Deviation inequalities and moderate deviations for the symmetric exclusion process

First, under a geometric ergodicity assumption, we provide some limit theorems and some probability inequalities for the bifurcating Markov chains (BMC). The BMC model was introduced by Guyon to detect cellular aging from cell lineage, and our aim is thus to complete his asymptotic results. The deviation inequalities are then applied to derive first result on the moderate deviation principle (MDP) for a functional of the BMC with a restricted range of speed, but with a function which can be unbounded. Next, under a uniform geometric ergodicity assumption, we provide deviation inequalities for the BMC and apply them to derive a second result on the MDP for a bounded functional of the BMC with a larger range of speed. As statistical applications, we provide superexponential convergence in probability and deviation inequalities (for either the Gaussian setting or the bounded setting), and the MDP for least square estimators of the parameters of a first-order bifurcating autoregressive process.

Some deviation inequalities and moderate deviation principles for the maximum likelihood estimators of parameters in an Ornstein-Uhlenbeck process with linear drift are established by the logarithmic Sobolev inequality and the exponential martingale method.

ABSTRACTIn this article, we study the deviation inequalities, moderate deviation principle (MDP) and Berry–Esseen bounds for certain Gaussian functionals arising from the Ornstein-Uhlenbeck process without tears. As an application, several asymptotic properties for the minimum distance estimator are obtained. The main methods include the MDP and deviation inequality for multiple Wiener–Itô integrals.

In this paper, we consider the self-normalized asymptotic properties of the parameter estimators in the fractional Ornstein–Uhlenbeck process. The deviation inequalities, Cramér-type moderate deviations and Berry–Esseen bounds are obtained. The main methods include the deviation inequalities and moderate deviations for multiple Wiener–Itô integrals [P. Major, Tail behavior of multiple integrals and U-statistics, Probab. Surv. 2 (2005) 448–505; On a multivariate version of Bernsteins inequality, Electron. J. Probab. 12 (2007) 966–988; M. Schulte and C. Thäle, Cumulants on Wiener chaos: Moderate deviations and the fourth moment theorem, J. Funct. Anal. 270 (2016) 2223–2248], as well as the Delta methods in large deviations [F. Q. Gao and X. Q. Zhao, Delta method in large deviations and moderate deviations for estimators, Ann. Statist. 39 (2011) 1211–1240]. For applications, we propose two test statistics which can be used to construct confidence intervals and rejection regions in the hypothesis testing for the drift coefficient. It is shown that the Type II errors tend to be zero exponential when using the proposed test statistics.

Heavy-tailed errors impair the accuracy of the least squares estimate, which can be spoiled by a single grossly outlying observation. As argued in the seminal work of Peter Huber in 1973 [Ann. Statist.1 (1973) 799-821], robust alternatives to the method of least squares are sorely needed. To achieve robustness against heavy-tailed sampling distributions, we revisit the Huber estimator from a new perspective by letting the tuning parameter involved diverge with the sample size. In this paper, we develop nonasymptotic concentration results for such an adaptive Huber estimator, namely, the Huber estimator with the tuning parameter adapted to sample size, dimension, and the variance of the noise. Specifically, we obtain a sub-Gaussian-type deviation inequality and a nonasymptotic Bahadur representation when noise variables only have finite second moments. The nonasymptotic results further yield two conventional normal approximation results that are of independent interest, the Berry-Esseen inequality and Cramér-type moderate deviation. As an important application to large-scale simultaneous inference, we apply these robust normal approximation results to analyze a dependence-adjusted multiple testing procedure for moderately heavy-tailed data. It is shown that the robust dependence-adjusted procedure asymptotically controls the overall false discovery proportion at the nominal level under mild moment conditions. Thorough numerical results on both simulated and real datasets are also provided to back up our theory.

In this paper, we study explicitly the deviation properties, including the deviation inequalities and Cramér-type moderate deviations, for some quadratic functionals of linear self-attracting diffusion process. As applications, Cramér-type moderate deviations for the log-likelihood ratio process and drift parameter estimator are obtained. The main methods consist of the deviation inequalities and Cramér-type moderate deviations for multiple Wiener–Itô integrals, as well as the asymptotic analysis techniques.

