In this paper, we consider the linear and circular consecutive $k$-out-of-$n$ systems consisting of arbitrarily dependent components. Under the condition that at least $n-r+1$ components ($r\le n$) of the system are working at time $t$, we study the reliability properties of the residual lifetime of such systems. Also, we present some stochastic ordering properties of residual lifetime of consecutive $k$-out-of-$n$ systems. In the following, we investigate the inactivity time of the component with lifetime...

For lower-semicontinuous and convex stochastic processes $Z_n$ and nonnegative random variables ${ϵ}_n$ we investigate the pertaining random sets $A(Z_n,{ϵ}_n)$ of all ${ϵ}_n$-approximating minimizers of $Z_n$. It is shown that, if the finite dimensional distributions of the $Z_n$ converge to some $Z$ and if the ${ϵ}_n$ converge in probability to some constant $c$, then the $A(Z_n,{ϵ}_n)$ converge in distribution to $A(Z,c)$ in the hyperspace of Vietoris. As a simple corollary we obtain an extension of several argmin-theorems in the literature. In particular,...

We consider “nonconventional” averaging setup in the form $\frac{\mathrm{d}{X}^{\epsilon }\left(t\right)}{\mathrm{d}t}=\epsilon B(X^{\epsilon }\left(t\right)$, $𝛯\left(q_1\left(t\right)\right),𝛯\left(q_2\left(t\right)\right),...,𝛯\left(q_{\ell }\left(t\right)\right))$ where $𝛯\left(t\right)$, $t\ge 0$ is either a stochastic process or a dynamical system with sufficiently fast mixing while $q_j\left(t\right)={\alpha }_{j}t$, ${\alpha }_{1}lt;{\alpha }_{2}lt;\cdots lt;{\alpha }_k$ and $q_j$, $j=k+1,...,\ell$ grow faster than linearly. We show that the properly normalized error term in the “nonconventional” averaging principle is asymptotically Gaussian.

For $X,Y\in {𝐌}_{n,m}$ it is said that $X$ is gut-majorized by $Y$, and we write $X{\prec }_{\mathrm{gut}}Y$, if there exists an $n$-by-$n$ upper triangular g-row stochastic matrix $R$ such that $X=RY$. Define the relation ${\sim }_{\mathrm{gut}}$ as follows. $X{\sim }_{\mathrm{gut}}Y$ if $X$ is gut-majorized by $Y$ and $Y$ is gut-majorized by $X$. The (strong) linear preservers of ${\prec }_{\mathrm{gut}}$ on ${ℝ}^n$ and strong linear preservers of this relation on ${𝐌}_{n,m}$ have been characterized before. This paper characterizes all (strong) linear preservers and strong linear preservers of ${\sim }_{\mathrm{gut}}$ on ${ℝ}^n$ and ${𝐌}_{n,m}$.

For $X,Y\in {𝐌}_{n,m}$, it is said that $X$ is majorized by $Y$ (and it is denoted by $X{\prec }_{gt}Y$) if there exists a tridiagonal g-doubly stochastic matrix $A$ such that $X=AY$. In this paper, the linear preservers and strong linear preservers of ${\prec }_{gt}$ are characterized on ${𝐌}_{n,m}$.

We consider a family of nonlinear stochastic heat equations of the form ${\partial }_{t}u=ℒu+\sigma \left(u\right)\dot{W}$, where $\dot{W}$ denotes space–time white noise, $ℒ$ the generator of a symmetric Lévy process on $𝐑$, and $\sigma$ is Lipschitz continuous and zero at 0. We show that this stochastic PDE has a random-field solution for every finite initial measure $u_0$. Tight a priori bounds on the moments of the solution are also obtained. In the particular case that $ℒf=c{f}^{\text{'}\text{'}}$ for some $cgt;0$, we prove that if $u_0$ is a finite measure of compact support, then the...

An $m\times n$ matrix $R$ with nonnegative entries is called row stochastic if the sum of entries on every row of $R$ is 1. Let ${𝐌}_{m,n}$ be the set of all $m\times n$ real matrices. For $A,B\in {𝐌}_{m,n}$, we say that $A$ is row Hadamard majorized by $B$ (denoted by $A{\prec }_{RH}B)$ if there exists an $m\times n$ row stochastic matrix $R$ such that $A=R\circ B$, where $X\circ Y$ is the Hadamard product (entrywise product) of matrices $X,Y\in {𝐌}_{m,n}$. In this paper, we consider the concept of row Hadamard majorization as a relation on ${𝐌}_{m,n}$ and characterize the structure of all linear operators $T:{𝐌}_{m,n}\to {𝐌}_{m,n}$ preserving...

