Local Functional Limit Theorems of Increments for Brownian Motion

In this paper, we present a local Csorgo- Revesz type functional limit theorem for increments of Brownian motion and give its convergence rate. The results also extend the functional forms of Levy＇s modulus of continuity for Brownian motion.

Large Deviations for Parameter Estimators of Some Time Inhomogeneous Diffusion Process

The goal of this paper is to study large deviations for estimator and score function of some time inhomogeneous diffusion process. Large deviation in the non-steepness case with explicit rate functions is obtained by using parameter-dependent change of measure.

Moderate Deviations for Random Sums of Heavy-Tailed Random Variables

Let {Xn;n≥ 1} be a sequence of independent non-negative random variables with common distribution function F having extended regularly varying tail and finite mean μ = E（X1） and let {N（t）; t ≥0} be a random process taking non-negative integer values with finite mean λ（t） = E（N（t）） and independent of {Xn; n ≥1}. In this paper, asymptotic expressions of P（（X1 ＋… ＋XN（t）） -λ（t）μ 〉 x） uniformly for x ∈[γb（t）, ∞） are obtained, where γ〉 0 and b（t） can be taken to be a positive function with limt→∞ b（t）/λ（t） = 0.

Large Deviations and Moderate Deviations for Kernel Density Estimators of Directional Data

Let fn be the non-parametric kernel density estimator of directional data based on a kernel function K and a sequence of independent and identically distributed random variables taking values in d-dimensional unit sphere Sd-1. It is proved that if the kernel function is a function with bounded variation and the density function f of the random variables is continuous, then large deviation principle and moderate deviation principle for {sup x∈sd-1 ｜fn（x） - E（fn（x））｜, n ≥ 1} hold.

Precise Deviations for Discrete Ensembles

We consider precise deviations for discrete ensembles.Forβ=2 case,we first establish an asymptotic formula of the Christoffel–Darboux kernel of the discrete orthogonal polynomials on an infinite regular lattice with weight e^(−NV(x)).Then we use the asymptotic formula to get the precise deviations of the extreme value for corresponding ensemble.

The Support of the Solution for Stochastic Differential Equations Driven by G-Brownian Motion

By a linear interpolation approximation method, we obtain a characterization of the support of the solution for stochastic differential equations driven by G-Brownian motion.

Laws of the Iterated Logarithm for Locally Square Integrable Martingales

Three types of laws of the iterated logarithm （LIL） for locally square integrable martingales with continuous parameter are considered by a discretization approach. By this approach, a lower bound of LIL and a number of FLIL are obtained, and Chung LIL is extended.