Weakly approaching sequences of random distributions

Abstract

We introduce the notion of weakly approaching sequences of distributions, which is a generalization of the well-known concept of weak convergence of distributions. The main difference is that the suggested notion does not demand the existence of a limit distribution. A similar definition for conditional (random) distributions is presented. Several properties of weakly approaching sequences are given. The tightness of some of them is essential. The Cramér-Lévy continuity theorem for weak convergence is generalized to weakly approaching sequences of (random) distributions. It has several applications in statistics and probability. A few examples of applications to resampling are given.

Approach to Analysis of Self-Selected Interval Data

“…Then the empirical distribution of differencesθ c n −θ n , c = 1, ..., R, will imitate the distribution of deviationθ n − θ 0 . The proof of this fact based on the Central Limit Resampling Theorem, Belyaev (2003Belyaev ( , 2007, Belyaev and Sjöstedt-de Luna (2000). A detailed theory for consistent estimation deviations of the ML-estimators of parameters based on resamplings is given in Nilsson (1998).…”

Approaching in Distribution with Applications to Resampling of Stochastic Processes

“…Aiming our statistical applications, we mainly consider weak and conditionally weak approaching in probability although conditional a.s.-approaching can be introduced and studied in a similar way (cf. Belyaev, 1996). It follows directly by the de®nitions and the Lebesgue dominated convergence theorem, that (X n j Z n ) 23 ad(P) Y n implies X n 6 ad Y n .…”

“…Spectral resampling for stationary random processes is considered in Nordgaard (1992) (see also Hjorth, 1994). For realvalued and vector-valued random elements, approaching and conditional approaching were introduced and investigated in Belyaev (1995Belyaev ( , 1996Belyaev ( , 1997, and Belyaev & Sjo Èstedt (1996, respectively. The theory of weak convergence and the weak approaching of sequences have a lot of useful applications in probability theory and statistics.…”

Maximum likelihood estimation for survey data with informative interval censoring

“…For obtaining asymptotic distributional results about √ n( θ − θ 0 ) we will use the notion of weakly approaching sequences of distributions (Belyaev and Sjöstedt-de Luna 2000), which is a generalization of the well-known concept of weak convergence of distributions but without the need to have a limiting distribution. Two sequences of random variables, {X n } n≥1 and {Y n } n≥1 , are said to have weakly approaching distribution laws, {L(X n )} n≥1 and {L(Y n )} n≥1 , if for every bounded continuous function ϕ(·), E ϕ(X n )−E ϕ(Y n ) −→ 0 as n −→ ∞.…”