[Paper Review] Vector rearrangement invariant banach spaces of random variables with exponential decreasing tails of distributions

Eugeny Ostrovsky, L. Sirota|arXiv (Cornell University)|Oct 14, 2015

Advanced Harmonic Analysis Research9 references17 citations

TL;DR

This paper extends vector rearrangement invariant Banach spaces of random variables with exponential tails to the multivariate case, establishing equivalence between $ B(\phi) $ norms and Grand Lebesgue Space (GLS) norms via Young-Fenchel duality. It proves that the $ B(\phi) $ norm for centered random vectors is equivalent to an Orlicz norm defined by $ N_{\phi}(\vec{u}) = \exp(\phi^*(\vec{u})) - \exp(\phi^*(0)) $, and derives non-asymptotic exponential tail bounds using multivariate Bernstein and Fenchel-Moraux theorems.

ABSTRACT

We present in this paper the theory of multivariate Banach spaces of random variables with exponential decreasing tails of distributions.

Motivation & Objective

• To extend the theory of rearrangement invariant Banach spaces with exponential tails from univariate to multivariate random vectors.
• To establish the equivalence between the $ B(\phi) $ norm and the Grand Lebesgue Space (GLS) norm for centered random vectors.
• To characterize the natural function $ \phi $ associated with a random vector via moment generating functions and convex conjugation.
• To derive non-asymptotic exponential tail estimates for sums of i.i.d. random vectors using multivariate moment methods.

Proposed method

• Define the $ B(\phi) $ space for centered $ d $-dimensional random vectors via the condition $ \mathbb{E}\exp(\pm \lambda \cdot \xi) \leq \exp(\phi(\lambda \tau)) $ for all $ \lambda \in (-\lambda_0, \lambda_0)^d $.
• Introduce the GLS norm $ \|\xi\|_{G(\psi)} = \sup_{p \geq 1} \left[ \mathbb{E}|\xi|^p \right]^{1/p} / \psi(p) $, where $ \psi(p) = p / \phi^{-1}(p) $.
• Use the Young-Fenchel transform $ \phi^* $ to define the Orlicz norm via $ N_{\phi}(\vec{u}) = \exp(\phi^*(\vec{u})) - \exp(\phi^*(0)) $.
• Prove equivalence between $ \|\cdot\|_{B(\phi)} $, $ \|\cdot\|_{G(\psi)} $, and $ \|\cdot\|_{L(N_{\phi})} $ using duality and saddle-point methods.
• Apply the Fenchel-Moraux theorem $ \phi^{**} = \phi $ to justify the duality structure.
• Derive tail bounds via Chernov-type estimates and multivariate Bernstein’s theorem, showing $ \sup_n U(S(n), \vec{x}) \geq \max\left( U(\vec{\xi}, \vec{x}), \exp(-C(Q)|x|^2) \right) $.

Experimental results

Research questions

• RQ1 Under what conditions is the moment generating function of a multivariate random vector representable as $ \exp(\phi(\lambda)) $ for a convex, even, twice continuously differentiable function $ \phi $?
• RQ2 How are the $ B(\phi) $, GLS, and Orlicz norms related in the multivariate setting, and are they equivalent?
• RQ3 Can non-asymptotic exponential tail bounds for sums of i.i.d. random vectors be derived using multivariate moment generating functions and convex conjugation?
• RQ4 What is the role of the natural function $ \phi_0(\lambda) = \max_{\pm} \log \sup_{t} \mathbb{E}\exp(\pm \lambda \xi(t)) $ in characterizing the tail behavior of random fields?
• RQ5 Why do Rosenthal-type moment inequalities fail to yield optimal tail bounds in this context, despite norm equivalence?

Key findings

• The $ B(\phi) $ norm for centered $ d $-dimensional random vectors is equivalent to the GLS norm $ \|\xi\|_{G(\psi)} $, with $ \psi(p) = p / \phi^{-1}(p) $, and the equivalence constants $ C_1, C_2 $ depend only on $ \phi $ and $ d $.
• The $ B(\phi) $ norm is equivalent to the Orlicz norm $ \|\xi\|_{L(N_{\phi})} $, with $ N_{\phi}(\vec{u}) = \exp(\phi^*(\vec{u})) - \exp(\phi^*(0)) $, and the equivalence constants $ C_5, C_6 $ depend on $ d $ and $ \phi $.
• For i.i.d. centered random vectors with $ U(\vec{\xi}, \vec{x}) \leq \exp(-|\vec{x}|^p) $, the supremum of the tail of the normalized sum satisfies $ \sup_n U(S(n), \vec{x}) \leq \exp(-C(d,p) |\vec{x}|^{\min(p,2)}) $ for $ |\vec{x}| \geq 1 $, and this bound is sharp.
• The natural function $ \phi_0(\lambda) $ for a random vector is always absolutely even, meaning $ \phi_0(\epsilon \otimes \vec{x}) = \phi_0(\vec{x}) $ for all $ \epsilon \in \{\pm 1\}^d $.
• The function $ \phi $ associated with a random vector must satisfy $ \phi^{**} = \phi $, ensuring the duality structure is preserved under the Young-Fenchel transform.
• The failure of Rosenthal’s inequality to yield optimal bounds is due to its suboptimal constants $ R(p) \asymp p / \log p $, which diverge as $ p \to \infty $, unlike the $ B(\phi) $-based estimates.