Khinchin-type inequalities via Hadamard's factorisation

Abstract

We prove Khinchin-type inequalities with sharp constants for type L random variables and all even moments. Our main tool is Hadamard's factorisation theorem from complex analysis, combined with Newton's inequalities for elementary symmetric functions. Besides the case of independent summands, we also treat ferromagnetic dependencies in a nonnegative external magnetic field (thanks to Newman's generalisation of the Lee-Yang theorem). Lastly, we compare the notions of type L, ultra sub-Gaussianity (introduced by Nayar and Oleszkiewicz) and strong logconcavity (introduced by Gurvits), with the latter two being equivalent.

Sharp bounds on $p$-norms for sums of independent uniform random variables, $0 < p < 1$

“…Particularly challenging, interesting and conducive to new methods is the question of sharp constants in such inequalities. We only mention in passing several classical as well as recent references, [1,9,11,15,17,18,23]. This paper finishes the pursuit of sharp constants in L p −L 2 Khinchin inequalities for sums of independent uniform random variables, addressing the range 0 < p < 1.…”

Haagerup's phase transition at polydisc slicing

“…We break the integral on the left hand side into the sum of 4 integrals A 1 + • • • + A 4 over (0, 1), (1,5), (5,10) and (10, ∞). For the first one, we use (21), .…”

“…Beyond their original use, most notably, such inequalities have played an important role in Banach space theory (in connection with topics such as unconditional convergence or type and cotype), see [13,22,34,50]. Considerable work has been devoted to the pursuit of sharp constants in Khinichin-type inequalities, see for instance [3,6,15,16,19,21,31,32,33,37,38,39,40,41,42,44,46,49,51], in particular for sums of random vectors uniform on Euclidean spheres [4,9,10,26,29] (as a natural generalisation of Rademacher and Steinhaus random variables, intimately related to uniform convergence in real and complex Banach spaces, respectively). This paper continues that line of research.…”

Distributional stability of the Szarek and Ball inequalities

“…The archetype was Khinchin's result asserting that all L p norms of Rademacher sums a j ε j are comparable to its L 2 -norm, established in his work [22] on the law of the iterated logarithm (and perhaps discovered independently by Littlewood in [26]). Due to the intricacies of the methods involved, sharp Khinchin inequalities are known only for a handful of distributions, most notably random signs ( [14,29]), but also uniforms ( [4,5,6,8,18,21,25]), type L ( [17,32]), Gaussian mixtures ( [1,10]), marginals of ℓ p -balls ( [3,11]), or distributions with good spectral properties ( [23,33]). The present work makes a first step towards more general distributions satisfying only a closeness-type assumption instead of imposing structural properties.…”