On positivity of orthogonal series and its applications in probability

On a complete characterization of a.s. convergence of multiple orthogonal series

We present a relation between convergence of multiple and single orthogonal series. This relation implies a complete characterization of all multiple sequences $(a_{n_1...n_d})_{n_1,...,n_d\in\bb N}$ such that for all orthonormal $(Φ_{n_1...n_d})$ multiple orthogonal series $\sum_{n_1,...,n_d\in\bb N}a_{n_1...n_d}Φ_{n_1...n_d}$ are a.s.\ convergent.

循环删除散步和总积极性

Loop-erased walks and total positivity

我们考虑矩阵的元素枚举的权重的散步在平面有向加权图（不一定非循环）。这些矩阵是完全非负的，更准确地说，他们所有的子式是正式的幂级数的边缘权重与非负系数。对这种现象的组合解释涉及环擦除行走。应用包括平面域上布朗运动的击中矩阵的全正性。

Orthogonal series and limit theorems for canonical U- and V-statistics of stationary connected observations

The limit behavior is studied for the distributions of normalized U- and V-statistics of an arbitrary order with canonical (degenerate) kernels, based on samples of increasing sizes from a stationary sequence of observations satisfying classical mixing conditions. The corresponding limit distributions are represented as infinite multilinear forms of a centered Gaussian sequence with a known covariance matrix.

si-过程及相关过程的正交级数与Karhunen-Loève分解

Orthogonal series for si- and related processes, Karhunen-Loève decompositions

本文利用de Branges整函数Hilbert空间理论研究si-过程及相关过程的级数展开，给出了新的Karhunen-Loève型分解。

关于a.e.系数的完全刻划。收敛正交级数

On complete characterization of coefficients of a.e. converging orthogonal series

我们刻画了数序列$(a_n)$，使得$\sum_{n\geq 1} a_nΦ_n$ a.e.收敛.对于任何$L_2$ -空间中的任何规范正交系$(Φ_n)$。在我们的准则中，我们使用集合$B =\{\sum_{m\geq n} |a_m|^2; n\geq 1\}$和它的信息函数$$h_B(t) = -\log_3(β-α)$$作为$t\in (α, β]$，$[α, β]\cap B =\{α, β\}.$

The complete characterization of a.s. convergence of orthogonal series

In this paper we prove the complete characterization of a.s. convergence of orthogonal series in terms of existence of a majorizing measure. It means that for a given $(a_n)^{\infty}_{n=1}$, $a_n>0$, series $\sum^{\infty}_{n=1}a_nφ_n$ is a.e. convergent for each orthonormal sequence $(φ_n)^{\infty}_{n=1}$ if and only if there exists a measure $m$ on \[T=\{0\}\cup\Biggl\{\sum^m_{n=1}a_n^2,m\geq 1\Biggr\}\] such that \[\sup_{t\in T}\int^{\sqrt{D(T)}}_0(m(B(t,r^2)))^{-{1}/{2}}\,dr<\infty,\] where $D(T)=\sup_{s,t\in T}|s-t|$ and $B(t,r)=\{s\in T:|s-t|\leq r\}$. The presented approach is based on w…