On the Stability of Geometric Extremes

Extremes of geometric variables with applications to branching processes

We obtain limit theorems for the row extrema of a triangular array of zero-modified geometric random variables. Some of this is used to obtain limit theorems for the maximum family size within a generation of a simple branching process with varying geometric offspring laws.

极端事件几何对齐下的谱坍缩

Spectral Collapse Under Geometric Alignment of Extreme Events

本文证明高维半鞅的二次协变矩阵发生谱坍缩（最大特征值与迹的比值趋于1，有效秩趋于1）当且仅当跳跃方向在加权意义下几何对齐且背景扩散渐近可忽略，并给出了随机情形下的概率收敛。

Rates of convergence for extremes of geometric random variables and marked point processes

We use the Stein-Chen method to study the extremal behaviour of the problem of extremes for univariate and bivariate geometric laws. We obtain a rate for the convergence to the Gumbel distribution of the law of the maximum of i. i. d. geometric random variables, and show that convergence is faster when approximating by a discretised Gumbel. We similarly find a rate of convergence for the law of maxima of bivariate Marshall-Olkin geometric random pairs when approximating by a discrete limit law. We introduce marked point processes of exceedances (MPPEs), both with univariate and bivariate Marsh…

几何布朗运动的极值理论与短期限期权定价

Extreme value theory for geometric Brownian motion and pricing of short maturity options

研究几何布朗运动在运行最大值取大值条件下的极限分布，证明样本路径强收敛并给出收敛速率，应用于短期限障碍期权定价，得到含定量误差的渐近公式。

Matrix geometric approach for random walks: stability condition and equilibrium distribution

In this paper, we analyse a sub-class of two-dimensional homogeneous nearest neighbour (simple) random walk restricted on the lattice using the matrix geometric approach. In particular, we first present an alternative approach for the calculation of the stability condition, extending the result of Neuts drift conditions [30] and connecting it with the result of Fayolle et al. which is based on Lyapunov functions [13]. Furthermore, we consider the sub-class of random walks with equilibrium distributions given as series of product-forms and, for this class of random walks, we calculate the eigen…

球面空间中的几何不等式、稳定性结果与Kendall问题

Geometric inequalities, stability results and Kendall's problem in spherical space

本文研究球面空间中泊松随机镶嵌胞腔的Kendall形状问题，通过建立球面凸体尺寸泛函与命中泛函的新几何不等式及稳定性结果，在高强度渐近机制下获得了尺寸泛函的偏差不等式与渐近分布。