Li-Xin Zhang
We prove a new Donsker's invariance principle for independent and identically distributed random variables under the sub-linear expectation. As applications, the small deviations and Chung's law of the iterated logarithm are obtained.
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A small ball problem and Chung's law of iterated logarithm for a hypoelliptic Brownian motion in Heisenberg group are proven. In addition, bounds on the limit in Chung's law are established.
This paper gives sufficent and necessary conditions on a kind of limit results to hold on the precise convergent rate of an infinite series of probabilities on the Chung type law of the iterated logarithm.
Chung's Law of the Iterated Logarithm for a Class of Stochastic Heat Equations
该研究针对一类系数依赖解的乘性噪声驱动随机热方程，结合小球概率文献与冻结系数技术，建立了其解的Chung型重对数律，且几乎必然可确定其中的极限常数。
We establish a Chung-type law of the iterated logarithm and the exact local and uniform moduli of continuity for a large class of anisotropic Gaussian random fields with a harmonizable-type integral representation and the property of strong local nondeterminism. Compared with the existing results in the literature, our results do not require the assumption of stationary increments and provide more precise upper and lower bounds for the limiting constants. The results are applicable to the solutions of a class of linear stochastic partial differential equations driven by a fractional-colored Ga…
Small ball probabilities and Chung's law of the iterated logarithm for Gaussian Volterra processes with power-type kernels
该研究针对带幂型核的高斯Volterra过程，推导其小球概率双边估计，进而证明各固定点、原点及无穷远处的Chung重对数律，明确参数对过程特性的决定作用。
Khinchin's and Chung's Laws of the Iterated Logarithm at Time Zero for the Linear Stochastic Fractional Diffusion Equation
该研究针对带零初值的线性随机分数阶扩散方程，建立了时间零点的辛钦型重对数律，在特定附加条件下证明了钟氏型重对数律，将随机热方程的初始时间重对数律推广至一类时间分数阶随机扩散方程。
We prove Chung-type laws of the iterated logarithm for general Lévy processes at zero. In particular, we provide tools to translate small deviation estimates directly into laws of the iterated logarithm. This reveals laws of the iterated logarithm for Lévy processes at small times in many concrete examples. In some cases, exotic norming functions are derived.
Consider the first exit time $T_{a,b}$ from a finite interval $[-a,b]$ for an homogeneous fluctuating functional $X$ of a linear Brownian motion. We show the existence of a finite positive constant $\k$ such that $$\lim_{t\to\infty}t^{-1}\log \p[ T_{ab} > t] = -\k.$$ Following Chung's original approach, we deduce a "liminf" law of the iterated logarithm for the two-sided supremum of $X$. This extends and gives a new point of view on a result of Khoshnevisan and Shi.
Chung-type laws of the iterated logarithm for $m$-fold weighted integrated fractional processes
本文研究了m重加权积分分数过程的Chung型迭代对数定律，推导了极限行为并给出了具体常数，同时探讨了随机化play-the-winner规则的迭代对数定律。
Chung's LIL for the linear stochastic fractional heat equation at origin
针对带零初值、由时空白噪声驱动的线性随机分数阶热方程，本文建立了其解在原点处的Chung重对数律。
We prove a Chung-type law of the iterated logarithm for a multiparameter extension of the fractional Brownian motion which is not increment stationary. This multiparameter fractional Brownian motion behaves very differently at the origin and away from the axes, which also appears in the Hausdorff dimension of its range and in the measure of its pointwise Hölder exponents. A functional version of this Chung-type law is also provided.
We show some Chung-type $\liminf$ law of the iterated logarithm results at zero for a class of (pure-jump) Feller or Lévy-type processes. This class includes all Lévy processes. The norming function is given in terms of the symbol of the infinitesimal generator of the process. In the Lévy case, the symbol coincides with the characteristic exponent.
We introduce a class of iterated processes called $α$-time Brownian motion for $0<α\leq 2$. These are obtained by taking Brownian motion and replacing the time parameter with a symmetric $α$-stable process. We prove a Chung-type law of the iterated logarithm (LIL) for these processes which is a generalization of LIL proved in \cite{hu} for iterated Brownian motion. When $α=1$ it takes the following form $$ \liminf_{T\to\infty}T^{-1/2}(\log \log T) \sup_{0\leq t\leq T}|Z_{t}|=π^{2}\sqrt{λ_{1}} a.s. $$ where $λ_{1}$ is the first eigenvalue for the Cauchy process in the interval $[-1,1].$ We also…
Concentration inequalities for strong laws and laws of the iterated logarithm
本文为独立同分布随机变量和推导集中不等式，得到多个强大数定律的非渐近推广及重对数不等式，并给出路径wise类比。
Invariance principle for the capacity and the cardinality of the range of stable random walks
本文针对整数格上的 α-稳定随机游走，在维度与稳定性指数满足特定条件下证明了值域容量和基数的几乎必然不变性原理，并由此导出相应的重对数律。
Strong Invariance Principle for Random Dirichlet Series
该研究证明了随机素数狄利克雷级数可由布朗运动逼近，得到其相关重对数律，还建立了特定参数下两类随机狄利克雷级数的逼近关系及另一类级数的强不变性原理。
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