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Register a new userFrom Schoenberg to Pick-Nevanlinna: Toward a complete picture of the variogram class
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We show that a large subclass of variograms is closed under products and that some desirable stability properties, such as the product of special compositions, can be obtained within the proposed setting. We introduce new classes of kernels of Schoenberg-L{e}vy type and demonstrate some important properties of rotationally invariant variograms.
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It is well known that an $n times n$ Wishart matrix with $d$ degrees of freedom is close to the appropriately centered and scaled Gaussian Orthogonal Ensemble (GOE) if $d$ is large enough. Recent work of Bubeck, Ding, Eldan, and Racz, and independently Jiang and Li, shows that the transition happens when $d = Theta ( n^{3} )$. Here we consider this critical window and explicitly compute the total variation distance between the Wishart and GOE matrices when $d / n^{3} to c in (0, infty)$. This shows, in particular, that the phase transition from Wishart to GOE is smooth.
The spectral gap $gamma$ of an ergodic and reversible Markov chain is an important parameter measuring the asymptotic rate of convergence. In applications, the transition matrix $P$ may be unknown, yet one sample of the chain up to a fixed time $t$ may be observed. Hsu, Kontorovich, and Szepesvari (2015) considered the problem of estimating $gamma$ from this data. Let $pi$ be the stationary distribution of $P$, and $pi_star = min_x pi(x)$. They showed that, if $t = tilde{O}bigl(frac{1}{gamma^3 pi_star}bigr)$, then $gamma$ can be estimated to within multiplicative constants with high probability. They also proved that $tilde{Omega}bigl(frac{n}{gamma}bigr)$ steps are required for precise estimation of $gamma$. We show that $tilde{O}bigl(frac{1}{gamma pi_star}bigr)$ steps of the chain suffice to estimate $gamma$ up to multiplicative constants with high probability. When $pi$ is uniform, this matches (up to logarithmic corrections) the lower bound of Hsu, Kontorovich, and Szepesvari.
We consider the class $Psi_d$ of continuous functions $psi colon [0,pi] to mathbb{R}$, with $psi(0)=1$ such that the associated isotropic kernel $C(xi,eta)= psi(theta(xi,eta))$ ---with $xi,eta in mathbb{S}^d$ and $theta$ the geodesic distance--- is positive definite on the product of two $d$-dimensional spheres $mathbb{S}^d$. We face Problems 1 and 3 proposed in the essay Gneiting (2013b). We have considered an extension that encompasses the solution of Problem 1 solved in Fiedler (2013), regarding the expression of the $d$-Schoenberg coefficients of members of $Psi_d$ as combinations of $1$-Schoenberg coefficients. We also give expressions for the computation of Schoenberg coefficients of the exponential and Askey families for all even dimensions through recurrence formula. Problem 3 regards the curvature at the origin of members of $Psi_d$ of local support. We have improved the current bounds for determining this curvature, which is of applied interest at least for $d=2$.
This paper has been temporarily withdrawn, pending a revised version taking into account similarities between this paper and the recent work of del Barrio, Gine and Utzet (Bernoulli, 11 (1), 2005, 131-189).
We establish a central limit theorem for (a sequence of) multivariate martingales which dimension potentially grows with the length $n$ of the martingale. A consequence of the results are Gaussian couplings and a multiplier bootstrap for the maximum of a multivariate martingale whose dimensionality $d$ can be as large as $e^{n^c}$ for some $c>0$. We also develop new anti-concentration bounds for the maximum component of a high-dimensional Gaussian vector, which we believe is of independent interest. The results are applicable to a variety of settings. We fully develop its use to the estimation of context tree models (or variable length Markov chains) for discrete stationary time series. Specifically, we provide a bootstrap-based rule to tune several regularization parameters in a theoretically valid Lepski-type method. Such bootstrap-based approach accounts for the correlation structure and leads to potentially smaller penalty choices, which in turn improve the estimation of the transition probabilities.
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