Tests for white noise via asymptotically independent U-statistics in high-dimensions
Abstract
Existing tests for high-dimensional white noise are tailored to the two ends of the sparsity spectrum: maximum-type statistics excel against a few dominant correlations and sum-type statistics against many weak ones, leaving the intermediate regime without a systematic tool. We propose a family of U-statistics built from the entry-wise ℓa-norms of the sample autocovariance matrices, in which the even order a interpolates between the sum-type (a=2) and maximum-type (a→∞) endpoints. As p and T diverge jointly, martingale difference theory yields joint asymptotic normality across orders together with asymptotic independence, so that several orders can be combined after a simple calibration. Neither cross-sectional independence nor a mixing condition is required, only a spectral condition on bSig0; the derivation links cross-sectional correlation to a graph structure that prices each moment edge by edge. Simulations confirm accurate size and power that is stable across sparse, intermediate and dense alternatives, and show that the low-order members of the family retain their level under conditional heteroscedasticity, where the existing tests distort severely.
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BibTeX
@misc{xu2026tests,
title = {{Tests for white noise via asymptotically independent U-statistics in high-dimensions}},
author = {Yuanya Xu},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2605.04968},
eprint = {2605.04968},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2605.04968},
note = {CC BY-NC-ND 4.0},
}
RIS
TY - UNPB
TI - Tests for white noise via asymptotically independent U-statistics in high-dimensions
AU - Yuanya Xu
PY - 2026
DA - 2026/10/04
PB - arXiv
DO - 10.48550/arxiv.2605.04968
UR - https://arxiv.org/abs/2605.04968
AB - Existing tests for high-dimensional white noise are tailored to the two ends of the sparsity spectrum: maximum-type statistics excel against a few dominant correlations and sum-type statistics against many weak ones, leaving the intermediate regime without a systematic tool. We propose a family of U-statistics built from the entry-wise ℓ_a-norms of the sample autocovariance matrices, in which the even order a interpolates between the sum-type (a=2) and maximum-type (a→∞) endpoints. As p and T diverge jointly, martingale difference theory yields joint asymptotic normality across orders together with asymptotic independence, so that several orders can be combined after a simple calibration. Neither cross-sectional independence nor a mixing condition is required, only a spectral condition on bSig_0; the derivation links cross-sectional correlation to a graph structure that prices each moment edge by edge. Simulations confirm accurate size and power that is stable across sparse, intermediate and dense alternatives, and show that the low-order members of the family retain their level under conditional heteroscedasticity, where the existing tests distort severely.
ER -