The Experts below are selected from a list of 285 Experts worldwide ranked by ideXlab platform

Lisa D. Peterson - One of the best experts on this subject based on the ideXlab platform.

  • Convergence in Distribution of point processes on Polish spaces to a simple limit
    Statistics & Probability Letters, 2011
    Co-Authors: Lisa D. Peterson
    Abstract:

    Abstract Let ξ , ξ 1 , ξ 2 , … be a sequence of point processes on a complete and separable metric space ( S , d ) with ξ simple. We assume that P { ξ n B = 0 } → P { ξ B = 0 } and lim sup n → ∞ P { ξ n B > 1 } ≤ P { ξ B > 1 } for all B in some suitable class B , and show that this assumption determines if the sequence { ξ n } converges in Distribution to ξ . This is an extension to general Polish spaces of the weak Convergence theory for point processes on locally compact Polish spaces found in Kallenberg (1996) .

  • Convergence in Distribution of random compact sets in Polish spaces
    Statistics & Probability Letters, 2008
    Co-Authors: Hussain Elalaoui-talibi, Lisa D. Peterson
    Abstract:

    Abstract Let φ , φ 1 , φ 2 , … be a sequence of random compact sets on a complete and separable metric space ( S , d ) . We assume that P { φ n ∩ B = ∅ } → P { φ ∩ B = ∅ } for all B in some suitable class B and show that this assumption determines if the sequence { φ n } converges in Distribution to φ . This is an extension to general Polish spaces of the weak Convergence theory for random closed sets on locally compact Polish spaces found in Norberg [1984. Convergence and existence of random set Distributions. Ann. Probab. 12, 726–732.]

Arunava Mukherjea - One of the best experts on this subject based on the ideXlab platform.

  • Convergence in Distribution of Products of I.I.D. Nonnegative Matrices
    Journal of Theoretical Probability, 1997
    Co-Authors: Subhankar Dhar, Arunava Mukherjea
    Abstract:

    Let (X i) be a sequence of m × m i.i.d. stochastic matrices with Distribution μ. Then μ n is the Distribution of X n X n−1 ...X 1. Simple sufficient conditions for the weak Convergence of (μ n ) are presented here. An extremely simple (and verifiable) necessary and sufficient condition is provided for m= 3. The method for m= 3 works for m> 3 even though calculations are more involved for higher values of m. We also discuss the purity of the limit Distribution for m≥2.

  • Convergence in Distribution of a Markov Process Generated by I.I.D. Random Matrices
    Diffusion Processes and Related Problems in Analysis Volume II, 1992
    Co-Authors: Arunava Mukherjea
    Abstract:

    1. The aim of this paper is to study a discrete time Markov process (η n ) on the state space S = {0, 1} V , V a countable set (of sites), where the transition rule is governed by a |V|-dimensional random matrix X with nonnegative integers as entries. Thus, we have: $${\eta _{n + 1}} = {X_{n + 1}}{\eta _n},{\text{ }}n \geqslant 0$$ (1.1) where (X j ) is an i.i.d sequence of copies of X (independent of η 0). We are interested in the asymptotic behavior of (η n ). We will also discuss in the last section a finite dimensional analog of this problem when S = [0, ∞)V, |V| = d < ∞, and X is a d × d nonnegative matrix. But, in what follows (in the first four sections), the discussion is restricted to the infinite dimensional context.

  • Convergence in Distribution of products of d × d random matrices
    Journal of Mathematical Analysis and Applications, 1991
    Co-Authors: Arunava Mukherjea
    Abstract:

    Abstract in this paper we present limit theorems on the Convergence in Distribution of products of i.i.d. matrices based on easily verifiable conditions on the support of the Distribution.

Pietro Rigo - One of the best experts on this subject based on the ideXlab platform.

