Metric Entropy and the Small Ball Problem for Gaussian Measures

Kuelbs, J.; Li, W.V.

doi:10.1006/jfan.1993.1107

Cited by 151 publications

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“…Kuelbs and Li [7] have discovered a tight connection between the Small Ball probabilities and the properties of the reproducing kernel Hilbert space corresponding to the process, which in the case of the Brownian Sheet is WM …”

Section: The Principal Conjecture and The Main Resultsmentioning

confidence: 99%

On the small ball inequality in three dimensions

Bilyk

Lacey

2008

Duke Math. J.

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Abstract. Let h R denote an L ∞ normalized Haar function adapted to a dyadic rectangle R ⊂ [0,1] 3 . We show that there is a positive η < 1 2 so that for all integers n, and coefficients α(R) we have This is an improvement over the 'trivial' estimate by an amount of n −η , while the Small Ball Conjecture says that the inequality should hold with η = 1 2 . There is a corresponding lower bound on the L ∞ norm of the Discrepancy function of an arbitrary distribution of a finite number of points in the unit cube in three dimensions. The prior result, in dimension 3, is that of József Beck [1], in which the improvement over the trivial estimate was logarithmic in n. We find several simplifications and extensions of Beck's argument to prove the result above. The Principal Conjecture and the Main ResultsIn one dimension, the class of dyadic intervals in the unit interval is D ≔ {[j2Each dyadic interval has a left and right half, which are also dyadic. Define the Haar functionsNote that we use an L ∞ normalization of these functions, which will make some formulas seem odd to a reader accustomed to the L 2 normalization. In dimension d, a dyadic rectangle in the unit cube [0,1] d is a product of dyadic intervals, thus an element of D d . A Haar function associated to R is defined as a product of the Haar functions associated with each side of R, namelyThis is the usual 'tensor' definition.We will concentrate on rectangles with fixed volume. This is the 'hyperbolic' assumption, that pervades the subject. Our concern is the following Theorem and Conjecture concerning a lower bound on the L ∞ norm of sums of hyperbolic Here, the sum on the right is taken over all rectangles with area at least 2 −n .1.3. Small Ball Conjecture. For dimension d ≥ 3 we have the inequalityThis conjecture is, by one square root of n, better than the trivial estimate available from the Cauchy-Schwartz inequality, see § 2. As well, see that section for an explanation as to why the conjecture is sharp. The case of d = 2 (with a sum over |R| = 2 −n on the right-hand side) was resolved by Talagrand [13]. Temlyakov has given an easier proof of the inequality in its present form [14], [16], which resonates with the ideas of Roth [9], Schmidt [10], and Halász [6].Perhaps, it is worthwhile to explain the nomenclature 'Small Ball' at this point. The name comes from the probability theory. Assume that X t : T → R is a canonical Gaussian process indexed by a set T. The Small Ball Problem is concerned with estimates of P(sup t∈T |X t | < ε) as ε goes to zero, i.e the probability that the random process takes values in an L ∞ ball of small radius. The reader is advised to consult a paper by Kuelbs and Li In dimension d = 2, this conjecture has been resolved by Talagrand in the already cited paper [13], in which he used a version of (1.2) for continuous wavelets in place of Haars to prove the lower bound in the inequality above. In higher dimensions, the upper bounds are established and the known lower bounds miss the conjecture by a single power of the logarit...

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Section: The Principal Conjecture and The Main Resultsmentioning

confidence: 99%

On the small ball inequality in three dimensions

Bilyk

Lacey

2008

Duke Math. J.

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“…Section 2 deals with the case d = 1 and points to the main difficulties in the higher dimensional case. Section 3 presents the useful connection with small ball probabilities, which is based on the duality relation (stated above) and fundamental links between metric entropy and small ball probabilities for Gaussian measures (discovered in Kuelbs and Li [9] and completed in Li and Linde [10]). In fact, the main significance of this paper is the discovery of this connection and using it to obtain previously unknown estimates on metric entropy of distribution functions.…”

Section: B(t K ε)mentioning

confidence: 99%

“…Roughly put, the smaller the value of γ(εT • ), the larger the covering number N (B l 2 , T • , ε). The remarkable discovery in [9] (completed in [10]) is that there is a tight connection between these two quantities: upper (lower) bound on one implies lower (upper) bound for the other. In particular, the statement is log γ(εT…”

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confidence: 99%

Metric entropy of high dimensional distributions

Blei

Gao

2007

Proc. Amer. Math. Soc.

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“…The so-called small ball problem for µ consists of finding a good approximation to the following: µ p (r) , µ ω ∈ B : p(ω) 6 r , (1.1) * Research supported by grants from the National Science Foundation and the National Security Agency as r → 0 + . A major breakthrough is the recent work of Kuelbs and Li [11] relating µ p to a combinatorial problem on the reproducing kernel Hilbert space corresponding to µ. The latter route typically leads one to long-standing open problems in functional analysis; cf.…”

Section: Introductionmentioning

confidence: 99%

“…The latter route typically leads one to long-standing open problems in functional analysis; cf. [11] for details.…”

Section: Introductionmentioning

confidence: 99%