scholarly journals On Weak Limiting Distributions for Random Walks on a Spider

Symmetry ◽  
2020 ◽  
Vol 12 (12) ◽  
pp. 2000
Author(s):  
Youngsoo Seol
Keyword(s):  
Brownian Motion ◽  
Random Walks ◽  
Classical Case ◽  
Weak Limit ◽  
Skew Brownian Motion ◽  
Finite Dimensional ◽  
Finite Dimensional Distribution ◽  
Dimensional Distribution ◽  
Functional Central Limit ◽  
Special Case

In this article, we study random walks on a spider that can be established from the classical case of simple symmetric random walks. The primary purpose of this article is to establish a functional central limit theorem for random walks on a spider and to define Brownian spider as the resulting weak limit. In special case, random walks on a spider can be characterized as skew random walks. It is well known for skew Brownian motion as the resulting weak limit of skew random walks. We first will study the tightness and then it will be shown for the convergence of finite dimensional distribution for random walks on a spider.

Weak convergence of random walks, conditioned to stay away from small sets

2013 ◽  
Vol 50 (1) ◽  
pp. 122-128
Author(s):  
Zsolt Pajor-Gyulai ◽  
Domokos Szász
Keyword(s):  
Brownian Motion ◽  
Random Walk ◽  
Weak Convergence ◽  
Random Walks ◽  
Continuous Process ◽  
Random Variables ◽  
Weak Limit ◽  
Limit Laws ◽  
Diffusive Limit

Let {Xn}n∈ℕ be a sequence of i.i.d. random variables in ℤd. Let Sk = X1 + … + Xk and Yn(t) be the continuous process on [0, 1] for which Yn(k/n) = Sk/n1/2 for k = 1, … n and which is linearly interpolated elsewhere. The paper gives a generalization of results of ([2]) on the weak limit laws of Yn(t) conditioned to stay away from some small sets. In particular, it is shown that the diffusive limit of the random walk meander on ℤd: d ≧ 2 is the Brownian motion.

The point-process approach to age- and time-dependent branching processes

1983 ◽  
Vol 15 (01) ◽  
pp. 1-20
Author(s):  
Marek Kimmel
Keyword(s):  
Point Process ◽  
Branching Process ◽  
Cell Kinetics ◽  
Branching Processes ◽  
Finite Dimensional ◽  
Finite Dimensional Distribution ◽  
Symbolic Method ◽  
Age Dependent ◽  
Dimensional Distribution ◽  
Varying Environment

The multitype age-dependent branching process in varying environment is treated as a random stream (point process) of the birth and death events. We derive the point-process version of the Bellman–Harris equation, which provides us with a symbolic method of writing the equations for the arbitrary finite-dimensional distribution of the process. We also derive a new recurrence relation, the ‘principle of last generation'. This result is applied to obtain new moment equations for the process with immigration also. General considerations are illustrated by a simple cell kinetics model.

The point-process approach to age- and time-dependent branching processes

1983 ◽  
Vol 15 (1) ◽  
pp. 1-20 ◽  
Author(s):  
Marek Kimmel
Keyword(s):  
Point Process ◽  
Branching Process ◽  
Cell Kinetics ◽  
Branching Processes ◽  
Finite Dimensional ◽  
Finite Dimensional Distribution ◽  
Symbolic Method ◽  
Age Dependent ◽  
Dimensional Distribution ◽  
Varying Environment

The multitype age-dependent branching process in varying environment is treated as a random stream (point process) of the birth and death events. We derive the point-process version of the Bellman–Harris equation, which provides us with a symbolic method of writing the equations for the arbitrary finite-dimensional distribution of the process. We also derive a new recurrence relation, the ‘principle of last generation'. This result is applied to obtain new moment equations for the process with immigration also. General considerations are illustrated by a simple cell kinetics model.

Archimedean copulas, exchangeability, and max-stability

1994 ◽  
Vol 31 (2) ◽  
pp. 383-390 ◽  
Author(s):  
Rocco Ballerini
Keyword(s):  
Random Variables ◽  
Distribution Functions ◽  
Archimedean Copula ◽  
Archimedean Copulas ◽  
Monotone Transformation ◽  
Stable Property ◽  
Finite Dimensional ◽  
Finite Dimensional Distribution ◽  
Exchangeable Sequence ◽  
Dimensional Distribution

An exchangeable sequence of random variables is constructed with all finite-dimensional distribution functions having an Archimedean copula (as defined by Schweizer and Sklar (1983)). Through a monotone transformation of this exchangeable sequence, we obtain and characterize the class of exchangeable sequences possessing the max-stable property as defined by De Haan and Rachev (1989). Several parametric examples are given.

