2022
DOI: 10.1007/s10959-022-01156-2
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High-Dimensional Central Limit Theorems for Homogeneous Sums

Abstract: This paper develops a quantitative version of de Jong’s central limit theorem for homogeneous sums in a high-dimensional setting. More precisely, under appropriate moment assumptions, we establish an upper bound for the Kolmogorov distance between a multi-dimensional vector of homogeneous sums and a Gaussian vector so that the bound depends polynomially on the logarithm of the dimension and is governed by the fourth cumulants and the maximal influences of the components. As a corollary, we obtain high-dimensio… Show more

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Cited by 5 publications
(3 citation statements)
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“…The presence of the factor (log m) 1/4 is consistent with the fact that, for the standard Gaussian measure on R m , the isoperimetric constant associated with all hyper-rectangles of R m is bounded from above by √ log m, see [3,14]. An estimate analogous to (23) is established by different methods in [12,Corollary 3.1].…”
Section: A Proof and Discussion Of Relation (3)mentioning
confidence: 65%
“…The presence of the factor (log m) 1/4 is consistent with the fact that, for the standard Gaussian measure on R m , the isoperimetric constant associated with all hyper-rectangles of R m is bounded from above by √ log m, see [3,14]. An estimate analogous to (23) is established by different methods in [12,Corollary 3.1].…”
Section: A Proof and Discussion Of Relation (3)mentioning
confidence: 65%
“…Since we have max Thus we obtain (2.3). Finally, we have by Lemma 2.1 in Koike (2019) tr(A 4 j ) | W 4 j − 3( W 2 j ) 2 | + C 0 M A j 2 H.S. M(A j )…”
Section: Proof Of Applicationsmentioning
confidence: 84%
“…We can modify Assumption 2(i) to for any , a standard assumption in the literature on ultra-high-dimensional data analysis. This assumption ensures subexponential upper bounds for the tail probabilities of the statistics in question when , as discussed in [ 23 , 24 ]. The requirement of sub-Gaussian properties in Assumption 2(i) is made for the sake of simplicity.…”
Section: Theoretical Resultsmentioning
confidence: 99%