An almost sure central limit theorem for products of sums of partial sums under association

Let be a strictly stationary positively or negatively associated sequence of positive random variables with EX₁=μ>0, and VarX₁=σ²<∞. Denote , and γ=σ/μ the coefficient of variation. Under suitable conditions, we show that

     

Endomorphisms of free modules as sums of four quadratic endomorphisms

We will prove that every endomorphism of a free right R-module of infinite rank can be decomposed as a sum of four endomorphisms which satisfy some fixed polynomial identities.     

Higher order differentiability of solutions to backward stochastic differential equations

In this paper, we consider the differentiability in the sense of the Malliavin calculus of solutions to backward stochastic differential equations (BSDEs for short). It is known that a solution is differentiable in the sense of the Malliavin calculus and the derivative is also a solution to a linear BSDE. Under additional conditions, we will show that the higher order differentiability of a solution to a BSDE and that it also becomes a solution to a linear BSDE.     

Reverse order law for generalized inverses of multiple operator product

In this paper, we consider the reverse order law for generalized inverses of operators on Hilbert spaces. We derive necessary and sufficient conditions for various inclusions concerning the reverse order law for generalized inverses of multiple operator product. We extend the finite dimensional results from (Wei M. Reverse order laws for generalized inverses of multiple matrix products. Linear Algebra Appl. 1999;293:13.) to infinite dimensional settings.     

Metrics on the set of semimartingale filtrations

Let X be a stochastic process with cadlag paths. We consider the set of all filtrations under which X is a semimartingale, and we define a distance between such filtrations with the help of the bounded variation parts in the corresponding decompositions of X. We show that this distance is a complete metric on the set of semimartingale filtrations if the given process X is continuous.