On Weak Tail Domination of Random Vectors

Motivated by a question of Krzysztof Oleszkiewicz we study a notion of weak tail domination of random vectors. We show that if the dominating random variable is sufficiently regular weak tail domination implies strong tail domination. In particular positive answer to Oleszkiewicz question would follow from the so-called Bernoulli conjecture. Introduction. This note is motivated by the following problem about Rademacher series, posed by Krzysztof Oleszkiewicz (private comunication): Problem. Suppose that (εi) is a Rademacher sequence (i.e. sequence of independent symmetric ±1 r.v.’s) and xi, yi are vectors in some Banach space F such that the series ∑ i xiεi and ∑ i yiεi are a.s. convergent and ∀x∗∈F ∗∀u>0 P (∣∣∣x∗(∑ i xiεi )∣∣∣ ≥ t) ≤ P(∣∣∣x∗(∑ i yiεi )∣∣∣ ≥ t). Does it imply that E ∥∥∥∑ i xiεi ∥∥∥ ≤ LE∥∥∥∑ i yiεi ∥∥∥, for some universal constant L <∞? Motivated by the above question we introduce a notion of weak tail domination of random vectors. We show that if the dominating vector has a regular distribution (including Gaussian case), weak tail domination yields strong tail domination (Theorem 1). In particular Oleszkiewicz question has positive answer provided that the so-called Bernoulli Conjecture is satisfied. Finally we show that in general weak tail domination does not yield comparison of means or medians of norms even if the distribution of dominated vector is Gaussian. ∗Research partially supported by MEiN Grant 1 PO3A 012 29.