Can an infinite left-product of nonnegative matrices be expressed in terms of infinite left-products of stochastic ones?

$G^{+}$ 方法的应用：新型非负矩阵类及其结果

$G^{+}$ Method in Action: New Classes of Nonnegative Matrices with Results

本文提出 $G^{+}$ 方法并引入和正矩阵、$\left[ Δ\right] $-正矩阵及 $g_{k}^{+}$-矩阵等新矩阵类，据此证明非负矩阵乘积的若干新旧结果，并给出 $k$-连通图的充要条件。

On positivity of orthogonal series and its applications in probability

We give necessary and sufficient conditions for an orthogonal series to converge in the mean-squares to a nonnegative function. We present many examples and applications, in analysis and probability. In particular, we give necessary and sufficient conditions for a Lancaster-type of expansion $% \sum_{n\geq 0}c_{n}α_{n}(x)β_{n}(y)$ with two sets of orthogonal polynomials $\left\{ α_{n}\right\} $ and $\left\{ β_{n}\right\} $ to converge in means-squares to a nonnegative bivariate function. In particular, we study the properties of the set $C(α,β)$ of the sequences $\left\{ c_{n}\right\} ,$ for w…

Optimal binomial, Poisson, and normal left-tail domination for sums of nonnegative random variables

Let $X_1,\dots,X_n$ be independent nonnegative random variables (r.v.'s), with $S_n:=X_1+\dots+X_n$ and finite values of $s_i:=E X_i^2$ and $m_i:=E X_i>0$. Exact upper bounds on $E f(S_n)$ for all functions $f$ in a certain class $\mathcal{F}$ of nonincreasing functions are obtained, in each of the following settings: (i) $n,m_1,\dots,m_n,s_1,\dots,s_n$ are fixed; (ii) $n$, $m:=m_1+\dots+m_n$, and $s:=s_1+\dots+s_n$ are fixed; (iii)~only $m$ and $s$ are fixed. These upper bounds are of the form $E f(η)$ for a certain r.v. $η$. The r.v. $η$ and the class $\mathcal{F}$ depend on the choice of on…

Inverse Exponential Decay: Stochastic Fixed Point Equation and ARMA Models

We study solutions to the stochastic fixed point equation $X\stackrel{d}{=}AX+B$ when the coefficients are nonnegative and $B$ is an "inverse exponential decay" (IED) random variable. We provide theorems on the left tail of $X$ which complement well-known tail results of Kesten and Goldie. We generalize our results to ARMA processes with nonnegative coefficients whose noise terms are from the IED class. We describe the lower envelope for these ARMA processes.

三对角随机矩阵的负指数、匹配和负特征值

Negative index, matchings, and nonnegative eigenvalues of tridiagonal stochastic matrices

利用基于匹配的惯性原理，研究稀疏图支撑的随机矩阵负特征值个数，证明三对角非负随机矩阵的负特征值个数不超过⌊n/2⌋，并给出特征值排序的下界。

半正定矩阵超鞅

Positive Semidefinite Matrix Supermartingales

本文研究半正定矩阵超鞅与后向子鞅的收敛性及极大不等式，并据此给出随机对称矩阵在多种尾部条件下的新型集中不等式。