Mean square convergence of weak martingales

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.

Download Full-text

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

Advances in Applied Probability ◽

10.1017/s000186780004074x ◽

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.

Download Full-text

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

Proceedings of the Royal Society of Edinburgh Section A Mathematics ◽

10.1017/s0308210500013342 ◽

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.

Download Full-text

Strong Convergence Analysis of Split-Step θ-Scheme for Nonlinear Stochastic Differential Equations with Jumps

Advances in Applied Mathematics and Mechanics ◽

10.4208/aamm.2015.m1208 ◽

2016 ◽

Vol 8 (6) ◽

pp. 1004-1022 ◽

Cited By ~ 1

Author(s):

Xu Yang ◽

Weidong Zhao

Keyword(s):

Differential Equations ◽

Theoretical Result ◽

Strong Convergence ◽

Stochastic Differential Equations ◽

Rate Of Convergence ◽

Numerical Experiments ◽

Mean Square ◽

Nonlinear Stochastic Differential Equations ◽

Mean Square Convergence ◽

The Mean

AbstractIn this paper, we investigate the mean-square convergence of the split-step θ-scheme for nonlinear stochastic differential equations with jumps. Under some standard assumptions, we rigorously prove that the strong rate of convergence of the split-step θ-scheme in strong sense is one half. Some numerical experiments are carried out to assert our theoretical result.

Download Full-text