Abstract
We provide necessary and sufficient conditions for hypercontractivity of the minima of nonnegative, i.i.d. random variables and of both the maxima of minima and the minima of maxima for such r.v.'s. It turns out that the idea of hypercontractivity for minima is closely related to small ball probabilities and Gaussian correlation inequalities.
Highlights
Let h be a real valued function defined on ∪∞ n=1 IRn and let 0 < p < q
Among other things we prove an analogous theorem to that of de la Pena, Montgomery-Smith and Szulga (Theorem 3.3) giving a sufficient condition for P (X ≤ ct) ≤ δP (Y ≤ t) for all t ≤ c X p for some 0 < c, δ < 1
We combine our theorem with a version of that of de la Pena, Montgomery-Smith and Szulga (Theorem 5.4) to give a sufficient condition for the comparison of the tail distributions of X and Y to hold for all t ∈ IR+. Another application of the technique we develop here is contained in Corollary 5.2, where we give a sufficient condition for a random variable X to be hypercontractive with respect to another interesting function(s) - the k-order statistic(s)
Summary
Let h be a real valued function defined on ∪∞ n=1 IRn and let 0 < p < q. By Lemma 2.3 it is enough to prove that there exists σ > 0, such that for each concave g: IR+ → IR+, (Egq(σX))1/q ≤ Eg(X), where we may assume that g is constant on (M, ∞) To prove this inequality for such a g we first note that by Theorem 3.4, X is {q, 1}-min-hypercontractive, there exists σ > 0 such that (Ehqt (σX))1/q ≤ Eht(X) for each t ≥ 0, where ht is given by ht(x) = x ∧ t. Let ρ be any positive number, and let τ be such that the inequality in Theorem 3.4 (iii) holds for ε = pq−1(1 − rq) for all t ≤ t0 = ρ X q. Taking into account the remarks before Theorems 3.4 and 4.6 we check that given p, q the constant σ depends only on the min and max hypercontractivity constants of X. Let X, Y be nonnegative r.v.’s such that X ∈ min Hp,q(C1)∩max Hp,q(C2) and there exist constants B1 and B2 such that mn(Y ) q ≤ B1 mn(X) q and Mn(Y ) q ≤ B2 Mn(X) q for all n, there exists a constant D, depending only on p, q, C1, C2, B1 and B2, such that P (Y ≤ t) ≥ P (DX ≤ t) for all t ∈ IR+
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