Moderate deviation principle for $ m $-dependent random variables under the sub-linear expectation

<abstract><p>Let $ \{X_n, n\geq1\} $ be a sequence of $ m $-dependent strictly stationary random variables in a sub-linear expectation $ (\Omega, \mathcal{H}, \mathbb{E}) $. In this article, we give the definition of $ m $-dependent sequence of random variables under sub-linear expectation spaces taking values in $ \mathbb{R} $. Then we establish moderate deviation principle for this kind of sequence which is strictly stationary. The results in this paper generalize the result that in the case of independent identically distributed samples. It provides a basis to discuss the moderate deviation principle for other types of dependent sequences.</p></abstract>

Moderate deviations for linear eigenvalue statistics of β-ensembles

We establish a moderate deviation principle for linear eigenvalue statistics of [Formula: see text]-ensembles in the one-cut regime with a real-analytic potential. The main ingredient is to obtain uniform estimates for the correlators of a family of perturbations of [Formula: see text]-ensembles using the loop equations.

A Study of SDEs Driven by Brownian Motion and Fractional Brownian Motion

In this thesis, we mainly study some properties for certain stochastic di↵er-ential equations.The types of stochastic di↵erential equations we are interested in are (i) stochastic di↵erential equations driven by Brownian motion, (ii) stochastic functional di↵erential equations driven by fractional Brownian motion, (iii) McKean-Vlasov stochastic di↵erential equations driven by Brownian motion,(iv) McKean-Vlasov stochastic di↵erential equations driven by fractional Brownian motion.The properties we investigate include the weak approximation rate of Euler-Maruyama scheme, the central limit theorem and moderate deviation principle for McKean-Vlasov stochastic di↵erential equations. Additionally, we investigate the existence and uniqueness of solution to McKean-Vlasov stochastic di↵erential equations driven by fractional Brownian motion, and then the Bismut formula of Lion’s derivatives for this model is also obtained.The crucial method we utilised to establish the weak approximation rate of Euler-Maruyama scheme for stochastic equations with irregular drift is the Girsanov transformation. More precisely, giving a reference stochastic equa-tions, we construct the equivalent expressions between the aim stochastic equations and associated numerical stochastic equations in another proba-bility spaces in view of the Girsanov theorem.For the Mckean-Vlasov stochastic di↵erential equation model, we first construct the moderate deviation principle for the law of the approxima-tion stochastic di↵erential equation in view of the weak convergence method. Subsequently, we show that the approximation stochastic equations and the McKean-Vlasov stochastic di↵erential equations are in the same exponen-tially equivalent family, and then we establish the moderate deviation prin-ciple for this model.Based on the result of Well-posedness for Mckean-Vlasov stochastic di↵er-ential equation driven by fractional Brownian motion, by using the Malliavin analysis, we first establish a general result of the Bismut type formula for Lions derivative, and then we apply this result to the non-degenerate case of this model.

Functional central limit theorems and moderate deviations for Poisson cluster processes

In this paper, we consider functional limit theorems for Poisson cluster processes. We first present a maximal inequality for Poisson cluster processes. Then we establish a functional central limit theorem under the second moment and a functional moderate deviation principle under the Cramér condition for Poisson cluster processes. We apply these results to obtain a functional moderate deviation principle for linear Hawkes processes.

Moderate deviation principle for multivalued stochastic differential equations

In this paper, we prove a moderate deviation principle for the multivalued stochastic differential equations whose proof are based on recently well-developed weak convergence approach. As an application, we obtain the moderate deviation principle for reflected Brownian motion.