Maximal Inequalities Research Articles

Maximal Inequalities for Fractional Brownian Motion with Variable Drift

Let BH be a fractional Brownian motion with H∈ (0, 1) and g be a deterministic function. We study the asymptotic behaviour of the tail probability as for fixed x and as for fixed T. Our results partially generalise those obtained by Prakasa Rao in [1].

Euler semigroup, Hardy\u2013Sobolev and Gagliardo\u2013Nirenberg type inequalities on homogeneous groups

In this paper we describe the Euler semigroup {e^{-tmathbb {E}^{*}mathbb {E}}}_{t>0} on homogeneous Lie groups, which allows us to obtain various types of the Hardy–Sobolev and Gagliardo–Nirenberg type inequalities for the Euler operator mathbb {E}. Moreover, the sharp remainder terms of the Sobolev type inequality, maximal Hardy inequality and |cdot |-radial weighted Hardy–Sobolev type inequality are established.

Functional Limit Theorems for Shot Noise Processes with Weakly Dependent Noises

We study shot noise processes when the shot noises are weakly dependent, satisfying the ρ-mixing condition. We prove a functional weak law of large numbers and a functional central limit theorem for this shot noise process in an asymptotic regime with a high intensity of shots. The deterministic fluid limit is unaffected by the presence of weak dependence. The limit in the diffusion scale is a continuous Gaussian process whose covariance function explicitly captures the dependence among the noises. The model and results can be applied in financial and insurance risks with dependent claims as well as queueing systems with dependent service times. To prove the existence of the limit process, we employ the existence criterion that uses a maximal inequality requiring a set function with a superadditivity property. We identify such a set function for the limit process by exploiting the ρ-mixing condition. To prove the weak convergence, we establish the tightness property and the convergence of finite dimensional distributions. To prove tightness, we construct two auxiliary processes and apply an Ottaviani-type inequality for weakly dependent sequences.

Moderate maximal inequalities for the Ornstein-Uhlenbeck process

The maximal inequalities for diffusion processes have drawn increasing attention in recent years. Here we prove the moderate maximal inequality for the Ornstein-Uhlenbeck process, which includes the $L^p$ maximal inequality as a special case and generalizes the $L^1$ maximal inequality obtained by Graversen and Peskir [Proc. Amer. Math. Soc. 128(10):3035-3041, 2000]. As a corollary, we also obtain a new moderate maximal inequality for continuous local martingales, which can be viewed as an extension of the classical Burkholder-Davis-Gundy inequality.

Approximating sums of products of dependent random variables

Stochastic approximation of a given time series {∑j=1kXjYj} by a linear combination of simpler sequences {∑j=1kXj} and {∑j=1kYj} is treated uniformly over k∈{1,…,n}. A maximal inequality is proven in order to find a sharp bound on Value-at-Risk of max1≤k≤n|∑j=1kXjYj|.

Variants of a theorem of Helson on general Dirichlet series

A result of Helson on λ-Dirichlet series ∑ane−λns states that, whenever (an) is 2-summable and λ=(λn) satisfies a certain condition introduced by Bohr, then for almost all homomorphism ω:(R,+)→T the Dirichlet series ∑anω(λn)e−λns converges on the open right half plane [Re>0]. For ordinary Dirichlet series ∑ann−s Hedenmalm and Saksman related this result with the famous Carleson-Hunt theorem on pointwise convergence of Fourier series, and Bayart extended it within his theory of Hardy spaces Hp of such series. The aim here is to prove variants of Helson's theorem within our recent theory of Hardy spaces Hp(λ),1≤p≤∞, of λ-Dirichlet series. To be more precise, in the reflexive case 1<p<∞ we extend Helson's result to Dirichlet series in Hp(λ) without any further condition on the frequency λ, and in the non-reflexive case p=1 to the wider class of frequencies satisfying the so-called Landau condition (more general than Bohr's condition). In both cases we add relevant maximal inequalities. Finally, we try to indicate how our study of Helson's theorem can be employed to understand that for a fixed but arbitrary frequency λ certain key stones of the structure theory of λ-Dirichlet series in fact turn out to be equivalent.

Littlewood-Paley Characterizations of Hardy-Type Spaces Associated with Ball Quasi-Banach Function Spaces

Let X be a ball quasi-Banach function space on \({{\mathbb {R}}}^n\). In this article, assuming that the powered Hardy–Littlewood maximal operator satisfies some Fefferman–Stein vector-valued maximal inequality on X and is bounded on the associated space, the authors establish various Littlewood–Paley function characterizations of the Hardy space \(H_X({{\mathbb {R}}}^n)\) associated with X, under some weak assumptions on the Littlewood–Paley functions. To this end, the authors also establish a useful estimate on the change of angles in tent spaces associated with X. All these results have wide applications. Particularly, when \(X:=M_r^p({{\mathbb {R}}}^n)\) (the Morrey space), \(X:=L^{\vec {p}}({{\mathbb {R}}}^n)\) (the mixed-norm Lebesgue space), \(X:=L^{p(\cdot )}({{\mathbb {R}}}^n)\) (the variable Lebesgue space), \(X:=L_\omega ^p({{\mathbb {R}}}^n)\) (the weighted Lebesgue space) and \(X:=(E_\Phi ^r)_t({{\mathbb {R}}}^n)\) (the Orlicz-slice space), the Littlewood–Paley function characterizations of \(H_X({{\mathbb {R}}}^n)\) obtained in this article improve the existing results via weakening the assumptions on the Littlewood–Paley functions and widening the range of \(\lambda \) in the Littlewood–Paley \(g_\lambda ^*\)-function characterization of \(H_X({\mathbb {R}}^n)\).

