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Invariance principles for homogeneous sums: universality of Gaussian Wiener chaos
 Ann. Probab
, 2010
"... Abstract: We compute explicit bounds in the normal and chisquare approximations of multilinear homogenous sums (of arbitrary order) of general centered independent random variables with unit variance. Our techniques combine an invariance principle by Mossel, O’Donnell and Oleszkiewicz with a refine ..."
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Cited by 16 (14 self)
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Abstract: We compute explicit bounds in the normal and chisquare approximations of multilinear homogenous sums (of arbitrary order) of general centered independent random variables with unit variance. Our techniques combine an invariance principle by Mossel, O’Donnell and Oleszkiewicz with a refinement of some recent results by Nourdin and Peccati, about the approximation of laws of random variables belonging to a fixed (Gaussian) Wiener chaos. In particular, we show that chaotic random variables enjoy the following form of universality: (a) the normal and chisquare approximations of any homogenous sum can be completely characterized and assessed by first switching to its Wiener chaos counterpart, and (b) the simple upper bounds and convergence criteria available on the Wiener chaos extend almost verbatim to the class of homogeneous sums. These results partially rely on the notion of “low influences ” for functions defined on product spaces, and provide a generalization of central and noncentral limit theorems proved by Nourdin, Nualart and Peccati. They also imply a further drastic simplification of the method of moments and cumulants – as applied to the proof of probabilistic limit theorems – and yield substantial generalizations, new proofs and new insights into some classic findings by de Jong and Rotar’. Our tools involve the use of Malliavin calculus, and of both the Stein’s
GRAPHONS, CUT NORM AND DISTANCE, COUPLINGS AND REARRANGEMENTS
"... Abstract. We give a survey of basic results on the cut norm and cut metric for graphons (and sometimes more general kernels), with emphasis on the equivalence problem. The main results are not new, but we add various technical complements. We allow graphons on general probability spaces whenever pos ..."
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Cited by 8 (5 self)
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Abstract. We give a survey of basic results on the cut norm and cut metric for graphons (and sometimes more general kernels), with emphasis on the equivalence problem. The main results are not new, but we add various technical complements. We allow graphons on general probability spaces whenever possible. We also give some new results for {0,1}valued graphons. 1.
Monotone graph limits and quasimonotone graphs
, 2011
"... Abstract. The recent theory of graph limits gives a powerful framework for understanding the properties of suitable (convergent) sequences (Gn) of graphs in terms of a limiting object which may be represented by a symmetric function W on [0, 1], i.e., a kernel or graphon. In this context it is natur ..."
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Cited by 5 (2 self)
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Abstract. The recent theory of graph limits gives a powerful framework for understanding the properties of suitable (convergent) sequences (Gn) of graphs in terms of a limiting object which may be represented by a symmetric function W on [0, 1], i.e., a kernel or graphon. In this context it is natural to wish to relate specific properties of the sequence to specific properties of the kernel. Here we show that the kernel is monotone (i.e., increasing in both variables) if and only if the sequence satisfies a ‘quasimonotonicity ’ property defined by a certain functional tending to zero. As a tool we prove an inequality relating the cut and L 1 norms of kernels of the form W1 − W2 with W1 and W2 monotone that may be of interest in its own right; no such inequality holds for general kernels.
HIGHER MOMENTS OF BANACH SPACE VALUED RANDOM VARIABLES
"... Abstract. We define the k:th moment of a Banach space valued random variable as the expectation of its k:th tensor power; thus the moment (if it exists) is an element of a tensor power of the original Banach space. We study both the projective and injective tensor products, and their relation. Moreo ..."
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Cited by 2 (0 self)
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Abstract. We define the k:th moment of a Banach space valued random variable as the expectation of its k:th tensor power; thus the moment (if it exists) is an element of a tensor power of the original Banach space. We study both the projective and injective tensor products, and their relation. Moreover, in order to be general and flexible, we study three different types of expectations: Bochner integrals, Pettis integrals and Dunford integrals. One of the problems studied is whether two random variables with the same injective moments (of a given order) necessarily have the same projective moments; this is of interest in applications. We show that this holds if the Banach space has the approximation property, but not in general. Several sections are devoted to results in special Banach spaces, including Hilbert spaces, C pK q and Dr0, 1s. The latter space is nonseparable, which complicates the arguments, and we prove various preliminary results on e.g. measurability in Dr0, 1s that we need. One of the main motivations of this paper is the application to Zolotarev metrics and their use in the contraction method. This is sketched in an appendix. 1.
Rademacher Chaos: Tail Estimates vs Limit Theorems
"... We study Rademacher chaos indexed by a sparse set which has a fractional combinatorial dimension. We obtain tail estimates for nite sums and a normal limit theorem as the size tends to in nity. The tails for nite sums may be much larger that the tails of the limit. ..."
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We study Rademacher chaos indexed by a sparse set which has a fractional combinatorial dimension. We obtain tail estimates for nite sums and a normal limit theorem as the size tends to in nity. The tails for nite sums may be much larger that the tails of the limit.
Sparse Interactions: Identifying HighDimensional Multilinear Systems via Compressed Sensing
"... Abstract—This paper investigates the problem of identifying sparse multilinear systems. Such systems are characterized by multiplicative interactions between the input variables with sparsity meaning that relatively few of all conceivable interactions are present. This problem is motivated by the st ..."
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Abstract—This paper investigates the problem of identifying sparse multilinear systems. Such systems are characterized by multiplicative interactions between the input variables with sparsity meaning that relatively few of all conceivable interactions are present. This problem is motivated by the study of interactions among genes and proteins in living cells. The goal is to develop a sampling/sensing scheme to identify sparse multilinear systems using as few measurements as possible. We derive bounds on the number of measurements required for perfect reconstruction as a function of the sparsity level. Our results extend the notion of compressed sensing from the traditional notion of (linear) sparsity to more refined notions of sparsity encountered in nonlinear systems. In contrast to the linear sparsity models, in the multilinear case the pattern of sparsity may play a role in the sensing requirements. I.