In this paper, we establish some deviation inequalities and the moderate deviation principles for the least squares estimators of the parameters in the threshold autoregressive model under the assumption that the noise random variable satisfies a logarithmic Sobolev inequality.

In this article, for some quadratic functionals of linear self-repelling diffusion process, we study the asymptotic properties, including the deviation inequalities and Cramér-type moderate deviations. The main methods consist of the deviation inequalities for multiple Wiener-Itô integrals, as well as the asymptotic analysis techiniques. As applications, (self-normalized) Cramér-type moderate deviations for the log-likelihood ratio process and drift parameter estimator are obtained.

This paper concerns the asymptotic properties of the quadratic functionals and associated ordinary least squares estimator in the explosive first-order Gaussian autoregressive process. By the deviation inequalities for multiple Wiener-Itô integrals and asymptotic analysis techniques, Cramér-type moderate deviations are achieved under the explosive and mildly explosive frameworks. As applications, the global and local powers for the unit root test are shown to approach one at exponential rates. Simulation experiments are conducted to confirm the theoretical results.

We study the asymptotic behaviours for the trajectory fitting estimator in the Ornstein–Uhlenbeck process with linear drift. Deviation inequality, moderate deviations, Berry–Esseen bound and the law of iterated logarithm (LIL) of this estimator can be obtained. Moreover, as an application of the Berry–Esseen bound, we can get the precise rate in LIL. The main method of this paper is the deviation inequality for multiple Wiener-Itô integrals [P. Major, On a multivariate version of Bernsteins inequality. Electron. J. Prob. 12 (2007), pp. 966–988; P. Major, Tail behavior of multiple integrals and U-statistics. Probab. Surv. 2 (2005), pp. 448–505].

ABSTRACTIn this article, we consider asymptotic behaviors for functionals of dynamical systems with small random perturbations. First, we present a deviation inequality for Gaussian approximation of dynamical systems with small random perturbations under Hölder norms and establish the moderate deviation principle and the central limit theorem for the dynamical systems by the deviation inequality. Then, applying these results to forward-backward stochastic differential equations and diffusions in small time intervals, combining the delta method in large deviations, we give a moderate deviation principle for solutions of forward-backward stochastic differential equations with small random perturbations, and obtain the central limit theorem, the moderate deviation principle and the iterated logarithm law for functionals of diffusions in small time intervals.

考虑如下非遍历α-布朗桥过程 dX_t= -α/(T-t)X_t dt + d W_t，X₀=0， t∈[0,T), 其中0<α<1/2, T∈(0,∞)固定，W={W_t: t≥ 0}是标准的布朗运动。利用渐近分析的技巧及多重Wiener-Itô积分的偏差性质，本文研究了二次泛函∫^t₀1/(T-s) X_sd W_s和∫^t₀1/(T-s)² X_s²d s 的偏差不等式以及Cramér-型中偏差。作为应用，得到了对数似然率过程以及参数α极大似然估计的（自正则化）Cramér-型中偏差。

Small noise fluctuations of the CIR model driven by [formula omitted]-stable noises

Using the macroscopic fluctuation theory of Bertini, De Sole, Gabrielli, Jona-Lasinio, and Landim, one can show that the statistics of the current of the symmetric simple exclusion process (SSEP) connected to two reservoirs on an arbitrary large finite domain in dimension d are the same as in the one-dimensional case. Numerical results on squares support this claim while results on cubes exhibit some discrepancy. We argue that the results of the macroscopic fluctuation theory should be recovered by increasing the size of the contacts. The generalization to other diffusive systems is straightforward.

The dynamics of an asymmetric tracer in the symmetric simple exclusion process (SEP) is mapped, in the continuous scaling limit, to the local current through the origin in the zero-range process with a biased bond. This allows us to study the hydrodynamics of the SEP with an asymmetric tracer with a step initial condition, leading to the average displacement as a function of the bias and the densities on both sides. We then derive the cumulant generating function of the process in the high-density limit, by using the macroscopic fluctuation theory and obtain agreement with the microscopic results of Poncet et al (2021). For more general initial conditions, we show that the tracer variance in the high-density limit depends only on the generalized susceptibility in the initial condition.