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

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.

💡 Research Summary

The paper develops a comprehensive large‑deviation theory for sums of independent, identically distributed (i.i.d.) random vectors whose marginal distributions have stretched‑exponential (Weibull or semi‑exponential) tails. In one dimension, the classical “one‑big‑jump” principle, originating from Nagaev’s work in the 1960s, states that a large deviation of the empirical mean is essentially caused by a single unusually large observation. Extending this principle to higher dimensions is non‑trivial because several coordinates may need to be large simultaneously, and dependence among coordinates can create more complex mechanisms.

Key definitions.
For a fixed exponent α∈(0,1) the authors introduce a multivariate stretched‑exponential tail via a lower‑semicontinuous, α‑homogeneous function J:ℝ⁺ᵏ→