Failure of Convex-Hull Bounds under Log-Convex Tails

Abstract: Fix $0&lt;r\&lt;1$, and let $X_1,X_2,\dots$ be independent symmetric Weibull$(r)$ random variables, that is, \[ \textsf{P}(|X_i|>t)=e^{-t^r},\qquad t\ge 0. ] We prove that there is no constant $C_r$, depending only on $r$, with the following universal property: for every finite set $T\subset \R<sup>N$ there exists a sequence $(y_k)<em>{k\ge 1}\subset \R<sup>N$ such that [ T-T\subset conv{y_k:k\ge 1}, \qquad |X{y_k}|{L{\log(k+2)}}\le C_r\,\bx(T) \quad (k\ge 1), ] where $X_t=\sum_i t_i X_i$ and $\bx(T)=\textsf{E}\sup_{t\in T}X_t$. This gives a negative answer to a question of Latała concerning the validity of convex-hull bounds for canonical Weibull processes. In fact, the failure persists even when the auxiliary vectors appearing in the convex hull are allowed to be arbitrary.

• The paper demonstrates the failure of convex-hull moment covering for canonical processes with log-convex tails using explicit counterexamples.
• It employs probabilistic constructions and VC dimension theory to show that heavy-tailed Weibull process moments defy universal moment bounds.
• The results necessitate rethinking chaining methods and inspire new research into bounding suprema in high-dimensional heavy-tailed settings.

Negative Resolution of the Convex-Hull Bound for Canonical Weibull Processes with Log-Convex Tails

Problem Overview and Context

The geometry and moment growth of random processes indexed by finite sets in high-dimensional spaces has far-reaching implications in probability theory and functional analysis, particularly under heavy-tailed distributions. A fundamental challenge is estimating the expected supremum $b_X(T) = \mathbb{E} \sup_{t \in T} X_t$ for $X_t = \langle t, X \rangle$, where $X$ is a random vector with independent, centered coordinates. While Talagrand’s generic chaining theorem provides sharp two-sided bounds for Gaussian processes, the extension to non-Gaussian or heavy-tailed settings is highly nontrivial and sensitive to the tail behavior of the coordinates.

For canonical processes generated by independent variables with log-concave tails, substantial progress has been made, notably via Latała’s chaining-based convex-hull method, which establishes a deterministic covering of the increment set $T-T$ by a convex hull of auxiliary vectors with controlled $L_p$-moments. However, the validity of this convex-hull reduction for canonical processes with log-convex tails—epitomized by symmetric Weibull($r$) variables with $0 < r < 1$—remained unresolved until this work.

Main Results and Their Methodological Foundation

This paper decisively demonstrates the failure of the convex-hull moment covering principle for canonical processes with log-convex tails, concretely refuting a conjecture of Latała. Specifically, the authors construct, for every $0 < r < 1$, finite sets $T \subset \mathbb{R}^N$ such that no absolute constant $C_r$ (depending only on $X_t = \langle t, X \rangle$0) can ensure existence of a uniformly covering sequence $X_t = \langle t, X \rangle$1 both

• covering $X_t = \langle t, X \rangle$2 in the (absolutely) convex hull, and
• satisfying moment bounds: $X_t = \langle t, X \rangle$3 for all $X_t = \langle t, X \rangle$4,

even when the auxiliary vectors $X_t = \langle t, X \rangle$5 are allowed to be arbitrary.

The proof is probabilistic and hinges on the rapidly growing moments of canonical Weibull($X_t = \langle t, X \rangle$6) random variables. The main steps involve:

• Construction of the index set $X_t = \langle t, X \rangle$7 via random sign vectors in high dimension, ensuring $X_t = \langle t, X \rangle$8 remains controlled.
• Reduction of the covering problem to the smallness of certain "exposed" sets within the discrete cube $X_t = \langle t, X \rangle$9, parameterized by the auxiliary sequence and governed by their log-convex moment growth.
• Application of VC dimension theory to show that, for the range of parameters of interest, the probability that any admissible (even sample-dependent) sequence $X$0 can cover all points in $X$1 while respecting the prescribed moment constraint decays exponentially with the dimension—implying existence of $X$2 where such a covering is impossible.

This refutation holds even when passing to the absolutely convex hull, a more permissive structure than the convex hull, underscoring that the obstruction is not geometric but rather arises from the incompatibility of tail growth and moment control constraints.

Numerical and Structural Claims

A quantitative counterexample is constructed: for large parameters $X$3, sets $X$4 are formed in $X$5 dimensions (for $X$6) with $X$7 points, with $X$8, yet no sequence $X$9 can both cover $T-T$0 and satisfy the moment bound $T-T$1 for $T-T$2. These explicit scalings sharply illustrate the failure of the convex-hull-covering principle in the regime of interest.

Theoretical and Practical Implications

The breakdown of the convex-hull principle invalidates the most direct extension of the chaining paradigm from log-concave to log-convex (superheavy-tailed) canonical processes. For researchers, this signals an essential divergence in the geometry of increments and the behavior of suprema for such processes. The findings close a conjecture and necessitate new techniques for bounding suprema, as methods successful for Gaussian or log-concave variables fail fundamentally in the Weibull ($T-T$3) context. The result also warns against indiscriminate use of convex-hull type reductions in high-dimensional heavy-tailed settings prevalent in applied mathematics, statistics, and theoretical computer science.

In the broader context of random process theory, the work underscores the deep interplay between moment growth, geometric covering, and combinatorial complexity (via VC theory), especially as one moves away from sub-Gaussian or even sub-exponential regimes.

Prospects for Future Research

The results prompt several directions:

• Development of finer extremal inequalities or structural decompositions for canonical processes under log-convex or irregular moment growth, possibly exploiting tail-dependent combinatorial or geometric representations.
• Exploration of whether alternative chaining notions, or hybrid convex-geometric-probabilistic bounds, may be effective in this setting.
• Further investigation into the threshold between log-concave and log-convex regimes where convex-hull bounds may break down as a function of moment growth, and implications for empirical processes and applications in statistical learning with heavy-tailed noise.

Conclusion

This paper rigorously establishes the nonexistence of universal convex-hull moment bounds for canonical processes with log-convex tails, using canonical Weibull($T-T$4) random variables as an explicit, natural family. The results decisively settle an open problem raised by Latała and clarify the geometric and probabilistic obstructions underlying suprema of heavy-tailed processes in high dimension. The work opens the field to new methods for bounding process suprema beyond the scope of chaining and convex-hull techniques.