[論文レビュー] Large deviations for sums of multivariate stretched-exponential random variables: the few-big-jumps principle

Gantert, Nina, N. Gantert+1 more|arXiv (Cornell University)|Feb 1, 2026

Random Matrices and Applications被引用数 0

ひとこと要約

Extends large deviation theory to sums of multivariate stretched-exponential random vectors, establishing a few-big-jumps principle where deviations are typically caused by at most k independent vectors.

ABSTRACT

Large deviations for sums of i.i.d.\ random variables with stretched-exponential tails (also called Weibull or semi-exponential tails) have been well understood since the 60's, going back to Nagaev's seminal work. Many extensions in the $1$-dimensional setting have been developed since then, showing that such deviations are typically governed by a single big jump. In higher dimensions, a corresponding theory has remained largely undeveloped. This work provides such a multivariate extension and establishes large deviation results for sums of i.i.d.\ random vectors in $\mathbb{R}^k$ under fairly general assumptions. Roughly speaking, for some $α\in(0,1)$, the log-probability of one random vector divided by $x$ exceeding a threshold $t$ in all components behaves asymptotically, for large $x$, as $x^α$ times a negative infimum of a function $\mathcal{J}$. We prove large deviation results for sums of i.i.d.\ copies, where the rate function is given by a minimization of at most $k$ summands of $\mathcal{J}$. This establishes a few-big-jumps principle that generalizes the classical $1$-dimensional phenomenon: the deviation is typically realized by \emph{at most} $k$ independent vectors. The results are applied to absolute powers of multivariate Gaussian vectors as well as to various other examples. They also allow us to study random projections of high-dimensional $\ell_p^N$-balls, revealing interesting insights about the appearance of light- and heavy-tailed distributions in high-dimensional geometry.

研究の動機と目的

• Motivate and formalize multivariate stretched-exponential tails beyond the one-dimensional setting.
• Develop a large deviation principle for sums of i.i.d. R^k-valued random vectors with a general tail rate.
• Introduce a rate function IJ that arises from minimizing over at most k summands of a base rate J.
• Demonstrate the theory with concrete multivariate examples and geometric implications in high dimensions.

提案手法

• Define multivariate stretched-exponential tails via a rate function J and α-homogeneity with α in (0,1).
• Establish a large deviation principle for the normalized sum 1/x_N N^{-1} ∑ X_i with speed x_N^α under growth x_N → ∞ and x_N N^{-1/(2-α)} → ∞.
• Construct the rate function IJ(t) as a minimization of sums of J over up to k terms, representing the few-big-jumps principle.
• Prove matching lower and upper bounds to obtain P(1/x_N ∑ X_i ≥ t) ≈ exp(-x_N^α IJ(t)).
• Provide moderate-deviation results and Gaussian-range results for the empirical mean under stretched-exponential tails.
• Apply the framework to absolute powers of multivariate Gaussians and to projections of high-dimensional ℓ_N^p balls.]
• research_questions: ["How can multivariate stretched-exponential tails be defined and characterized in R^k?","What is the large deviation principle for sums of i.i.d. R^k-valued random vectors with such tails?","How does the few-big-jumps principle generalize to higher dimensions, and what is the form of the rate function IJ?","What are concrete applications and implications for multivariate Gaussian constructions and high-dimensional geometry?","How do moderate deviations and Gaussian-range results extend in this multivariate stretched-exponential setting?"],
• key_findings: ["A multivariate stretched-exponential tail framework is established with a rate J and α in (0,1), leading to a componentwise rate IJ defined by a minimization over up to k summands of J.","The main large deviation result shows that, for x_N → ∞ and x_N N^{-1/(2-α)} → ∞, P(1/x_N ∑ X_i ≥ t) decays as exp(-x_N^α IJ(t)).","Deviations in the multivariate case are typically realized by at most k independent vector deviations, though one vector can influence multiple coordinates; the rate IJ is not necessarily convex or concave. ","A moderate-deviation principle is provided in the Gaussian regime with a quadratic rate function, under appropriate tail assumptions.","Applications include the absolute powers of multivariate Gaussian vectors, multivariate Weibull and generalized Gaussian models, Gaussian scale mixtures, and polynomial images of Gaussians, with implications for high-dimensional geometry via ℓ_N^p-ball projections.","Examples demonstrate that the framework captures both light- and heavy-tailed behavior in high-dimensional projections and relates to rate functions via infima over sums of J"],
• table_headers: []
• table_rows: []

実験結果

リサーチクエスチョン

• RQ1 研究の質問1: How can multivariate stretched-exponential tails be defined and characterized in R^k?
• RQ2 研究の質問2: What is the large deviation principle for sums of i.i.d. R^k-valued random vectors with such tails?
• RQ3 研究の質問3: How does the few-big-jumps principle generalize to higher dimensions, and what is the form of the rate function IJ?
• RQ4 研究の質問4: What are concrete applications and implications for multivariate Gaussian constructions and high-dimensional geometry?
• RQ5 研究の質問5: How do moderate deviations and Gaussian-range results extend in this multivariate stretched-exponential setting?

主な発見

• A multivariate stretched-exponential tail framework is established with a rate J and α in (0,1), leading to a componentwise rate IJ defined by a minimization over up to k summands of J.
• The main large deviation result shows that, for x_N → ∞ and x_N N^{-1/(2-α)} → ∞, P(1/x_N ∑ X_i ≥ t) decays as exp(-x_N^α IJ(t)).
• Deviations in the multivariate case are typically realized by at most k independent vector deviations, though one vector can influence multiple coordinates; the rate IJ is not necessarily convex or concave.
• A moderate-deviation principle is provided in the Gaussian regime with a quadratic rate function, under appropriate tail assumptions.
• Applications include the absolute powers of multivariate Gaussian vectors, multivariate Weibull and generalized Gaussian models, Gaussian scale mixtures, and polynomial images of Gaussians, with implications for high-dimensional geometry via ℓ_N^p-ball projections.
• Examples demonstrate that the framework captures both light- and heavy-tailed behavior in high-dimensional projections and relates to rate functions via infima over sums of J