Invariance principle for non-homogeneous random walks

@article{Georgiou2019InvariancePF,
  title={Invariance principle for non-homogeneous random walks},
  author={Nicholas Georgiou and Aleksandar Mijatovi'c and Andrew R. Wade},
  journal={Electronic Journal of Probability},
  year={2019}
}
We prove an invariance principle for a class of zero-drift spatially non-homogeneous random walks in $\mathbb{R}^d$, which may be recurrent in any dimension. The limit $\mathcal{X}$ is an elliptic martingale diffusion, which may be point-recurrent at the origin for any $d\geq2$. To characterise $\mathcal{X}$, we introduce a (non-Euclidean) Riemannian metric on the unit sphere in $\mathbb{R}^d$ and use it to express a related spherical diffusion as a Brownian motion with drift. This… Expand
4 Citations
Projections of spherical Brownian motion
We obtain a stochastic differential equation (SDE) satisfied by the first $n$ coordinates of a Brownian motion on the unit sphere in $\mathbb{R}^{n+\ell}$. The SDE has non-Lipschitz coefficients butExpand
Invariance principles for local times in regenerative settings
Consider a stochastic process $\mathfrak{X}$, regenerative at a state $x$ which is instantaneous and regular. Let $L$ be a regenerative local time for $\mathfrak{X}$ at $x$. Suppose furthermore thatExpand
A note on the exact simulation of spherical Brownian motion
We describe an exact simulation algorithm for the increments of Brownian motion on a sphere of arbitrary dimension, based on the skew-product decomposition of the process with respect to the standardExpand
Cosmic CARNage II: the evolution of the galaxy stellar mass function in observations and galaxy formation models
We present a comparison of the observed evolving galaxy stellar mass functions with the predictions of eight semi-analytic models and one halo occupation distribution model. While most models areExpand

References

SHOWING 1-10 OF 35 REFERENCES
Convergence of integral functionals of one-dimensional diffusions
In this paper we describe the pathwise behaviour of the integral functional $\int_0^t f(Y_u)\,du$ for any $t\in[0,\zeta]$, where $\zeta$ is (a possibly infinite) exit time of a one-dimensionalExpand
Inversion, duality and Doob $h$-transforms for self-similar Markov processes
We show that any $\mathbb{R}^d\setminus\{0\}$-valued self-similar Markov process $X$, with index $\alpha>0$ can be represented as a path transformation of some Markov additive process (MAP)Expand
Anomalous recurrence properties of many-dimensional zero-drift random walks
Abstract Famously, a d-dimensional, spatially homogeneous random walk whose increments are nondegenerate, have finite second moments, and have zero mean is recurrent if d∈{1,2}, but transient if d≥3.Expand
A radial invariance principle for non-homogeneous random walks
Consider non-homogeneous zero-drift random walks in Rd, d≥2, with the asymptotic increment covariance matrix σ2(u) satisfying u⊤σ2(u)u=U and trσ2(u)=V in all in directions u∈Sd−1 for some positiveExpand
Eigenvalue expansions for diffusion hitting times
Consider a non-singular diffusion on an interval (ro, rt) and let r 0 < a < b < r~. Set zab to be the first time the diffusion hits b, starting at a, with moment generating function (m.g.f.)Expand
Excursion theory for rotation invariant Markov processes
SummaryLet (Xt,Px) be a rotation invariant (RI) strong Markov process onRd{0} having a skew product representation [|Xt|, $$\theta _{A_t }$$ ], where (θt) is a time homogeneous, RI strong MarkovExpand
Convergence of probability measures
The author's preface gives an outline: "This book is about weakconvergence methods in metric spaces, with applications sufficient to show their power and utility. The Introduction motivates theExpand
Continuous martingales and Brownian motion
0. Preliminaries.- I. Introduction.- II. Martingales.- III. Markov Processes.- IV. Stochastic Integration.- V. Representation of Martingales.- VI. Local Times.- VII. Generators and Time Reversal.-Expand
Positive Harmonic Functions and Diffusion
1. Existence and uniqueness for diffusion processes 2. The basic properties of diffusion processes 3. The spectral theory of elliptic operators on smooth bounded domains 4. Generalized spectralExpand
Convergence of Some Integrals Associated with Bessel Processes
We study the convergence of the Lebesgue integrals for the processes $f(\rho_t)$. Here, $(\rho_t,\,t\ge0)$ is the $\delta$-dimensional Bessel process started at $\rho_0\ge0$ and~f is a positive BorelExpand
...
1
2
3
4
...