Let $X(t)$, $t\in K$, be a centred Gaussian process with continuous sample paths on a compact metric space $K$, and let $M=\min_{t\in K}X(t)$. Let $\sigma_*^2$ denote the minimum covariance energy associated with $X$, and assume that $\sigma_*^2>0$. Motivated by the results of \cite{chakrabarty2018asymptotic} for smooth Gaussian processes, we show that, conditionally on $M>u$, the scaled overshoot $u(M-u)$ converges, as $u\to\infty$, to an exponential random variable with mean $\sigma_*^2$. Moreover, every weak subsequential limit of the conditional law of a measurable minimizer of $X$ is an optimal covariance-energy measure. In particular, if this measure is unique, then the conditional law converges weakly to it. The results are illustrated by stationary Gaussian processes, fractional Brownian motion, and fractional Brownian sheet.
The main innovation is to develop stochastic control theory within the branching OGP framework, significantly expanding the settings in which it locates an exact algorithmic threshold.
Brice Huang, Mark Sellke, Ni-Ke Sun· 3 citations· ⚡1
To each finite symmetric measure $\sigma$ on the real line, with support a compact subset of $(-2,2)$, we associate a measure $\mu$ on the unit circle by transporting mass at $\pm x$ to $e^{\pm i\theta(x)}$, $\theta(x)=2\arcsin(x/2)$. The linear prediction errors, Verblunsky coefficients, and Toeplitz determinants of $...
Let $Y_s=\sqrt{s}X+Z$, where $Z$ is standard Gaussian and independent of the real random variable $X$. We prove that, under the square-exponential moment condition $\mathbb{E}e^{\beta X^2}<\infty$ for some $\beta>0$, the scalar minimum mean-square error $\operatorname{mmse}_X(s)$ is analytic at zero signal-to-noise rat...
Let $P$ be a real polynomial of degree at most $m$ on $\mathbb{R}^d$, and let $X$ be standard Gaussian. Because Gaussian observations are invariant under $O(d)$, the natural inverse problem is to recover the orthogonal orbit of $P$; the law of $P(X)$ alone is generally insufficient. We prove that a prescribed finite fa...
For a centered, variance-one random variable $X$ with finite fourth moment, let $x$ be the vector of square-free degree-$d$ monomials in $n$ independent copies of $X$. At the critical scale $d^2/n \to \lambda \in (0,\infty)$, the normalized squared length of $x$ converges to the lognormal variable $R = \exp(\sqrt{\lamb...
We study the maximizers of the generalized volume product \[ \gamma_\sigma^n(A)\,\gamma_\sigma^n(A^\circ) \] among all measurable subsets $A\subset\mathbb{R}^n$, where $A^\circ$ denotes the polar set of $A$, and where $\gamma_\sigma^n$ denotes the centered Gaussian probability measure on $\mathbb{R}^n$ with covariance...
S. Artstein-Avidan, M. Fradelizi, K. Wyczesany· 0 citations
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