Non-comparability with respect to the convex transform order with applications

On a conjecture about the comparability of parallel systems with respect to convex transform order

In this paper we prove that two heterogeneous parallel systems with independent exponentially distributed components are comparable via the star transform order while the comparison via the convex transform fails. The latter conclusion provides a partial answer to a problem that remained open for a decade.

Convex transform order of Beta distributions with some consequences

The convex transform order is one way to make precise comparison between the skewness of probability distributions on the real line. We establish a simple and complete characterisation of when one Beta distribution is smaller than another according to the convex transform order. As an application, we derive monotonicity properties for the probability of Beta distributed random variables exceeding the mean or mode of their distribution. Moreover, we obtain a simple alternative proof of the mode-median-mean inequality for unimodal distributions that are skewed in a sense made precise by the conv…

一般闭凸状态空间仿射跳跃扩散的仿射变换公式

The affine transform formula for affine jump-diffusions with a general closed convex state space

在一般闭凸状态空间上，我们建立了仿射跳跃扩散方程指数矩的存在性和仿射变换公式的有效性。这推广了具有正则状态空间的仿射跳跃扩散的已知结果。关键的一步是利用相关的鞅问题的适定性，证明指数局部鞅的鞅性。通过解析延拓得到了复指数的仿射变换公式，特别是特征函数的仿射变换公式。我们的结果适用于广泛的一类仿射过程，包括那些与矩阵值的状态空间，最近在文献中获得了兴趣。

由凸集和凸函数构造的连分数

Continued fractions built from convex sets and convex functions

在具有对偶（或极性）变换的偏序半群中，可以定义连分数的推广。给出了确定项连分数收敛的一般充分条件。两个特殊的应用涉及的情况下，凸集的Minkowski加法和极性变换（其中也获得了充要条件的连分数收敛的常数项）和家庭的非负凸函数的Legendre-Fenchel和Artstein-Avidan-Milman变换。

Diagonal Minkowski classes, zonoid equivalence, and stable laws

We consider the family of convex bodies obtained from an origin symmetric convex body $K$ by multiplication with diagonal matrices, by forming Minkowski sums of the transformed sets, and by taking limits in the Hausdorff metric. Support functions of these convex bodies arise by an integral transform of measures on the family of diagonal matrices, equivalently, on Euclidean space, which we call $K$-transform. In the special case, if $K$ is a segment not lying on any coordinate hyperplane, one obtains the family of zonoids and the cosine transform. In this case two facts are known: the vector sp…

凸体的克拉默变换、半空间深度与阈值现象

Cramér transform, half-space depth and threshold phenomena for convex bodies

该研究探讨高维凸几何中克拉默变换与半空间深度的关系，证明相关尖锐比较式，推导矩、尾估计及随机凸包阈值准则，建立大偏差代价等的定量联系。