Abstract. A Bernstein-type exponential inequality for (generalized) canonical U -statistics of order 2 is obtained and the Rosenthal and Hoff-mannJørgensen inequalities for sums of independent random variables are extended to (generalized) U -statistics of any order whose kernels are either nonnegative or canonical.
We derive two-sided estimates on moments and tails of Gaussian chaoses, that
is, random variables of the form $\sum a_{i_1,...,i_d}g_{i_1}... g_{i_d}$,
where $g_i$ are i.i.d. ${\mathcal{N}}(0,1)$ r.v.'s. Estimates are exact up to
constants depending on $d$ only.Comment: Published at http://dx.doi.org/10.1214/009117906000000421 in the
Annals of Probability (http://www.imstat.org/aop/) by the Institute of
Mathematical Statistics (http://www.imstat.org
Let a ∈ [0, 1] and r ∈ [1, 2] satisfy relation r = 2/(2 − a). Let µ(dx) = c n r exp(−(|x1| r +|x2| r +. . .+|xn| r ))dx1dx2 . . . dxn be a probability measure on the Euclidean space (R n , · ). We prove that there exists a universal constant C such that for any smooth real function f on R n and anyWe prove also that if for some probabilistic measure µ on R n the above inequality is satisfied for any p ∈ [1, 2) and any smooth f then for any h : R n −→ R such that |h(x) − h(y)| ≤ x − y there is Eµ|h| < ∞ andfor t > 1, where K > 0 is some universal constant.Let us begin with few definitions.Definition 1 Let (Ω, µ) be a probability space and let f be a measurable, square integrable non-negative function on Ω. For p ∈ [1, 2) we define the p-variance of f bywhere Ent µ denotes a classical entropy functional (see [L] for a nice introduction to the subject). * Research partially supported by KBN Grant 2 P03A 043 151 Definition 2 Let E be a non-negative functional on some class C of non-negative functions from L 2 (Ω, µ). We will say that f ∈ C satisfies• the Poincaré inequality with constant• the logarithmic Sobolev inequality with constantLemma 1 For a fixed f ∈ C and p ∈ [1, 2) let
Then ϕ is a non-decreasing function.Proof. Hölder's inequality yields that α(t) = t ln(E µ f 1/t ) is a convex function for t ∈ (1/2, 1]. Hence also β(t) = e 2α(t) = (E µ f 1/t ) 2t is convex and therefore β(t)−β(1/2) t−1/2 is non-decreasing on (1/2, 1]. Observation thatcompletes the proof. 2Corollary 1 For f ∈ C the following implications hold true:• f satisfies the Poincaré inequality with constant C if and only if f satisfies I µ (0) with constant C,• if f satisfies the logarithmic Sobolev inequality with constant C then f satisfies I µ (1) with constant C,• if f satisfies I µ (1) with constant C then f satisfies the logarithmic Sobolev inequality with constant 2C,• if f satisfies I µ (a) with constant C and 0 ≤ α ≤ a ≤ 1 then f satisfies I µ (α) with constant C.Proof.• To prove the first part of Corollary 1 it suffices to note that p −→ V ar(p) µ (f ) is a non-increasing function.
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