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Zapiski Nauchnykh Seminarov POMI, 2012, Volume 408, Pages 187–196 (Mi znsl5500)  

This article is cited in 11 scientific papers (total in 11 papers)

Random determinants, mixed volumes of ellipsoids, and zeros of Gaussian random fields

D. N. Zaporozhetsa, Z. Kabluchkob

a St. Petersburg Department of the Steklov Mathematical Institute, St. Petersburg, Russia
b Institute of Stochastics, Ulm University, Ulm, Germany
References:
Abstract: Consider a $d\times d$ matrix $M$ whose rows are independent centered non-degenerate Gaussian vectors $\xi_1,\ldots,\xi_d$ with covariance matrices $\Sigma_1,\dots,\Sigma_d$. Denote by $\mathcal E_i$ the location-dispersion ellipsoid of $\xi_i$: $\mathcal E_i=\{\mathbf x\in\mathbb R^d\colon\mathbf x^\top\Sigma_i^{-1} \mathbf x\leqslant1\}$. We show that
$$ \mathbb E\,|\det M|=\frac{d!}{(2\pi)^{d/2}}V_d(\mathcal{E}_1,\dots,\mathcal E_d), $$
where $V_d(\cdot,\dots,\cdot)$ denotes the mixed volume. We also generalize this result to the case of rectangular matrices. As a direct corollary we get an analytic expression for the mixed volume of $d$ arbitrary ellipsoids in $\mathbb R^d$.
As another application, we consider a smooth centered non-degenerate Gaussian random field $X=(X_1,\dots,X_k)^\top\colon\mathbb R^d\to\mathbb R^k$. Using the Kac–Rice formula, we obtain the geometric interpretation of the intensity of zeros of $X$ in terms of the mixed volume of location-dispersion ellipsoids of the gradients of $X_i/\sqrt{\mathbf{Var}X_i}$. This relates the zero sets of equations to the mixed volumes in a way which resembles the well-known Bernstein theorem on the number of solutions of a typical system of algebraic equations.
Key words and phrases: Gaussian random determinant, Wishart matrix, Gaussian random parallelotope, mixed volumes of ellipsoids, location-dispersion ellipsoid, zeros of Gaussian random fields, Bernstein theorem, Kac–Rice formula.
Received: 10.10.2012
English version:
Journal of Mathematical Sciences (New York), 2014, Volume 199, Issue 2, Pages 168–173
DOI: https://doi.org/10.1007/s10958-014-1844-9
Bibliographic databases:
Document Type: Article
UDC: 519.2+514
Language: Russian
Citation: D. N. Zaporozhets, Z. Kabluchko, “Random determinants, mixed volumes of ellipsoids, and zeros of Gaussian random fields”, Probability and statistics. Part 18, Zap. Nauchn. Sem. POMI, 408, POMI, St. Petersburg, 2012, 187–196; J. Math. Sci. (N. Y.), 199:2 (2014), 168–173
Citation in format AMSBIB
\Bibitem{ZapKab12}
\by D.~N.~Zaporozhets, Z.~Kabluchko
\paper Random determinants, mixed volumes of ellipsoids, and zeros of Gaussian random fields
\inbook Probability and statistics. Part~18
\serial Zap. Nauchn. Sem. POMI
\yr 2012
\vol 408
\pages 187--196
\publ POMI
\publaddr St.~Petersburg
\mathnet{http://mi.mathnet.ru/znsl5500}
\mathscinet{http://mathscinet.ams.org/mathscinet-getitem?mr=3032216}
\transl
\jour J. Math. Sci. (N. Y.)
\yr 2014
\vol 199
\issue 2
\pages 168--173
\crossref{https://doi.org/10.1007/s10958-014-1844-9}
\scopus{https://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-84902275899}
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  • https://www.mathnet.ru/eng/znsl/v408/p187
  • This publication is cited in the following 11 articles:
    Citing articles in Google Scholar: Russian citations, English citations
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