Subgaussianity is hereditarily determined

Abstract

Let n n be a positive integer, let X = ( X 1 , … , X n ) \boldsymbol {X}=(X_1,\dots ,X_n) be a random vector in R n \mathbb {R}^n with bounded entries, and let ( θ 1 , … , θ n ) (\theta _1,\dots ,\theta _n) be a vector in R n \mathbb {R}^n . We show that the subgaussian behavior of the random variable θ 1 X 1 + ⋯ + θ n X n \theta _1 X_1+\dots +\theta _n X_n is essentially determined by the subgaussian behavior of the random variables ∑ i ∈ H θ i X i \sum _{i\in H} \theta _i X_i where H H is a random subset of { 1 , … , n } \{1,\dots ,n\} .