Weak convergences of probability measures: A uniform principle

2021

We establish a framework for the study of the effective theory of weak convergence of measures. We define two effective notions of weak convergence of measures on R: one uniform and one non-uniform. We show that these notions are equivalent. By means of this equivalence, we prove an effective version of the Portmanteau Theorem, which consists of multiple equivalent definitions of weak convergence of measures.

Statistics & Probability Letters

We propose a new approach to vague convergence of measures based on the general theory of boundedness due to Hu (1966). The article explains how this connects and unifies several frequently used types of vague convergence from the literature. Such an approach allows one to translate already developed results from one type of vague convergence to another. We further analyze the corresponding notion of vague topology and give a new and useful characterization of convergence in distribution of random measures in this topology.

Proceedings of the American Mathematical Society, 1978

Let A" be any topological space, and C(X) the space of bounded continuous functions on X. We give a nonstandard characterization of weak convergence of a net of bounded linear functionals on CiX) to a tight Baire measure on X. This characterization applies whether or not the net or the individual functionals in the net are tight. Moreover, the characterization is expressed in terms of the values of an associated net of countably additive measures on all Baire sets of X; no distinguished family, such as the family of continuity sets of the limit, is involved. As a corollary, we obtain a new proof that a tight set of measures is relatively weakly compact.

Filomat, 2012

It is proved that some classes of sequences of measurable functions satisfy certain selection principles related to special modes of convergence (convergence in measure, almost everywhere convergence, almost uniform convergence, mean convergence).

Journal of Mathematical Analysis and Applications, 1986

We discuss some basic properties of polar convergence in metric spaces. Polar convergence is closely connected with the notion of Delta-convergence of T.C. Lim known for several years. Possible existence of a topology which induces polar convergence is also investigated. Some applications of polar convergence follow.

Stochastics An International Journal of Probability and Stochastic Processes, 2006

Given a sequence (m n ) of random probability measures on a metric space S, consider the conditions: (i) m n /m (weakly) a.s. for some random probability measure m on S; (ii) m n (f) converges a.s. for all f2C b (S). Then, (i) implies (ii), while the converse is not true, even if S is separable. For (i) and (ii) to be equivalent, it is enough that S is Radon (i.e. each probability on the Borel sets of S is tight) or that the sequence (Pm n ) is tight, where Pm n ($)ZE(m n ($)). In particular, (i)5(ii) in case S is Polish. The latter result is still available if a.s. convergence is weakened into convergence in probability. In case SZT N with T Radon, a.s. convergence of m n (f), for those f2C b (S) which are finite products of elements of C b (T), is sufficient for (i). In case SZ R d and the limit m is given in advance, a.s. convergence of characteristic functions is enough for m n /m (weakly) a.s.

2019

In this article, we introduce the space $D([0,1];D)$ of functions defined on $[0,1]$ with values in the Skorohod space $D$, which are right-continuous and have left limits with respect to the $J_1$ topology. This space is equipped with the Skorohod-type distance introduced in Whitt (1980). Following the classical approach of Billingsley (1968, 1999), we give several criteria for tightness of probability measures on this space, by characterizing the relatively compact subsets of this space. In particular, one of these criteria has been used in the recent article Balan and Saidani (2018) for proving the existence of a $D$-valued $\alpha$-stable L\'evy motion. Finally, we give a criterion for weak convergence of random elements in $D([0,1];D)$, and a criterion for the existence of a process with sample paths in $D([0,1];D)$ based on its finite-dimensional distributions.

2021

We expand our effective framework for weak convergence of measures on the real line by showing that effective convergence in the Prokhorov metric is equivalent to effective weak convergence. In addition, we establish a framework for the study of the effective theory of vague convergence of measures. We introduce a uniform notion and a non-uniform notion of vague convergence, and we show that both these notions are equivalent. However, limits under effective vague convergence may not be computable even when they are finite. We give an example of a finite incomputable effective vague limit measure, and we provide a necessary and sufficient condition so that effective vague convergence produces a computable limit. Finally, we determine a sufficient condition for which effective weak and vague convergence of measures coincide. As a corollary, we obtain an effective version of the equivalence between classical weak and vague convergence of sequences of probability measures.

2021

Abstract. A uniformly continuously integrable sequence of real-valued measurable functions, defined on some probability space, is relatively compact in the σ(L, L) topology. In this paper, we link such a result to weak convergence theory of bounded measures as exposed in Billingsley (1968) and in Lo(2021) to offer a detailed and new proof using the ideas beneath the proof of prohorov’s theorem where the continuous integrability replaces the uniform or asymptotic tightness.