@article{cheng_xiao_2016, title={Excursion probability of Gaussian random fields on sphere}, volume={22}, ISSN={["1573-9759"]}, DOI={10.3150/14-bej688}, abstractNote={Let $X=\{X(x): x\in\mathbb{S}^N\}$ be a real-valued, centered Gaussian random field indexed on the $N$-dimensional unit sphere $\mathbb{S}^N$. Approximations to the excursion probability ${\mathbb{P}}\{\sup_{x\in\mathbb{S}^N}X(x)\ge u\}$, as $u\to\infty$, are obtained for two cases: (i) $X$ is locally isotropic and its sample functions are non-smooth and; (ii) $X$ is isotropic and its sample functions are twice differentiable. For case (i), the excursion probability can be studied by applying the results in Piterbarg (Asymptotic Methods in the Theory of Gaussian Processes and Fields (1996) Amer. Math. Soc.), Mikhaleva and Piterbarg (Theory Probab. Appl. 41 (1997) 367--379) and Chan and Lai (Ann. Probab. 34 (2006) 80--121). It is shown that the asymptotics of ${\mathbb{P}}\{\sup_{x\in\mathbb {S}^N}X(x)\ge u\}$ is similar to Pickands' approximation on the Euclidean space which involves Pickands' constant. For case (ii), we apply the expected Euler characteristic method to obtain a more precise approximation such that the error is super-exponentially small.}, number={2}, journal={BERNOULLI}, author={Cheng, Dan and Xiao, Yimin}, year={2016}, month={May}, pages={1113–1130} }
 @article{cheng_2016, title={Excursion probability of certain non-centered smooth Gaussian random fields}, volume={126}, ISSN={["1879-209X"]}, DOI={10.1016/j.spa.2015.10.003}, abstractNote={Let X={X(t),t∈T} be a non-centered, unit-variance, smooth Gaussian random field indexed on some parameter space T, and let Au(X,T)={t∈T:X(t)≥u} be the excursion set. It is shown that, as u→∞, the excursion probability P{supt∈TX(t)≥u} can be approximated by the expected Euler characteristic of Au(X,T), denoted by E{χ(Au(X,T))}, such that the error is super-exponentially small. The explicit formulae for E{χ(Au(X,T))} are also derived for two cases: (i) T is a rectangle and X−EX is stationary; (ii) T is an N-dimensional sphere and X−EX is isotropic.}, number={3}, journal={STOCHASTIC PROCESSES AND THEIR APPLICATIONS}, author={Cheng, Dan}, year={2016}, month={Mar}, pages={883–905} }
 @article{cheng_xiao_2016, title={THE MEAN EULER CHARACTERISTIC AND EXCURSION PROBABILITY OF GAUSSIAN RANDOM FIELDS WITH STATIONARY INCREMENTS}, volume={26}, ISSN={["1050-5164"]}, DOI={10.1214/15-aap1101}, abstractNote={Let $X=\{X(t),t\in {\mathbb{R}}^N\}$ be a centered Gaussian random field with stationary increments and $X(0)=0$. For any compact rectangle $T\subset {\mathbb{R}}^N$ and $u\in {\mathbb{R}}$, denote by $A_u=\{t\in T:X(t)\geq u\}$ the excursion set. Under $X(\cdot)\in C^2({\mathbb{R}}^N)$ and certain regularity conditions, the mean Euler characteristic of $A_u$, denoted by ${\mathbb{E}}\{\varphi(A_u)\}$, is derived. By applying the Rice method, it is shown that, as $u\to\infty$, the excursion probability ${\mathbb{P}}\{\sup_{t\in T}X(t)\geq u\}$ can be approximated by ${\mathbb{E}}\{\varphi(A_u)\}$ such that the error is exponentially smaller than ${\mathbb{E}}\{\varphi(A_u)\}$. This verifies the expected Euler characteristic heuristic for a large class of Gaussian random fields with stationary increments.}, number={2}, journal={ANNALS OF APPLIED PROBABILITY}, author={Cheng, Dan and Xiao, Yimin}, year={2016}, month={Apr}, pages={722–759} }
 @article{cheng_schwartzman_2015, title={Distribution of the height of local maxima of Gaussian random fields}, volume={18}, ISSN={["1572-915X"]}, DOI={10.1007/s10687-014-0211-z}, abstractNote={Let {f(t) : t ∈ T} be a smooth Gaussian random field over a parameter space T, where T may be a subset of Euclidean space or, more generally, a Riemannian manifold. We provide a general formula for the distribution of the height of a local maximum ℙ{f(t 0) > u|t 0 is a local maximum of f(t)} when f is non-stationary. Moreover, we establish asymptotic approximations for the overshoot distribution of a local maximum ℙ{f(t 0) > u+v|t 0 is a local maximum of f(t) and f(t 0) > v} as $v\to \infty $ . Assuming further that f is isotropic, we apply techniques from random matrix theory related to the Gaussian orthogonal ensemble to compute such conditional probabilities explicitly when T is Euclidean or a sphere of arbitrary dimension. Such calculations are motivated by the statistical problem of detecting peaks in the presence of smooth Gaussian noise.}, number={2}, journal={EXTREMES}, author={Cheng, Dan and Schwartzman, Armin}, year={2015}, month={Jun}, pages={213–240} }
 @article{cheng_2014, title={Double extreme on joint sets for Gaussian random fields}, volume={92}, ISSN={["1879-2103"]}, DOI={10.1016/j.spl.2014.05.001}, abstractNote={For a centered Gaussian random field X={X(t),t∈RN}, let T1 and T2 be two compact sets in RN such that I=T1∩T2≠0̸ and denote by χ(Au(I)) the Euler characteristic of the excursion set Au(I)={t∈I:X(t)≥u}. We show that under certain smoothness and regularity conditions, as u→∞, the joint excursion probability P{supt∈T1X(t)≥u,sups∈T2X(s)≥u} can be approximated by the expected Euler characteristic E{χ(Au(I))} such that the error is super-exponentially small.}, journal={STATISTICS & PROBABILITY LETTERS}, author={Cheng, Dan}, year={2014}, month={Sep}, pages={79–82} }