High-dimensional limit theorems for random vectors in ℓpn-balls. II

Kabluchko, Zakhar; Prochno, Joscha; Thäle, Christoph

doi:10.1142/s0219199719500731

Cited by 9 publications

(3 citation statements)

References 26 publications

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“…Quenched LDPs for projections of various radially symmetric measures on B n p were also recently studied in [18]. We also remark that LDPs for the ℓ q norm of a random element of B n p (with q = p) were established in [17].…”

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confidence: 76%

Large deviations for random projections of $\ell^{p}$ balls

Gantert¹,

Kim²,

Ramanan³

2017

Ann. Probab.

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Let p ∈ [1, ∞]. Consider the projection of a uniform random vector from a suitably normalized p ball in R n onto an independent random vector from the unit sphere. We show that sequences of such random projections, when suitably normalized, satisfy a large deviation principle (LDP) as the dimension n goes to ∞, which can be viewed as an annealed LDP. We also establish a quenched LDP (conditioned on a fixed sequence of projection directions) and show that for p ∈ (1, ∞] (but not for p = 1), the corresponding rate function is "universal", in the sense that it coincides for "almost every" sequence of projection directions. We also analyze some exceptional sequences of directions in the "measure zero" set, including the directions corresponding to the classical Cramér's theorem, and show that those directions yield LDPs with rate functions that are distinct from the universal rate function of the quenched LDP. Lastly, we identify a variational formula that relates the annealed and quenched LDPs, and analyze the minimizer of this variational formula. These large deviation results complement the central limit theorem for convex sets, specialized to the case of sequences of p balls.

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confidence: 76%

Large deviations for random projections of $\ell^{p}$ balls

Gantert¹,

Kim²,

Ramanan³

2017

Ann. Probab.

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“…A complete characterization of all distributions on B n p , which have the probabilistic representation ( 9), is hitherto unknown. Since it had been devised, authors have been using this generalized p-radial distribution in several works, for instance Kabluchko, Prochno, and Thäle [14] or Kaufmann, Sambale, and Thäle [16].…”

Section: Measures On Finite-dimensional ℓ P Spacesmentioning

confidence: 99%

Hölder's inequality and its reverse—A probabilistic point of view

Frühwirth

Prochno

2023

Mathematische Nachrichten

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In this article, we take a probabilistic look at Hölder's inequality, considering the ratio of terms in the classical Hölder inequality for random vectors in . We prove a central limit theorem for this ratio, which then allows us to reverse the inequality up to a multiplicative constant with high probability. The models of randomness include the uniform distribution on balls and spheres. We also provide a Berry–Esseen–type result and prove a large and a moderate deviation principle for the suitably normalized Hölder ratio.

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“…Moreover, they provided a large deviation principle (LDP) for Z q with Z ∼ C n,p and Z ∼ U n,p . In a follow-up paper [30], the same authors showed a CLT for Z q , where the distribution of Z is taken from a wider class of p-radial distributions P n,p,W , introduced by Barthe, Guédon, Mendelson and Naor [8], consisting of mixtures of U n,p and C n,p , combined via a measure W on [0, ∞). This class contains both U n,p and C n,p , but also distributions corresponding with geometrically interesting projections (see, e.g., [30,Introduction,(iii)]).…”

Section: Introductionmentioning

confidence: 99%

Sharp asymptotics for q-norms of random vectors in high-dimensional ℓpn-balls

Kaufmann¹

2021

Modern Stochastics: Theory and Applications

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Sharp large deviation results of Bahadur-Ranga Rao type are provided for the qnorm of random vectors distributed on the n p -ball B n p according to the cone probability measure or the uniform distribution for 1 ≤ q < p < ∞, thereby furthering previous large deviation results by Kabluchko, Prochno and Thäle in the same setting. These results are then applied to deduce sharp asymptotics for intersection volumes of different n p -balls in the spirit of Schechtman and Schmuckenschläger, and for the length of the projection of an n p -ball onto a line with uniform random direction. The sharp large deviation results are proven by providing convenient probabilistic representations of the q-norms, employing local limit theorems to approximate their densities, and then using geometric results for asymptotic expansions of Laplace integrals to integrate these densities and derive concrete probability estimates.

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High-dimensional limit theorems for random vectors in ℓpn-balls. II

Cited by 9 publications

References 26 publications

Large deviations for random projections of $\ell^{p}$ balls

Large deviations for random projections of $\ell^{p}$ balls

Hölder's inequality and its reverse—A probabilistic point of view

Sharp asymptotics for q-norms of random vectors in high-dimensional ℓpn-balls

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