The result in (Vershynin 2018, Lemma 2.7.6, ii)) and the log-mgf bound in Lemma 1 gives ||Y||ψ1≤C (√v+b ) =C( √ λ (1−θ)3/2 + |2θ+ 1| 3(1−θ)2 + (1−θ)3r 12λ Br(θ,λ))

SincethedefinitionofOrlicznorm ||Z||ψ1 = inf{K >0 :E(exp(|Z|/K))≤2}, requires E(exp(|Z|/K))≤2, the log-mgf bound in Lemma 1 can be used to derive, through Chernoff bound, tail-integral representation, a bound for the Orlicz norm · 2018