scholarly journals THE SMOLYANOV SURFACE MEASURE ON TRAJECTORIES IN A RIEMANNIAN MANIFOLD

2004 ◽  
Vol 07 (03) ◽  
pp. 461-471 ◽  
Author(s):  
NADEZDA A. SIDOROVA
Keyword(s):  
Brownian Motion ◽  
Riemannian Manifold ◽  
Compact Riemannian Manifold ◽  
Direct Proof ◽  
Weak Limit ◽  
Natural Object ◽  
Surface Measure ◽  
Definition Of ◽  
Special Case ◽  
Restrictive Definition

It has been shown in Refs. 2–6 that two natural definitions of surface measures, on the space of continuous paths in a compact Riemannian manifold embedded into ℝn, introduced in the paper by Smolyanov1 are equivalent; this means that there exists a natural object — the surface measure, which we call the Smolyanov surface measure. Moreover, it has been shown2–6 that this surface measure is equivalent to the Wiener measure and the corresponding density has been found. But the known proof of the equivalence of the two definitions of the surface measure is rather nonexplicit; in fact the densities of the measures corresponding to the two different definitions were found independently and only a posteriori it was discovered that those densities coincided. Our aim is to give a direct proof of this fact. We introduce a more restrictive definition of the surface measure as the weak limit of a standard Brownian motion in ℝn conditioned to be in the tubular ε-neighborhood of the manifold at times 0=t0<t1<⋯<tn-1<tn= 1 as both ε and the diameter of the partition tend to zero. Letting ε and then the diameter of the partition tend to zero and vice versa, we arrive at the two definitions above. We prove the existence of the Smolyanov surface measure using our definition, show that this measure is equivalent to the law of a Brownian motion on the manifold, and compute the corresponding density in terms of the curvature of the manifold. As a special case of this, we again obtain the results of Refs. 2–6.

Archimedean copulas, exchangeability, and max-stability

1994 ◽  
Vol 31 (02) ◽  
pp. 383-390 ◽  
Author(s):  
Rocco Ballerini
Keyword(s):  
Random Variables ◽  
Distribution Functions ◽  
Archimedean Copula ◽  
Archimedean Copulas ◽  
Monotone Transformation ◽  
Stable Property ◽  
Finite Dimensional ◽  
Finite Dimensional Distribution ◽  
Exchangeable Sequence ◽  
Dimensional Distribution

An exchangeable sequence of random variables is constructed with all finite-dimensional distribution functions having an Archimedean copula (as defined by Schweizer and Sklar (1983)). Through a monotone transformation of this exchangeable sequence, we obtain and characterize the class of exchangeable sequences possessing the max-stable property as defined by De Haan and Rachev (1989). Several parametric examples are given.

scholarly journals Stability Estimates for Finite-Dimensional Distributions of Time-Inhomogeneous Markov Chains

Mathematics ◽  
2020 ◽  
Vol 8 (2) ◽  
pp. 174
Author(s):  
Vitaliy Golomoziy ◽  
Yuliya Mishura
Keyword(s):  
Markov Chains ◽  
Transition Probabilities ◽  
Absolute Difference ◽  
Stability Estimates ◽  
Finite Dimensional ◽  
Finite Dimensional Distribution ◽  
Inhomogeneous Markov Chain ◽  
Dimensional Distribution ◽  
General State Space ◽  
The Stability

This paper is devoted to the study of the stability of finite-dimensional distribution of time-inhomogeneous, discrete-time Markov chains on a general state space. The main result of the paper provides an estimate for the absolute difference of finite-dimensional distributions of a given time-inhomogeneous Markov chain and its perturbed version. By perturbation, we mean here small changes in the transition probabilities. Stability estimates are obtained using the coupling method.

Poisson law for Axiom A diffeomorphisms

1993 ◽  
Vol 13 (3) ◽  
pp. 533-556 ◽  
Author(s):  
Masaki Hirata
Keyword(s):  
Point Process ◽  
Gibbs Measure ◽  
Poisson Point Process ◽  
Axiom A ◽  
Lipschitz Continuous ◽  
Return Times ◽  
Finite Dimensional ◽  
Finite Dimensional Distribution ◽  
Dimensional Distribution ◽  
Continuous Potential

AbstractLet f be an Axiom A diffeomorphism, Ω its non wandering set, µ the Gibbs measure for the Lipschitz continuous potential. We consider the (suitably normalized) return times of the orbit to the ε-neighborhood of a point z ∈ Ω and prove that for µ-a.e. z the sequence of the normalized return times converges to the Poisson point process in finite dimensional distribution as ε → 0.

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