Probabilistic estimates for tensor products of random vectors

Alonso–Gutiérrez, David; Prochno, Joscha; Passenbrunner, Markus

doi:10.1090/proc/12883

Cited by 3 publications

(4 citation statements)

References 24 publications

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“…One of the main objectives of this paper is to provide a complete answer to this question. In fact, we shall present necessary and sufficient conditions on a quasi-Banach symmetric function space X such that the deterministic equivalence (1.3) holds for every quasi-Banach symmetric sequence space E. This result strengthens and complements all above mentioned papers [21,12,11,16,19,17,4,5,1,7,2]. We also mention that Astashkin and Tikhomirov [9, Corollary 2] established that the inequality…”

Section: Introductionsupporting

confidence: 66%

“…[18]). Note that properties (2) and ( 4) of a ∆-norm imply that for any α ∈ C, there exists a positive constant M such…”

Section: Preliminariesmentioning

confidence: 99%

“…where Φ is an Orlicz function depending on N and the distribution function of f 1 (here A ≈ N B means that the ratio A/B is bounded below and above by constants which depend only on N ). Moreover, the generalizations of (1.2) to Musielak-Orlicz norms and tensor products of random variables were studied in [1] and [2], respectively. Further, answering Yehoram Gordon's question whether a formula similar to (1.2) exists for arbitrary sequences of independent random variables and not just for scalar multiples of independent identically distributed random variables, Montgomery-Smith [19] produced a positive answer in the more general setting of symmetric Banach function and Orlicz sequence spaces.…”

Section: Introductionmentioning

confidence: 99%

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Rosenthal's inequalities: $Δ-$norms and quasi-Banach symmetric sequence spaces

Jiao¹,

Xie²,

Sukochev³

et al. 2019

Preprint

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Let X be a symmetric quasi-Banach function space with Fatou property and let E be an arbitrary symmetric quasi-Banach sequence space. Suppose that (f k ) k≥0 ⊂ X is a sequence of independent random variables. We present a necessary and sufficient condition on X such that the quantityadmits an equivalent characterization in terms of disjoint copies of (f k ) n k=0 for every n ≥ 0; in particular, we obtain the deterministic description of n k=0 f k e k ℓq Lp for all 0 < p, q < ∞, which is the ultimate form of Rosenthal's inequality. We also consider the case of a ∆-normed symmetric function space X, defined via an Orlicz function Φ satisfying the ∆ 2 -condition. That is, we provide a formula for "E-valued Φ-moments", namely the quantity E Φ (f k ) k≥0 E , in terms of the sum of disjoint copies of f k , k ≥ 0.

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Section: Introductionsupporting

confidence: 66%

“…[18]). Note that properties (2) and ( 4) of a ∆-norm imply that for any α ∈ C, there exists a positive constant M such…”

Section: Preliminariesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Rosenthal's inequalities: $Δ-$norms and quasi-Banach symmetric sequence spaces

Jiao¹,

Xie²,

Sukochev³

et al. 2019

Preprint

View full text Add to dashboard Cite

show abstract

“…In [31], also using a combinatorial approach, the Lorentz spaces isomorphic to a subspace of L 1 were characterized by Schütt. In the works [2,27,29] embeddings of certain matrix spaces into L 1 were obtained.…”

Section: Introductionmentioning

confidence: 99%

Embeddings of Orlicz-Lorentz spaces into $L_1$

Prochno¹

2019

Preprint

Self Cite

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In this article, we show that Orlicz-Lorentz spaces ℓ n M,a , n ∈ N with Orlicz function M and weight sequence a are uniformly isomorphic to subspaces of L 1 if the norm • M,a satisfies certain Hardy-type inequalities. This includes the embedding of some Lorentz spaces d n (a, p). Our approach is based on combinatorial averaging techniques and we prove a new result of independent interest that relates suitable averages with Orlicz-Lorentz norms.

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Embeddings of Orlicz–Lorentz spaces into $L_1$

Prochno

2021

St. Petersburg Math. J.

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Probabilistic estimates for tensor products of random vectors

Cited by 3 publications

References 24 publications

Rosenthal's inequalities: $Δ-$norms and quasi-Banach symmetric sequence spaces

Rosenthal's inequalities: $Δ-$norms and quasi-Banach symmetric sequence spaces

Embeddings of Orlicz-Lorentz spaces into $L_1$

Embeddings of Orlicz–Lorentz spaces into $L_1$

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