
A. A. Olosunde
Keyword(s):  
Density Function ◽  
Shape Parameter ◽  
Normal Density ◽  
Normal Density Function ◽  
Moment Distribution

Normal moment distribution is a family of elliptical density, the density which depends on the shape parameter \(\alpha\), such that when \(\alpha=0\), it corresponds to the normal density function. Several properties of this class of density function and characterizations are established.

An age dependent branching process with variable lifetime distribution: The generation size

1974 ◽  
Vol 6 (02) ◽  
pp. 291-308 ◽  
Author(s):  
Robert Fildes
Keyword(s):  
Branching Process ◽  
Random Variable ◽  
Lifetime Distribution ◽  
Mean Square ◽  
Initial Particle ◽  
Age Dependent ◽  
The Mean ◽  
Watson Process ◽  
Number Of Particles

In a branching process with variable lifetime, introduced by Fildes (1972) define Yjk (t) as the number of particles alive in generation k at time t when the initial particle is born in generation j. A limit law similar to that derived in the Bellman-Harris process is proved where it is shown that Yjk (t) suitably normalised converges in mean square to a random variable which is the limit random variable of Znm–n in the Galton-Watson process (m is the mean number of particles born).

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