We consider the stochastic differential equation $X_t=X₀+{\int }_0^t(A_s+B_s{X}_s)ds+{\int }_0^t{C}_{s}d{Y}_s$, where $A_t$, $B_t$, $C_t$ are nonrandom continuous functions of t, X₀ is an initial random variable, $Y=(Y_t,t\ge 0)$ is a Gaussian process and X₀, Y are independent. We give the form of the solution ($X_t$) to (0.1) and then basing on the results of Plucińska [Teor. Veroyatnost. i Primenen. 25 (1980)] we prove that ($X_t$) is a quasi-diffusion proces.

Let ${𝕄}_{n,m}$ be the set of all $n\times m$ real or complex matrices. For $A,B\in {𝕄}_{n,m}$, we say that $A$ is row-sum majorized by $B$ (written as $A{\prec }^{\mathrm{rs}}B$) if $R\left(A\right)\prec R\left(B\right)$, where $R\left(A\right)$ is the row sum vector of $A$ and $\prec$ is the classical majorization on ${ℝ}^n$. In the present paper, the structure of all linear operators $T:{𝕄}_{n,m}\to {𝕄}_{n,m}$ preserving or strongly preserving row-sum majorization is characterized. Also we consider the concepts of even and circulant majorization on ${ℝ}^n$ and then find the linear preservers of row-sum majorization of these relations on ${𝕄}_{n,m}$. ...

A $0/1$-simplex is the convex hull of $n+1$ affinely independent vertices of the unit $n$-cube $I^n$. It is nonobtuse if none of its dihedral angles is obtuse, and acute if additionally none of them is right. Acute $0/1$-simplices in $I^n$ can be represented by $0/1$-matrices $P$ of size $n\times n$ whose Gramians $G=P^{\top }P$ have an inverse that is strictly diagonally dominant, with negative off-diagonal entries. In this paper, we will prove that the positive part $D$ of the transposed inverse $P^{-\top }$ of $P$ is doubly stochastic and has the...

Let $X$ be a $\mu$-symmetric Hunt process on a LCCB space $𝙴$. For an open set $𝙶\subseteq 𝙴$, let ${\tau }_{𝙶}$ be the exit time of $X$ from $𝙶$ and $A^{𝙶}$ be the generator of the process killed when it leaves $𝙶$. Let $r:[0,\infty [\to [0,\infty [$ and $R\left(t\right)={\int }_0^{t}r\left(s\right)\mathrm{d}s$. We give necessary and sufficient conditions for ${𝔼}_{\mu }R\left({\tau }_{𝙶}\right)lt;\infty$ in terms of the behavior near the origin of the spectral measure of $-A^{𝙶}$. When $r\left(t\right)=t^l$, $l\ge 0$, by means of this condition we derive the Nash inequality for the killed process. In the diffusion case this permits to show that the existence of moments of order $l+1$ for ${\tau }_{𝙶}$...

For each n ≥ 1, let $v_{n,k},k\ge 1$ and $u_{n,k},k\ge 1$ be mutually independent sequences of nonnegative random variables and let each of them consist of mutually independent and identically distributed random variables with means v̅ₙ and u̅̅ₙ, respectively. Let $X{ₙ}^B\left(t\right)=(1/cₙ){\sum }_{j=1}^{\left[nt\right]}(v_{n,j}-v̅ₙ)$, $X{ₙ}^A\left(t\right)=(1/cₙ){\sum }_{j=1}^{\left[nt\right]}(u_{n,j}-u̅̅ₙ)$, t ≥ 0, and $Xₙ=X{ₙ}^B-X{ₙ}^A$. The main result gives conditions under which the weak convergence $Xₙ\stackrel{}{\to }X$, where X is a Lévy process, implies $X{ₙ}^B\stackrel{}{\to }{X}^B$ and $X{ₙ}^A\stackrel{}{\to }{X}^A$, where $X^B$ and $X^A$ are mutually independent Lévy processes and $X=X^B-X^A$.

In a Tychonoff space $X$, the point $p\in X$ is called a $C^{*}$-point if every real-valued continuous function on $C\setminus \left\{p\right\}$ can be extended continuously to $p$. Every point in an extremally disconnected space is a $C^{*}$-point. A classic example is the space ${𝐖}^{*}={\omega }_1+1$ consisting of the countable ordinals together with ${\omega }_1$. The point ${\omega }_1$ is known to be a $C^{*}$-point as well as a $P$-point. We supply a characterization of $C^{*}$-points in totally ordered spaces. The remainder of our time is aimed at studying when a point in a product space...