On the increments of the local time of a Wiener sheet

{W(x, y), x≥0, y≥0} be a Wiener process and let η(u, (x, y)) be its local time. The continuity of η in (x, y) is investigated, i.e., an upper estimate of the process η(μ, [x, x + α) × [y, y + β)) is given when αβ is small.     

Asymptotic expansions in functional limit theorems

Asymptotic expansions for a class of functional limit theorems are investigated. It is shown that the expansions in this class fit into a common scheme, defined by a sequence of functions h_n (ε₁,…, ε_n), n ≥ 1, of “weights” (for n observations), which are smooth, symmetric, compatible and have vanishing first derivatives at zero. Then $\text{h}{}_{\mathrm{n}}\text{(n}{}^{\mathrm{<mtext>?1</mtext><mtext>2</mtext>}}\text{,…, n}{}^{\mathrm{<mtext>?1</mtext><mtext>2</mtext>}}\text{)}$ admits an asymptotic expansion in powers of $\text{n}{}^{\mathrm{<mtext>?1</mtext><mtext>2</mtext>}}$. Applications to quadratic von Mises functionals, the C.L.T. in Banach spaces, and the invariance principle are discussed.     

Orientor field equations in Banach spaces

This paper focuses on the interplay between seminormality requirements and convergence hypotheses on trajectories in the lower closure theorems for orientor field equations. With the use of a weak form of seminormality, called the intermediate property (Q*), we obtain lower closure theorems (and thereby closure theorems) for orientor field equations in Banach spaces, under the assumption of strong convergence of some coordinates of the trajectories, while only weak convergence is assumed in the others. In the Euclidean case, this requirement of property (Q*) is further reduced to mere Kuratowski property (K), under the usual growth-type conditions. Finally, in the appendix, property (Q*) is investigated in detail.This research has been partially supported by Research Project AFOSR-71-2122 at the University of Michigan, Ann Arbor, Michigan.     

局部化函子的扩张问题

本文考虑了经典ｐ局部化函子在相应的范畴的最小正交偶扩张与最大正交偶扩张是否联系局部化函子的问题．这里ｐ是素数或零．     

下半连续函数的Proximal—次微分与广义中值定理

该文对定义在Hilbert空间E上的一般下半连续函数证明了如［9］中形式的逼近中值定理在Proximal-次微分意义下也成立．若E＝［a，b］R，则得到了不等式形式的中值定理．作为应用给出了函数凸性、Lipschitz性质及常数性质的Proximal-次微分刻划．     

Linear systems and normality

This paper is concerned with responses of linear systems to non-Gaussian random excitation and with the measurement of the departure of the responses from Gaussian behavior. First, we show the classical Rosenblatt result and its nonapplicability to the most popular practical systems described by differential equations of first and second order. Then, using a simple measure of departure from normality (the asymmetry and excess coefficients) and performing numerical calculations, we give quantitative information about the effect of system parameters and the radius of correlation of the excitation process on the distance from normality.     

Boolean Valued and Stone Algebra Valued Measure Theories

In conventional generalization of the main results of classical measure theory to Stone algebra valued measures, the values that measures and functions can take are Booleanized, while the classical notion of a σ-field is retained. The main purpose of this paper is to show by abundace of illustrations that if we agree to Booleanize the notion of a σ-field as well, then all the glorious legacy of classical measure theory is preserved completely. Mathematics Subject Classification: 03C90, 28B15.