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On asymptotic constancy of diagonal elements of a random orthogonal projection. (English. Russian original) Zbl 1308.60016

Russ. Math. Surv. 69, No. 4, 755-756 (2014); translation from Usp. Mat. Nauk 69, No. 4, 179-180 (2014).
The paper provides a theorem for the diagonal elements of a random projection obtained from the natural setting of a linear regression analysis when the rank \(p\) of the projection (i.e., the number of parameters in the regression) is of the same order as the dimension \(n\) of the initial space (i.e., the number of i.i.d.observations). It is shown under weak conditions that, in average, the diagonal elements must be close to \(p/n\). This short note is a good first step towards the understanding of random projections whose ranks are of order of the dimension of the initial large vector space. Since the topic has important applications in statistics, it will certainly lead to many extensions: bounds for a specific diagonal element, or even non-diagonal elements. Eventually, a possible weakening of the technical (third) assumption may be expected.

MSC:

60B20 Random matrices (probabilistic aspects)
62J05 Linear regression; mixed models
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