An immigration super-critical branching diffusion process

1972 ◽  
Vol 9 (01) ◽  
pp. 13-23 ◽  
Author(s):  
J. Radcliffe
Keyword(s):  
Diffusion Process ◽  
Random Variable ◽  
Mean Square ◽  
Branching Diffusion ◽  
Mean Square Convergence

This paper is an extension of Davis (1965) by allowing immigration. Mean square convergence is proved for a random variable in a branching diffusion process allowing immigration.

An immigration super-critical branching diffusion process

1972 ◽  
Vol 9 (1) ◽  
pp. 13-23 ◽  
Author(s):  
J. Radcliffe
Keyword(s):  
Diffusion Process ◽  
Random Variable ◽  
Mean Square ◽  
Branching Diffusion ◽  
Mean Square Convergence

This paper is an extension of Davis (1965) by allowing immigration. Mean square convergence is proved for a random variable in a branching diffusion process allowing immigration.

The convergence of a position-dependent branching process used as an approximation to a model describing the spread of an epidemic

1976 ◽  
Vol 13 (2) ◽  
pp. 338-344 ◽  
Author(s):  
J. Radcliffe
Keyword(s):  
Branching Process ◽  
Random Variable ◽  
Mean Square ◽  
Malthusian Parameter ◽  
Markov Branching Process ◽  
Geographical Spread ◽  
Mean Square Convergence ◽  
Velocity Of Propagation ◽  
The Mean ◽  
Number Of Individuals

A supercritical position-dependent Markov branching process has been used as an approximation to a model describing the initial geographical spread of a measles epidemic (Bartlett (1956)). Let α be its Malthusian parameter, ß its velocity of propagation, Z(A, t) the number of individuals in the set A at time t, and A√(ßt) = [√(ßt) r: r ∈ A]. The mean square convergence of the random variable W(A, t)= e–αtZ(A√(ßt), t) to a limit variable W(A) is established.

The convergence of a position-dependent branching process used as an approximation to a model describing the spread of an epidemic

1976 ◽  
Vol 13 (02) ◽  
pp. 338-344 ◽  
Author(s):  
J. Radcliffe
Keyword(s):  
Branching Process ◽  
Random Variable ◽  
Mean Square ◽  
Malthusian Parameter ◽  
Markov Branching Process ◽  
Geographical Spread ◽  
Mean Square Convergence ◽  
Velocity Of Propagation ◽  
The Mean ◽  
Number Of Individuals

A supercritical position-dependent Markov branching process has been used as an approximation to a model describing the initial geographical spread of a measles epidemic (Bartlett (1956)). Letαbe its Malthusian parameter,ßits velocity of propagation,Z(A,t) the number of individuals in the setAat timet,andA√(ßt)= [√(ßt)r:r ∈ A]. The mean square convergence of the random variableW(A, t)= e–αtZ(A√(ßt),t) to a limit variableW(A) is established.

Non-Markovian effect of the fractional damping environment and Newton’s second law of motion

2018 ◽  
Vol 32 (19) ◽  
pp. 1850210
Author(s):  
Chun-Yang Wang ◽  
Zhao-Peng Sun ◽  
Ming Qin ◽  
Yu-Qing Xu ◽  
Shu-Qin Lv ◽  
...  
Keyword(s):  
Diffusion Process ◽  
Mean Square Displacement ◽  
Dynamical Analysis ◽  
Second Law ◽  
Mean Square ◽  
Fractional Damping ◽  
Dynamical Mechanism ◽  
Brownian Particles ◽  
The Mean ◽  
Dissipative Environments

We report, in this paper, a recent study on the dynamical mechanism of Brownian particles diffusing in the fractional damping environment, where several important quantities such as the mean square displacement (MSD) and mean square velocity are calculated for dynamical analysis. A particular type of backward motion is found in the diffusion process. The reason of it is analyzed intrinsically by comparing with the diffusion in various dissipative environments. Results show that the diffusion in the fractional damping environment obeys the Langevin dynamics which is quite different form what is expected.

