Ranges of Extremal Processes and Heavy-Tailed Random Walks in Spaces of Growing Dimension

We consider extremal processes and random walks generated by heavy-tailed random vectors taking values in $\mathbb{R}^d$ with the $\ell_p$ distance. We establish limit theorems for their paths in the triangular array setting when both the number of steps $n$ and the dimension $d$ grow to infinity. It is shown that it is possible to map the paths using isometries of $\ell_p$ such that the images converge and to identify the limit as derived from a Poisson cluster process. This also implies the convergence in distribution of the paths interpreted as finite metric spaces on the family of metric spaces equipped with the Gromov-Hausdorff distance. We also show that the images of the paths converge in distribution in the space of counting measures on the line equipped with the Hausdorff metric generated by a suitable variant of $\ell_p$ distance between counting measures.

💡 Research Summary

The paper investigates the asymptotic behaviour of extremal processes and heavy‑tailed random walks when both the number of observations n and the ambient dimension d grow to infinity. The authors work in ℓₚ spaces (1 ≤ p ≤ ∞) and consider i.i.d. random vectors X^{(d)}∈ℝ^{d}{+} whose coordinates are heavy‑tailed and possibly dependent across dimensions. The central object of study is the partial coordinate‑wise maxima M^{(d)}{n}=X^{(d)}{1}∨⋯∨X^{(d)}{n} and, in parallel, the random walk S^{(d)}{k}=X^{(d)}{1}+⋯+X^{(d)}_{k}. Both are viewed as finite metric sub‑sets of ℝ^{d} equipped with the ℓₚ norm.

A major difficulty is that for p≠2 the group of isometries of ℓₚ is very small (essentially permutations of coordinates together with a limited class of “spreading” maps). Consequently, classical techniques that rely on rich orthogonal transformations (as in the ℓ₂ case) cannot be applied. To overcome this, the authors encode each vector x∈ℝ^{d} by a counting measure m(x)=∑{i=1}^{d}δ{x_i} on ℝ. They introduce a distance ϑₚ between counting measures defined as the minimal ℓₚ distance over all finite permutations of the atoms. Lemma A.1 shows that the metric space (ℓₚ,ρ_{Π,p}) is isometric to (𝒩,ϑₚ), where 𝒩 denotes the space of locally finite counting measures. This representation makes the problem invariant under permutations and thus suitable for a triangular‑array setting where d=d(n)→∞.

The authors impose three structural conditions.

(A) Regular variation in the triangular array: there exists a normalising sequence a_n such that a_n^{-1}χ^{(d)} (where χ^{(d)}=m(X^{(d)})) converges vaguely on a suitable boundedness family Sₚ to a non‑trivial tail measure ν on 𝒩. This is the analogue of regular variation for vectors of growing dimension.

(B) Asymptotic non‑overlap of large components: for any ε,s>0 the probability that two independent copies have a coordinate where both exceed ε a_n while each also exceeds s a_n tends to zero. Equivalently, the coordinate‑wise minima of two independent vectors, after normalisation, vanish in probability. This replaces the orthogonality assumptions used in earlier ℓ₂ work.

(C) Negligibility of small components: a moment condition ensuring that contributions from coordinates smaller than s a_n become irrelevant as s↓0. For p<∞ it involves the p‑th power of the ℓₚ norm; for p=∞ it involves the sup‑norm.

Under (A)–(C) the authors construct a Poisson point process η=∑δ_{(t_k,μ_k)} on ℝ_{+}×𝒩 with intensity λ⊗ν (λ is Lebesgue measure). From η they define two limit objects:

1. Counting‑measure valued process υ_t=∑_{t_k≤t}μ_k, t≥0, whose range Y_T={υ_t:0≤t≤T} is a compact subset of 𝒩.

2. Crinkled subordinator Y_t=∑_{t_k≤t}‖μ_k‖ₚ e_k, t≥0, where {e_k} is the canonical basis of ℓₚ. Its closed range Y_T=cl{Y_t:0≤t≤T} lives in ℓₚ and is isometric to the interval