On the strong approximation and functional limit laws for the increments of the non-overlapping k-spacings processes

基于k-间隔过程的经验过程振动模的强极限

Strong limits related to the oscillation modulus of the empirical process based on the k-spacing process

近年来，在iid情形下，人们给出了经验过程振动模的几个强极限定理。我们表明，非常轻微的差异，这些强有力的结果也得到了一些代表性的减少经验过程的基础上（非重叠）的k-间距所产生的序列的独立随机变量（RV的）均匀分布在$(0,1)$。这就产生了上述过程的弱极限。我们的研究包括的情况下，步骤k是无界的。这些结果主要是由关于参数为k和1的Gamma函数增量的几个性质得到的。

间隔过程的高斯逼近及相关问题

Gaussian Approximations and Related Questions for the Spacings process

所有关于k-间距过程逼近高斯过程的结果都只使用了一种方法，即Shorack和Pyke的方法。这里，它表明，这种方法不能产生比$% \left( N/\log \log N\right) ^{-\frac{1}{4}}\left( \log N\right) ^{\frac{1}{2}% }$更好的速率。该速率的强界和弱界都是在k固定和$k\rightarrow +\infty $的情况下指定的。给出了Glivenko-Cantelli定理，并将Stute关于独立一致分布随机变量的经验过程增量的结果推广到了间距过程.本文还得到了Mason-Wellner-Shorack情形之一。

均匀经验过程和分位数过程增量的加权高斯逼近：固定端点扩展到有限计数尺度

Weighted Gaussian Approximations for Increments of the Uniform Empirical and Quantile Processes: Fixed-Endpoint Extensions to the Finite-Count Scale

针对固定端点t处的均匀经验过程和分位数过程及其增量，建立了加权高斯逼近，并将结果统一扩展到有限计数尺度λ/n。

The range of a self-similar additive gamma process is a scale invariant Poisson point process

It is shown that for a non-decreasing self-similar stochastic process $T$ with independent increments, the range of $T$ forms a Poisson point process with $σ$-finite intensity if and only if the one-dimensional distribution of $T(1)$ is of the gamma type. This follows from a general hold-jump description of such processes $T$, and implies the known result that the spacings between consecutive points of a scale invariant Poisson point process, with intensity $θx^{-1} dx$, are the points of another scale invariant Poisson point process with the same intensity.