We define a notion of approximate sufficiency and approximate ancillarity and show that such statistics are approximately independent pointwise under each value of the parameter. We do so without mentioning the somewhat nonintuitive concept of completeness, thus providing a more transparent version of Basu’s theorem. Two total variation inequalities are given, which we call approximate Basu the...

We consider a continuous, infinitely divisible random field in Rd given as an integral of a kernel function with respect to a Lévy basis with convolution equivalent Lévy measure. For a large class of such random fields we compute the asymptotic probability that the supremum of the field exceeds the level x as x→∞. Our main result is that the asymptotic probability is equivalent to the right tai...

Let $m_G$ denote the number of perfect matchings graph $G$. We introduce a combinatorial tools for determining parity and giving lower bound on power 2 dividing $m_G$. In particular, we certain vertex sets called channels, which correspond to elements in kernel adjacency matrix $G$ modulo $2$. A result Lov\'asz states that existence nontrivial channel is equivalent being even. give new proof th...

For a non-commutative ring R, we consider factorizations of polynomials in R[t] where t is central variable. A pseudo-root polynomial p(t)=p0+p1t+?pktk an element ??R, for which there exist q1,q2 such that p=q1(t??)q2. We investigate the rational relationships hold among pseudo-roots p(t) by using diamond operations cover graphs modular lattices. When R division ring, each finite subset S corre...

In a previous paper ([B-G1]), we defined the rectangular free convolution ⊞ λ . Here, we investigate the related notion of infinite divisibility, which happens to be closely related the classical infinite divisibility: there exists a bijection between the set of classical symmetric infinitely divisible distributions and the set of ⊞ λ -infinitely divisible distributions, which preserves limit t...

In this paper we consider the Z-module of integer-valued functions / defined on the nonnegative integers (respectively, on all integers) and characterize the submodule determined by the divisibility relation of the title and also, as a corollary, by the divisibility relation m+n\f(m)+f(n). Our results suggest some rather basic questions about such modules (equivalently, about infinite matrices ...