  • Convergence in Distribution of nonmeasurable random elements
    Annals of Probability, 2004
    Co-Authors: Patrizia Berti, Pietro Rigo
    Abstract:

    A notion of Convergence in Distribution for non (necessarily) measurable random elements, due to Hoffmann-Jorgensen, is characterized in terms of weak Convergence of finitely additive probability measures. A similar characterization is given for a strengthened version of such a notion. Further, it is shown that the empirical process for an exchangeable sequence can fail to converge, due to the nonexistence of any measurable limit, although it converges for an i.i.d. sequence. Because of phenomena of this type, Hoffmann-Jorgensen's definition is extended to the case of a nonmeasurable limit. in the extended definition, naturally suggested by the main results, the limit is a finitely additive probability measure.

Guillaume Poly - One of the best experts on this subject based on the ideXlab platform.

  • Convergence in Distribution norms in the CLT for non identical distributed random variables
    Electronic Journal of Probability, 2018
    Co-Authors: Vlad Bally, Lucia Caramellino, Guillaume Poly
    Abstract:

    We study the Convergence in Distribution norms in the Central Limit Theorem for non identical distributed random variables. We also consider local developments (Edgeworth expansion). This kind of results is well understood in the case of smooth test functions f. If one deals with measurable and bounded test functions (Convergence in total variation distance), a well known theorem due to Prohorov shows that some regularity condition for the law of the random variables Xn, n∈N, on hand is needed. Essentially, one needs that the law of Xn is locally lower bounded by the Lebesgue measure (Doeblin's condition). This topic is also widely discussed in the literature (see the book by Battacharaya and Rao). Our main contribution is to discuss Convergence in Distribution norms, that is to replace the test function f by some derivative ∂αf and to obtain upper bounds for εn(∂αf) in terms of the original function f.

  • Convergence in Distribution norms in the CLT for non identical distributed random variables
    Electronic Journal of Probability, 2018
    Co-Authors: Vlad Bally, Lucia Caramellino, Guillaume Poly
    Abstract:

    We study the Convergence in Distribution norms in the Central Limit Theorem for non identical distributed random variables that is \[ \varepsilon _{n}(f):={\mathbb{E} }\Big (f\Big (\frac 1{\sqrt n}\sum _{i=1}^{n}Z_{i}\Big )\Big )-{\mathbb{E} }\big (f(G)\big )\rightarrow 0 \] where $Z_{i}$, $i\in \mathbb{N} $, are centred independent random variables and $G$ is a Gaussian random variable. We also consider local developments (Edgeworth expansion). This kind of results is well understood in the case of smooth test functions $f$. If one deals with measurable and bounded test functions (Convergence in total variation distance), a well known theorem due to Prohorov shows that some regularity condition for the law of the random variables $Z_{i}$, $i\in{\mathbb {N}} $, on hand is needed. Essentially, one needs that the law of $Z_{i}$ is locally lower bounded by the Lebesgue measure (Doeblin’s condition). This topic is also widely discussed in the literature. Our main contribution is to discuss Convergence in Distribution norms, that is to replace the test function $f$ by some derivative $\partial _{\alpha }f$ and to obtain upper bounds for $\varepsilon _{n}(\partial _{\alpha }f)$ in terms of the infinite norm of $f$. Some applications are also discussed: an invariance principle for the occupation time for random walks, small balls estimates and expected value of the number of roots of trigonometric polynomials with random coefficients.

Dan A. Ralescu - One of the best experts on this subject based on the ideXlab platform.

  • Convergence in Distribution for Uncertain Random Variables
    IEEE Transactions on Fuzzy Systems, 2018
    Co-Authors: Rong Gao, Dan A. Ralescu
    Abstract:

    A random variable is a measurable function from an uncertainty space to the set of real numbers, which is used to model randomness. An uncertain variable is a measurable function from uncertainty space to the set of real numbers, which is used to describe uncertainty. However, randomness and uncertainty often simultaneously appear in a complex system. Uncertain random variable provides a useful tool to handle such a hybrid case. This concept integrates random variable and uncertain variable into a broader view. For uncertain random variables, a basic and important topic is to discuss the Convergence of its sequence. Specifically, this paper focuses on studying the Convergence in Distribution for a sequence of uncertain random variables without a common chance Distribution.