Let $X(t),t\in R^d$ be a centered Gaussian random field with continuous trajectories and set $ξ_u(t)= X(f(u)t),t\in R^d$ with $f$ some positive function. Classical results establish the tail asymptotics of $P\{ Γ(ξ_u) > u\}$ as $u\to \infty$ with $Γ(ξ_u)= \sup_{t \in [0,T ]^d} ξ_u(t),T>0$ by requiring that $f(u) \to 0$ with speed controlled by the local behaviour of the correlation function of $X$. Recent research shows that for applications more general continuous functionals than supremum should be considered and the Gaussian field can depend also on some additional parameter $τ_u \in K$, say $ξ_{u,τ_u}(t),t\in R^d$. In this contribution we derive uniform approximations of $P\{ Γ(ξ_{u,τ_u})> u\}$ with respect to $τ_u$ in some index set $K_u$, as $u\to\infty$. Our main result have important theoretical implications; two applications are already included in [10,11]. In this paper we present three additional ones, namely i) we derive uniform upper bounds for the probability of double-maxima, ii) we extend Piterbarg-Prisyazhnyuk theorem to some large classes of homogeneous functionals of centered Gaussian fields $ξ_{u}$, and iii) we show the finiteness of generalized Piterbarg constants.
Tail exponents of the three-dimensional uniform spanning tree and Abelian sandpile
Beijing International Center for Mathematical Research, Peking University（北京大学数学中心）；School of Mathematical Sciences, Peking University（北京大学数学科学学院）；Graduate School of Informatics, Kyoto University（京都大学情报学府）
Sharp moment estimates for martingales with uniformly bounded square functions
本文给出了平方函数一致有界的鞅的终止分布的指数矩和$p$ -矩$1\leqslant p \leqslant 2$的精确界。我们引入了相应的极值问题的Bellman函数，并将其减少到$\mathrm{BMO}([0,1])$上已知的Bellman函数。在尾部估计的情况下，类似的减少并不完全有效，所以我们提出了一个精细的超解，导致尖尾估计。
