Large deviations for sums of multivariate stretched-exponential random variables: the few-big-jumps principle

TL;DR

The paper develops a multivariate large deviations theory for sums of i.i.d. vectors with stretched-exponential tails, introducing a multivariate tail rate $\mathcal{J}$ and index $\alpha\in(0,1)$. A few-big-jumps principle is established: the large deviation probability decays like $\exp(-x_N^{\alpha}\mathcal{I}_{\mathcal{J}}(\mathbf{t}))$, with the rate function $\mathcal{I}_{\mathcal{J}}$ given by a minimization over at most $k$ summands of $\mathcal{J}$. The framework yields a comprehensive LDP, a moderate-deviation principle, and a suite of applications, including absolute powers of multivariate Gaussian vectors and high-dimensional projections of $\ell_p^N$-balls, revealing rich geometric and probabilistic structure. The results show that, unlike the 1D case, deviations in higher dimensions can be realized by multiple joint contributions, and the rate functions can be non-convex, influencing both theory and applications in high-dimensional probability and geometry.

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.

Theorems & Definitions (41)

• Definition 2.1: Multivariate stretched-exponential tails
• Theorem A: Large deviations for sums of stretched-exponential random vectors
• Remark 2.2
• Theorem B: Moderate deviations
• Remark 2.3
• Lemma 3.1
• Lemma 3.2
• Theorem 3.3
• Corollary 3.4
• Theorem 3.5