Some generalizations of Bailey's birth death and migration model

1970 ◽  
Vol 2 (01) ◽  
pp. 83-109 ◽  
Author(s):  
A. W. Davis
Keyword(s):  
Diffusion Process ◽  
Hypergeometric Functions ◽  
Spatial Location ◽  
Death Process ◽  
Birth And Death Process ◽  
Limiting Distributions ◽  
Migration Model ◽  
Branching Diffusion ◽  
And Migration ◽  
Natural Measures

Some results for a general Markov branching-diffusion process are presented, and applied to a model recently considered by Bailey. Moments of the limiting distributions of certain natural measures of the spatial location and dispersion of the population are shown to be expressible in terms of the LauricellaFD-type hypergeometric functions, when the population multiplies according to the simple birth and death process with λ > μ.

Asymptotic behaviour of immigration-branching processes by general set of types. II: Supercritical branching part

1975 ◽  
Vol 7 (03) ◽  
pp. 468-494
Author(s):  
H. Hering
Keyword(s):  
Branching Process ◽  
Branching Processes ◽  
Almost Sure Convergence ◽  
Transition Function ◽  
Mean Square ◽  
Second Moments ◽  
Mean Square Convergence ◽  
The Mean ◽  
Parameter Dependent ◽  
Dependent Probability

We construct an immigration-branching process from an inhomogeneous Poisson process, a parameter-dependent probability distribution of populations and a Markov branching process with homogeneous transition function. The set of types is arbitrary, and the parameter is allowed to be discrete or continuous. Assuming a supercritical branching part with primitive first moments and finite second moments, we prove propositions on the mean square convergence and the almost sure convergence of normalized averaging processes associated with the immigration-branching process.

24.—Mean-square Convergence of Non-harmonic Trigonometrical Series

1976 ◽  
Vol 75 (4) ◽  
pp. 297-323
Author(s):  
J. Cossar
Keyword(s):  
Fourier Series ◽  
Periodic Function ◽  
Mean Square ◽  
Almost Periodic ◽  
Periodic Functions ◽  
Real Numbers ◽  
Trigonometrical Series ◽  
Fixed Positive Number ◽  
Mean Square Convergence ◽  
The Difference

SynopsisThe series considered are of the form , where Σ | cn |2 is convergent and the real numbers λn (the exponents) are distinct. It is known that if the exponents are integers, the series is the Fourier series of a periodic function of locally integrable square (the Riesz-Fischer theorem); and more generally that if the exponents are not necessarily integers but are such that the difference between any pair exceeds a fixed positive number, the series is the Fourier series of a function of the Stepanov class, S2, of almost periodic functions.We consider in this paper cases where the exponents are subject to less stringent conditions (depending on the coefficients cn). Some of the theorems included here are known but had been proved by other methods. A fuller account of the contents of the paper is given in Sections 1-5.

scholarly journals LEARNING FROM THE EXPECTATIONS OF OTHERS

2008 ◽  
Vol 12 (3) ◽  
pp. 345-377 ◽  
Author(s):  
JIM GRANATO ◽  
ERAN A. GUSE ◽  
M. C. SUNNY WONG
Keyword(s):  
Asymmetric Information ◽  
Diffusion Process ◽  
Adaptive Learning ◽  
Information Diffusion ◽  
Multiple Equilibria ◽  
Forecast Error ◽  
Mean Square ◽  
Cobweb Model ◽  
Boomerang Effect ◽  
Forecast Efficiency

This paper explores the equilibrium properties of boundedly rational heterogeneous agents under adaptive learning. In a modified cobweb model with a Stackelberg framework, there is an asymmetric information diffusion process from leading to following firms. It turns out that the conditions for at least one learnable equilibrium are similar to those under homogeneous expectations. However, the introduction of information diffusion leads to the possibility of multiple equilibria and can expand the parameter space of potential learnable equilibria. In addition, the inability to correctly interpret expectations will cause a “boomerang effect” on the forecasts and forecast efficiency of the leading firms. The leading firms' mean square forecast error can be larger than that of following firms if the proportion of following firms is sufficiently large.

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