Random vectors satisfying Khinchine-Kahane type inequalities for linear and quadratic forms

We solve the field equations of modified gravity for [Formula: see text] model in metric formalism. Further, we obtain the fixed points of the dynamical system in phase-space analysis of [Formula: see text] models, both with and without the effects of radiation. The stability of these points is studied against the perturbations in a smooth spatial background by applying the conditions on the eigenvalues of the matrix obtained in the linearized first-order differential equations. Following this, these fixed points are used for analyzing the dynamics of the system during the radiation, matter and acceleration-dominated phases of the universe. Certain linear and quadratic forms of [Formula: see text] are determined from the geometrical and physical considerations and the behavior of the scale factor is found for those forms. Further, we also determine the Hubble parameter [Formula: see text], the Ricci scalar [Formula: see text] and the scale factor [Formula: see text] for these cosmic phases. We show the emergence of an asymmetry of time from the dynamics of the scalar field exclusively owing to the [Formula: see text] gravity in the Einstein frame that may lead to an arrow of time at a classical level.

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A Remark on Independence of Linear and Quadratic Forms Involving Independent Gaussian Variables

The Annals of Mathematical Statistics ◽

10.1214/aoms/1177731069 ◽

1945 ◽

Vol 16 (4) ◽

pp. 400-401 ◽

Cited By ~ 4

Author(s):

M. Kac

Keyword(s):

Quadratic Forms ◽

Linear And Quadratic Forms

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An invariance property of quadratic forms in random vectors with a selection distribution, with application to sample variogram and covariogram estimators

Annals of the Institute of Statistical Mathematics ◽

10.1007/s10463-008-0175-3 ◽

2008 ◽

Vol 62 (2) ◽

pp. 363-381 ◽

Cited By ~ 11

Author(s):

Reinaldo B. Arellano-Valle ◽

Marc G. Genton

Keyword(s):

Quadratic Forms ◽

Invariance Property ◽

Random Vectors

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On quadratic forms in multivariate generalized hyperbolic random vectors

Biometrika ◽

10.1093/biomet/asaa067 ◽

2020 ◽

Author(s):

Simon A Broda ◽

Juan Arismendi Zambrano

Keyword(s):

Least Squares ◽

Quadratic Forms ◽

Inversion Formula ◽

Expected Shortfall ◽

Distribution Functions ◽

Least Squares Estimator ◽

Tail Probabilities ◽

Random Vectors ◽

Two Stage ◽

Approximate Expressions

Summary This article presents exact and approximate expressions for tail probabilities and partial moments of quadratic forms in multivariate generalized hyperbolic random vectors. The derivations involve a generalization of the classic inversion formula for distribution functions (Gil-Pelaez, 1951). Two numerical applications are considered: the distribution of the two-stage least squares estimator and the expected shortfall of a quadratic portfolio.

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