Krzysztof Oleszkiewicz

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In this paper, we study functions with low influences on product probability spaces. The analysis of Boolean functions f {-1, 1}/sup n/ /spl rarr/ {-1, 1} with low influences has become a central problem in discrete Fourier analysis. It is motivated by fundamental questions arising from the construction of probabilistically checkable proofs in theoretical(More)
Let a ∈ [0, 1] and r ∈ [1, 2] satisfy relation r = 2/(2− a). Let μ(dx) = cr exp(−(|x1|+|x2|+. . .+|xn|))dx1dx2 . . . dxn be a probability measure on the Euclidean space (R, ‖ · ‖). We prove that there exists a universal constant C such that for any smooth real function f on R and any p ∈ [1, 2) Eμf 2 − (Eμ|f |) ≤ C(2− p)Eμ‖∇f‖. We prove also that if for(More)
We consider polytopes in Rn that are generated by N vectors in Rn whose coordinates are independent subgaussian random variables. (A particular case of such polytopes are symmetric random ±1 polytopes generated by N independent vertices of the unit cube.) We show that for a random pair of such polytopes the Banach-Mazur distance between them is essentially(More)
Let S = a1r1+a2r2+ · · ·+anrn be a weighted Rademacher sum. Friedgut, Kalai, and Naor have shown that if Var(|S|) is much smaller than Var(S), then the sum is largely determined by one of the summands. We provide a simple and elementary proof of this result, strengthen it, and extend it in various ways to a more general setting. ACM Classification: G.3 AMS(More)
We give a short proof of a result of G. Paouris on the tail behaviour of the Euclidean norm |X| of an isotropic log-concave random vector X ∈ R, stating that for every t ≥ 1, P ` |X| ≥ ct √ n ́ ≤ exp(−t √ n). More precisely we show that for any log-concave random vector X and any p ≥ 1, (E|X|) ∼ E|X|+ sup z∈Sn−1 (E|〈z, X〉|). AMS Classification: 46B06, 46B09(More)
Pay-as-bid is the most popular auction format for selling treasury securities. We prove the uniqueness of pure-strategy Bayesian-Nash equilibria in pay-as-bid auctions where symmetrically-informed bidders face uncertain supply, and we establish a tight sufficient condition for the existence of this equilibrium. Equilibrium bids have a convenient separable(More)