Weak convergence with random indices

Weak Convergence of Random Functions

Let {vij}, i, j = 1, 2, . . . , be i.i.d. symmetric random variables with E(v 11) < ∞, and for each n let Mn = 1 sVnV T n , where Vn = (vij), i = 1, 2, . . . , n, j = 1, 2, . . . , s = s(n), and n/s → y > 0 as n → ∞. Denote by OnΛnO n the spectral decomposition of Mn. Define X ∈ D[0, 1] by Xn(t) = √ n 2 ∑[nt] i=1(y 2 i − 1 n), where (y1, y2, . . . , yn) = O (± 1 √ n ,± 1 √ n , . . . ,± 1 √ n ) ...

Weak Convergence of Random Sets

In this paper the classical Portmanteau theorem which provides equivalent conditions of weak convergence of sequence of probability measures is extended on the space of the sequence of probability measures induced by random sets.

On weak convergence of functionals on smooth random functions ∗

The numbers of level crossings and extremes for random processes and fields play an important role in reliability theory and many engineering applications. In many cases for Gaussian processes the Poisson approximation for their asymptotic distributions is used. This paper extends an approach proposed in Rusakov and Seleznjev (1988) for smooth random processes on a finite interval. It turns out...

The Weak Convergence for Functions of Negatively Associated Random Variables

Weak Convergence and Weak* Convergence1

In this article, we deal with weak convergence on sequences in real normed spaces, and weak* convergence on sequences in dual spaces of real normed spaces. In the first section, we proved some topological properties of dual spaces of real normed spaces. We used these theorems for proofs of Section 3. In Section 2, we defined weak convergence and weak* convergence, and proved some properties. By...