Distinctness of the Eigenvalues of a Quadratic form in a Multivariate Sample

By M. Okamoto. 1973

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Multi-Level Cause-Effect Systems (2015, background): “;i)P(ijh)P(h) which we can rewrite in terms of the independent parameters as X h [ j 1;h;i j 2;h;i] h;i h= 0; (1) which is a polynomial constraint on P[ G;; ]. By a simple algebraic lemma (proven by Okamoto, 1973), if the above constraint is not trivial (that is, if there exists for which the constr”
Additive models with trend filtering (2017, background): “, d, and by Lemma 1 of Okamoto (1973), the roots of any polynomial (that is not identically zero) form a set of Lebesgue measure zero.”
Visual Causal Feature Learning (2014, background): “By a simple algebraic lemma (proven by Okamoto, 1973), if the above constraint is not trivial (that is, if there exists β for which the constraint does not hold), the subset of P [β;α, γ] on which it holds is measure zero.”
Strong completeness and faithfulness in Bayesian networks (1995, background): “The proof of this claim follows from the fact that the solution set to non-trivial polynomial constraints has measure zero (See Okamoto 1973).”
Principal Component Analysis of High-Frequency Data (2016, unknown): “Almost surely, the eigenvalues stacked in λ̂ikn n are distinct (see, e.g., Okamoto 1973) so that we have λ̂1,ikn n > λ̂2,ikn n > · · · > λ̂d,ikn n .”
Semiparametric estimation of fractional cointegrating subspaces (2006, background): “First, note that the eigenvalues of ̃D are distinct, with probability 1, by Okamoto [14].”
Semiparametric Estimation of Multivariate Fractional Cointegration (2003, unknown): “Because we are assuming that m ¸ q ¡ r, it follows from the theorem of Okamoto (1973) that both UU0 and VV0 are positive de nite with probability 1.”
Causal feature learning: an overview (2017, unknown): “By an algebraic lemma (proven by Okamoto 1973), the subset on which the constraint holds is measure zero if the constraint is not trivial.”
Random Coefficients on Endogenous Variables in Simultaneous Equations Models (2014, methodology): “This mapping from the parameters {πki/πii : k 6= i} to {γk : k 6= i} is one-to-one and differentiable and hence the joint distribution of {γk : k 6= i} can be written in terms of the joint distribution of {πki/πii : k 6= i} via the change of variables formula (e.g., Munkres (1991) theorem 17.2).”
A New Identification Condition for Recursive Models With Correlated Errors (2002, background): “And, from Okamoto (1973), we can further conclude that this set has a Lebesgue measure of zero.”
A Bayesian information criterion for singular models (2013, methodology): “ial equation g ij(! j) = 0 that does not hold for all points in j. Here, j = RN i Ri M is the parameter space of M j. Since g ij is a nonzero polynomial, its zero set has measure zero by the lemma in Okamoto (1973). Consequently, ( i(ˇ 0);m i(ˇ 0)) = ( ij;m ij) holds almost surely under P(ˇ 0 jM j;Y”
A matrix handbook for statisticians (2007, unknown): “Ostrowski, A. M. (1973).”
Global identifiability of linear structural equation models (2010, result): “In particular, the condition holds for almost every point (in Lebesgue measure), and some authors thus also speak of an almost everywhere identifiable model; compare the lemma in [10].”
A Simple Rule for the Selection of Principal Components (2003, background): “In addition, since the sample eigenvalues are distinct (Okamoto, 1973 ) and constitute an ordered sample, their distributions will differ.”