On the subgaussian comparison theorem

The aim of this expository note is to prove that any $1$-subgaussian random vector is dominated in the convex ordering by a universal constant times a standard Gaussian vector. This strengthens Talagrand’s celebrated subgaussian comparison theorem. The proof combines a tensorization argument due to J. Liu with ideas that date back to the work of Fernique.

💡 Research Summary

The paper presents a concise yet powerful extension of Talagrand’s sub‑gaussian comparison theorem. The authors prove that any random vector X in ℝⁿ satisfying the 1‑subgaussian condition (centered and with one‑dimensional marginals dominated by a standard Gaussian tail) is dominated in the convex order by a universal constant multiple of a standard Gaussian vector G ∼ N(0,Iₙ). In other words, there exists a constant c>0 such that for every convex function f:ℝⁿ→ℝ,
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