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Multiple integral representation for functionals of Dirichlet processes
 Bernoulli
, 2008
"... We point out that a proper use of the Hoeffding–ANOVA decomposition for symmetric statistics of finite urn sequences, previously introduced by the author, yields a decomposition of the space of squareintegrable functionals of a Dirichlet–Ferguson process, written L 2 (D), into orthogonal subspaces ..."
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We point out that a proper use of the Hoeffding–ANOVA decomposition for symmetric statistics of finite urn sequences, previously introduced by the author, yields a decomposition of the space of squareintegrable functionals of a Dirichlet–Ferguson process, written L 2 (D), into orthogonal subspaces of multiple integrals of increasing order. This gives an isomorphism between L 2 (D) and an appropriate Fock space over a class of deterministic functions. By means of a wellknown result due to Blackwell and MacQueen, we show that each element of the nth orthogonal space of multiple integrals can be represented as the L 2 limit of Ustatistics with degenerate kernel of degree n. General formulae for the decomposition of a given functional are provided in terms of linear combinations of conditioned expectations whose coefficients are explicitly computed. We show that, in simple cases, multiple integrals have a natural representation in terms of Jacobi polynomials. Several connections are established, in particular with Bayesian decision problems, and with some classic formulae concerning the transition densities of multiallele diffusion models, due to Littler and Fackerell, and Griffiths. Our results may also be used to calculate the best approximation of elements of L 2 (D) by means of Ustatistics of finite vectors of exchangeable observations.
ON NORMAL APPROXIMATIONS TO USTATISTICS
 SUBMITTED TO THE “ANNALS OF PROBABILITY”
, 2009
"... Let X1,..., Xn be i.i.d. random observations. Let S = L + T be a Ustatistic of order k ≥ 2, where L is a linear statistic having asymptotic normal distribution, and T is a stochastically smaller statistic. We show that the rate of convergence to normality for S can be simply expressed as the rate ..."
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Let X1,..., Xn be i.i.d. random observations. Let S = L + T be a Ustatistic of order k ≥ 2, where L is a linear statistic having asymptotic normal distribution, and T is a stochastically smaller statistic. We show that the rate of convergence to normality for S can be simply expressed as the rate of convergence to normality for the linear part L plus a correction term, (var T) ln 2 (var T), under the condition E T 2 < ∞. An optimal bound without this log factor is obtained under a lower moment assumption E T  α < ∞ for α < 2. Some other related results are also obtained in the paper. Our results extend, refine and yield a number of related known results in the literature.
Hoeffding spaces and Specht modules
, 2009
"... It is proved that each Hoeffding space associated with a random permutation (or, equivalently, with extractions without replacement from a finite population) carries an irreducible representation of the symmetric group, equivalent to a twoblock Specht module. ..."
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It is proved that each Hoeffding space associated with a random permutation (or, equivalently, with extractions without replacement from a finite population) carries an irreducible representation of the symmetric group, equivalent to a twoblock Specht module.
Hoeffding Decompositions and TwoColour Urn Sequences
, 2006
"... Let X = (X1, X2,...) be a nondeterministic infinite exchangeable sequence with values in {0, 1}. We show that X is Hoeffdingdecomposable if, and only if, X is either an i.i.d. sequence or a Pólya sequence. This completes the results established in Peccati [2004]. The proof uses several combinatori ..."
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Let X = (X1, X2,...) be a nondeterministic infinite exchangeable sequence with values in {0, 1}. We show that X is Hoeffdingdecomposable if, and only if, X is either an i.i.d. sequence or a Pólya sequence. This completes the results established in Peccati [2004]. The proof uses several combinatorial implications of the correspondence between Hoeffding decomposability and weak independence. Our results must be compared with previous characterizations of i.i.d. and Pólya sequences given by Hill et al. [1987] and Diaconis and Yilvisaker [1979].
Exchangeable Hoeffding decompositions over finite sets: a combinatorial characterization and counterexamples
, 2014
"... www.carloalberto.org/research/workingpapers ..."
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