The area of exponential random walk and partial sums of uniform order statistics

On the density for sums of independent exponential, Erlang and gamma variates

This paper re-examines the density for sums of independent exponential, Erlang and gamma random variables. By using a divided difference perspective, the paper provides a unified approach to finding closed-form formulae for such convolutions. In particular, the divided difference perspective for sums of Erlang variates suggests a new approach to finding the density for sums of independent gamma variates using fractional calculus.

Improved concentration bounds for sums of independent sub-exponential random variables

Known Bernstein-type upper bounds on the tail probabilities for sums of independent zero-mean sub-exponential random variables are improved in several ways at once. The new upper bounds have a certain optimality property.

独立双边指数随机变量和的尾部界

Tail bounds for sums of independent two-sided exponential random variables

本文为双边指数随机变量加权和建立匹配主项的尾部上下界，推广了Janson的单边指数结果。

不平衡步长强化随机游走的中心极限定理

A central limit theorem for unbalanced step-reinforced random walks

本文研究一类统一了象随机游走等模型的不平衡步长强化随机游走，通过建立与随机递归树键渗流的联系将其表示为随机加权和，推导出正态与稳定中心极限定理，进而获得该类游走的统一中心极限定理。

指数变量和的矩，超越CHS

Moments of sums of exponentials, beyond CHS

本文研究了固定方差下独立指数变量和的$L_p$-范数下界，并探讨了其Schur单调性条件，扩展了Hunter的正性定理。

随机变量自归一化和的大偏差概率

On large deviation probabilities for self-normalized sums of random variables

本文通过将自归一化随机游走的大偏差问题转化为二元随机游走问题，在更一般条件下证明了Shao的经典定理，并给出了精确的大偏差渐近及图形解释。

马尔可夫游走在α-稳定领域中保持正性的极限定理

On the $L^p$ Norms of Random Exponential Sums and the Metric Domination Principle over the Simple Random Walk

本文在α-稳定领域中研究了马尔可夫游走保持正性的极限定理，证明了严格正的调和函数存在性及稳定游走的分布收敛。

Sufficient conditions for exponential tightness of normed sums of independent random fields in the space of continuous functions

We formulate and prove in this report some sufficient conditions for exponential tightness (ET) of a family of independent identical distributed (i.i.d.) random fields (r.f.) (processes) in the space of continuous functions defined on certain metric compact set. We will use the entropic method as well as too modern approach through the majorizing measures (generic chaining).

黎曼流形上随机游走导出的加权张量和函数的尾部界

Tail Bounds for Functions of Weighted Tensor Sums Derived from Random Walks on Riemannian Manifolds

本文利用Mond-Pecaric方法建立新张量不等式，并结合Laplace-Beltrami算子谱信息，为流形上随机游走导出的加权张量和给出新的尾部界及转移矩阵列和估计。

Contraction principle for trajectories of random walks and Cramer's theorem for kernel-weighted sums

In 2013 A.A. Borovkov and A.A. Mogulskii proved a weaker-than-standard "metric" large deviations principle (LDP) for trajectories of random walks in $R^d$ whose increments have the Laplace transform finite in a neighbourhood of zero. We prove that general metric LDPs are preserved under uniformly continuous mappings. This allows us to transform the result of Borovkov and Mogulskii into standard LDPs. We also give an explicit integral representation of the rate function they found. As an application, we extend the classical Cram'er theorem by proving an LPD for kernel-weighted sums of i.i.d. ra…

Sums of twisted circulants

The rate of convergence of simple random walk on the Heisenberg group over $Z/nZ$ with a standard generating set was determined by Bump et al [1,2]. We extend this result to random walks on the same groups with an arbitrary minimal symmetric generating set. We also determine the rate of convergence of simple random walk on higher-dimensional versions of the Heisenberg group with a standard generating set. We obtain our results via Fourier analysis, using an eigenvalue bound for sums of twisted circulant matrices. The key tool is a generalization of a version of the Heisenberg Uncertainty Princ…

关于二项和、加性能量与懒惰随机游走

On binomial sums, additive energies, and lazy random walks

本研究证明了关于离散超立方体子集k-加性能量猜想所需的实数初等不等式（此前仅验证k≤100的情况），完成了推广Kane-Tao结果的证明，并给出该不等式的非对称懒惰随机游走解释。

Kloosterman paths and the shape of exponential sums

We consider the distribution of the polygonal paths joining partial sums of classical Kloosterman sums, as their parameter varies modulo a prime tending to infinity. Using independence of Kloosterman sheaves, we prove convergence in the sense of finite distributions to a specific random Fourier series. We also consider Birch sums, for which we can establish convergence in law in the space of continuous functions. We then derive some applications.

Randomness of character sums modulo $m$

Using a probabilistic model, based on random walks on the additive group $\mathbb{Z}/m\mathbb{Z}$, we prove that the values of certain real character sums are uniformly distributed in residue classes modulo $m$.

自然规范化下独立中心化随机变量之和的精确指数尾部估计，并应用于 U-统计量理论

Exact exponential tail estimation for sums of independent centered random variables, under natural norming, with applications to the theory of U-statistics

本文利用 GLS 理论推导自然规范化独立中心化随机变量之和的精确指数尾部，并给出 U 统计量中的精细尾部估计。