Showing papers by "Pietro Rigo published in 2006"
TL;DR: In this paper, the authors considered weak weak convergence of random probability measures on a metric space S and showed that for S = T ∞ with T Radon, a.s. convergence of μ n (f) is sufficient for (i) and (ii) implies (iii) while the converse is not true.
Abstract: Given a sequence (μ n ) of random probability measures on a metric space S, consider the conditions: (i) μ n →μ (weakly) a.s. for some random probability measure μ on S; (ii) μ n (f) converges a.s. for all f∈C 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 (P μ n ) is tight, where Pμ n (·)=E(μ n (·)). In particular, (i)⇔(ii) in case S is Polish. The latter result is still available if a.s. convergence is weakened into convergence in probability. In case S=T ∞ with T Radon, a.s. convergence of μ n (f), for those f∈C b (S) which are finite products of elements of C b (T), is sufficient for (i). In case and the limit μ is given in advance, a.s. convergence of characteristic functions is enough for μ n →μ (weakly) a.s. Almost sure weak convergence of random probability measures.
64 citations
TL;DR: In this paper, it was shown that the Skorohod space with separable probability measures can be taken for some H⊂ (0, 1) with outer Lebesgue measure 1.
Abstract: Let \((\Omega,\mathcal{A},P)\) be a probability space, S a metric space, μ a probability measure on the Borel σ-field of S, and \(X_n:\Omega\rightarrow S\) an arbitrary map, n = 1,2,.... If μ is tight and Xn converges in distribution to μ (in Hoffmann–Jorgensen’s sense), then X∼μ for some S-valued random variable X on \((\Omega,\mathcal{A},P)\). If, in addition, the Xn are measurable and tight, there are S-valued random variables \(\overset{\sim}{X}_n\) and X, defined on \((\Omega,\mathcal{A},P)\), such that \(\overset{\sim}{X}_n\sim X_n\), X∼μ, and \(\overset{\sim}{X}_{n_k}\rightarrow X\) a.s. for some subsequence (nk). Further, \(\overset{\sim}{X}_n\rightarrow X\) a.s. (without need of taking subsequences) if μ{x} = 0 for all x, or if P(Xn = x) = 0 for some n and all x. When P is perfect, the tightness assumption can be weakened into separability up to extending P to \(\sigma(\mathcal{A}\cup\{H\})\) for some H⊂Ω with P*(H) = 1. As a consequence, in applying Skorohod representation theorem with separable probability measures, the Skorohod space can be taken \(((0,1),\sigma(\mathcal{U}\cup\{H\}),m_H)\), for some H⊂ (0,1) with outer Lebesgue measure 1, where \(\mathcal{U}\) is the Borel σ-field on (0,1) and mH the only extension of Lebesgue measure such that mH(H) = 1. In order to prove the previous results, it is also shown that, if Xn converges in distribution to a separable limit, then Xnk converges stably for some subsequence (nk).
16 citations
TL;DR: In this paper, it was shown that the convergence in distribution of non-measurable random elements can be achieved by considering the random element X n as a random element of the space of real cadlag functions on R with finite limits.
9 citations
TL;DR: In this article, the 0-1 condition for a sub-sigma sub-field is shown to hold for some classically interesting sub-sub-$sigma-fields, including symmetric, invariant, and invariant to continuous time processes.
Abstract: Let $(\Omega,\mathcal{B},P)$ be a probability space, $\mathcal{A}\subset\mathcal{B}$ a sub-$\sigma$-field, and $\mu$ a regular conditional distribution for $P$ given $\mathcal{A}$. Necessary and sufficient conditions for $\mu(\omega)(A)$ to be 0--1, for all $A\in\mathcal{A}$ and $\omega\in A_0$, where $A_0\in\mathcal{A}$ and $P(A_0)=1$, are given. Such conditions apply, in particular, when $\mathcal{A}$ is a tail sub-$\sigma$-field. Let $H(\omega)$ denote the $\mathcal{A}$-atom including the point $\omega\in\Omega$. Necessary and sufficient conditions for $\mu(\omega)(H(\omega))$ to be 0--1, for all $\omega\in A_0$, are also given. If $(\Omega,\mathcal{B})$ is a standard space, the latter 0--1 law is true for various classically interesting sub-$\sigma$-fields $\mathcal{A}$, including tail, symmetric, invariant, as well as some sub-$\sigma$-fields connected with continuous time processes.