Let q ≥ 2 be a positive integer, B be a fractional Brownian motion with Hurst index H ∈ (0, 1), Z be an Hermite random variable of index q, and H q denote the Hermite polynomial having degree q. For any n ≥ 1, setThe aim of the current paper is to derive, in the case when the Hurst index verifies H > 1 − 1/(2q), an upper bound for the total variation distance between the laws L (Z n ) and L (Z), where Z n stands for the correct renormalization of V n which converges in distribution towards Z. Our results should be compared with those obtained recently by Nourdin and Peccati (2007) in the case when H < 1 − 1/(2q), corresponding to the situation where one has normal approximation.
In this short note, we show how to use concentration inequalities in order to build exact confidence intervals for the Hurst parameter associated with a one-dimensional fractional Brownian motion.
We consider a collection of Euclidean random balls in R d generated by a determinantal point process inducing interaction into the balls. We study this model at a macroscopic level obtained by a zooming-out and three different regimes -Gaussian, Poissonian and stable-are exhibited as in the Poissonian model without interaction. This shows that the macroscopic behaviour erases the interactions induced by the determinantal point process.
arXiv:1504.04513v2 [math.PR] 1 Jun 20171 Determinantal random balls modely − x ≤ r whose centers x ∈ R d and radii r ∈ R + are generated by a marked stationary determinantal point process (Dpp) Φ on R d × R + . In this section, we describe thoroughly the model and we refer to the Appendix A for more details on Dpp, in particular see the definition in Def. A.2. First, consider a stationary Dpp φ with a kernel K with respect to the Lebesgue measure Leb satisfying K(x, y) = K(x − y) (for simplicity, we use the same letter K for two different functions), moreover we assume that the map K given for all x ∈ R d and any f ∈ L 2 (R d , dx)satisfies the following hypothesis Hypothesis 1 The map K in (3) is a bounded symmetric integral operator K from L 2 (R d , dx) into L 2 (R d , dx) with spectrum included in [0, 1[. Moreover, K is locally trace-class, i.e. for all compact Λ ⊂ E, the restriction K Λ of K on L 2 (Λ, λ) is of trace-class.
Concentration inequalities are obtained on Poisson space, for random functionals with finite or infinite variance. In particular, dimension free tail estimates and exponential integrability results are given for the Euclidean norm of vectors of independent functionals. In the finite variance case these results are applied to infinitely divisible random variables such as quadratic Wiener functionals, including Lévy's stochastic area and the square norm of Brownian paths. In the infinite variance case, various tail estimates such as stable ones are also presented.
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