A counter-example linked to Gaussian convex hulls

Abstract: We consider the sequence of independent centered Gaussian random elements of a separable Banach space and their consecutive closed convex hulls. If inicial elements converge weakly to some limite, then, as shown in Davydov- Paulauskas (2024), its normalized convex hulls converge, with probability 1, to the concentration ellipsoid of the limiting distribution. The goal of the present note is to show that if the assumption of weak convergence of the initial sequence is relaxed, than the limit set can be an arbitrary convex compact set.

• The paper establishes that every symmetric convex compact set can emerge as the almost sure limit of normalized convex hulls from Gaussian vectors.
• It employs tailored sequences along dense directions in Banach spaces and maximal inequalities to ensure uniform convergence of support functions.
• The result cautions high-dimensional analyses by showing that without weak convergence, the geometry of random convex hulls may diverge from classical ellipsoidal limits.

Counter-Examples in Gaussian Convex Hull Asymptotics

Problem Context and Prior Work

This paper addresses the asymptotic behavior of normalized convex hulls formed by independent centered Gaussian random elements $\{X_n\}$ in a separable Banach space $B$. Classical results, such as that of Goodman [4], established that when $\{X_n\}$ share a common distribution, the normalized convex hulls converge almost surely (in the Hausdorff metric) to the concentration ellipsoid of the underlying Gaussian measure. This conclusion was further extended to more general contexts—including sequences in Skorokhod spaces [1], stationary weakly dependent Gaussian fields [2], and under weak convergence of measures [3]. Importantly, [3] also constructed examples showing that the absence of weak convergence can yield limit sets that diverge from the ellipsoidal archetype, such as centrally symmetric polytopes.

Main Contribution

The author establishes that, beyond previous constructions, the class of possible limit sets for normalized convex hulls of Gaussian samples is maximally broad. The main theorem demonstrates that every convex, compact, centrally symmetric set $V \subset B$ can be realized as the almost sure limit (in the Hausdorff metric) of a sequence of normalized convex hulls from independent Gaussian vectors, even though the input sequence does not converge weakly.

Formally, for any such $V$, there exists a sequence $\{ X_k \}$ of independent Gaussian vectors in $B$, with uniformly bounded second moments, such that if $W_n = \mathrm{conv}\{ X_1, ..., X_n \}$ and $b(n)$ is an appropriate normalization, then $b(n)^{-1} W_n \to V$ almost surely.

Technical Approach

The construction relies on tailored sequences of Gaussian vectors whose distributions are concentrated on lines directed by a dense subset of the unit sphere in $B$0. The indices are partitioned into disjoint sets $B$1 with prescribed asymptotic densities, each governing the frequency of a particular direction $B$2. Along each such direction, the vectors are scaled so that their support under the normalization matches that of $B$3. By carefully controlling the variances and using maximal inequalities for Gaussian processes, it is shown that the convex hull, under normalization, approaches $B$4 almost surely.

Key arguments involve controlling the support function $B$5 of the normalized convex hulls and showing (via a sandwiching argument and classical properties of support functions) that it converges uniformly to the support function of $B$6. The probabilistic arguments are grounded in exponential moment estimates and the Borel-Cantelli lemma. The technical prerequisites are ensured by Fernique’s theorem [5] concerning regularity and moment boundedness for Gaussian processes.

Implications

This result overturns the intuition that Gaussian convex hulls are inherently “asymptotically ellipsoidal” unless the sequence is strongly degenerate: the absence of weak convergence opens the door to arbitrary symmetrical convex compact sets as a.s. limits. In the context of probability theory and convex geometry, this sharply delimits the scope of geometric law of large numbers phenomena for random convex hulls, showing that weak convergence is not only sufficient but essentially necessary for ellipsoidal limit sets.

From a practical perspective, the result provides caution to those leveraging convex hulls of Gaussian samples in high-dimensional statistical analysis or random geometric algorithms, as the boundary structure of sample convex hulls may diverge dramatically from the covariance ellipsoid unless input distributions are tightly controlled. Theoretically, the paper underscores the richness of geometric limit operations and connects convergence in measure (weak convergence) with robust geometric convergence phenomena.

Prospects for Future Work

The construction suggests multiple avenues for further investigation. Extensions to non-symmetric convex sets, non-Gaussian input sequences, or explicit characterization of the rates of convergence are immediate possibilities. Additionally, links to small deviation and concentration inequalities for Gaussian processes in Banach spaces, as well as computational implications in the design of randomized algorithms for geometry, merit closer study. The question of universality classes for convex hull asymptotics under dependent or non-identically distributed Gaussian sequences remains largely open.

Conclusion

The paper establishes that under minimal constraints (independence and bounded second moments, without weak convergence), normalized convex hulls of Gaussian vectors in a Banach space can converge almost surely to any pre-specified convex, compact, centrally symmetric set. This result delimits the predictive scope of classical convex hull limit theorems and compels careful scrutiny of distributional assumptions when analyzing the geometry of random point clouds in infinite-dimensional settings.