Maximal inequalities and their applications to orthogonal and Hadamard matrices

Maximal inequalities for the signed vector summands are established. Probabilistic estimations for the sets of appropriate signs are given. By use of the “transference technique” appropriate maximal inequalities are derived for the permutations. One application for orthogonal and Hadamard matrices is suggested.

Newton’s method for nonlinear stochastic wave equations

Abstract We consider nonlinear stochastic wave equations driven by time-space white noise. The existence of solutions is proved by the method of successive approximations. Next we apply Newton’s method. The main result concerning its first-order convergence is based on Cairoli’s maximal inequalities for two-parameter martingales. Moreover, a second-order convergence in a probabilistic sense is demonstrated.

Weighted maximal inequalities for martingale transforms

Weighted maximal inequalities for martingale transforms

Weighted boundedness for Toeplitz type operator related to singular integral transform with variable Calderón-Zygmund kernel

&lt;abstract&gt; &lt;p&gt;In the paper, some weighted maximal inequalities for the Toeplitz operator related to the singular integral transform with variable CalderȮn-Zygmund kernel are proved. As an application, the boundedness of the operator on weighted Lebesgue space are obtained.&lt;/p&gt; &lt;/abstract&gt;

Moments of random multiplicative functions, II: High moments

We determine the order of magnitude of $\mathbb{E}|\sum_{n \leq x} f(n)|^{2q}$ up to factors of size $e^{O(q^2)}$, where $f(n)$ is a Steinhaus or Rademacher random multiplicative function, for all real $1 \leq q \leq \frac{c\log x}{\log\log x}$. In the Steinhaus case, we show that $\mathbb{E}|\sum_{n \leq x} f(n)|^{2q} = e^{O(q^2)} x^q (\frac{\log x}{q\log(2q)})^{(q-1)^2}$ on this whole range. In the Rademacher case, we find a transition in the behaviour of the moments when $q \approx (1+\sqrt{5})/2$, where the size starts to be dominated by "orthogonal" rather than "unitary" behaviour. We also deduce some consequences for the large deviations of $\sum_{n \leq x} f(n)$. The proofs use various tools, including hypercontractive inequalities, to connect $\mathbb{E}|\sum_{n \leq x} f(n)|^{2q}$ with the $q$-th moment of an Euler product integral. When $q$ is large, it is then fairly easy to analyse this integral. When $q$ is close to 1 the analysis seems to require subtler arguments, including Doob's $L^p$ maximal inequality for martingales.

On Rough maximal inequalities: an extension of Fefferman-Stein results

On Rough maximal inequalities: an extension of Fefferman-Stein results

On a maximal inequality and its application to SDEs with singular drift

In this paper we present a Doob type maximal inequality for stochastic processes satisfying the conditional increment control condition. If we assume, in addition, that the margins of the process have uniform exponential tail decay, we prove that the supremum of the process decays exponentially in the same manner. Then we apply this result to the construction of the almost everywhere stochastic flow to stochastic differential equations with singular time dependent divergence-free drift.

Variable exponent Triebel-Lizorkin-Morrey spaces

We introduce variable exponent versions of Morreyfied Triebel-Lizorkin spaces. To that end, we prove an important convolution inequality which is a replacement for the Hardy-Littlewood maximal inequality in the fully variable setting. Using it we obtain characterizations by means of Peetre maximal functions and use them to show the independence of the introduced spaces from the admissible system used.

Strong laws of large numbers for arrays of random variables and stable random fields

Strong laws of large numbers are established for random fields with weak or strong dependence. These limit theorems are applicable to random fields with heavy-tailed distributions including fractional stable random fields. The conditions for SLLN are described in terms of the p-th moments of the partial sums of the random fields, which are convenient to verify. The main technical tool in this paper is a maximal inequality for the moments of partial sums of random fields that extends the technique of Levental, Chobanyan and Salehi [6] for a sequence of random variables indexed by a one-parameter.

Maximal Inequalities of Noncommutative Martingale Transforms

AbstractIn this paper, we investigate noncommutative symmetric and asymmetric maximal inequalities associated with martingale transforms and fractional integrals. Our proofs depend on some recent advances on algebraic atomic decomposition and the noncommutative Gundy decomposition. We also prove several fractional maximal inequalities.

More on maximal inequalities for sub-fractional Brownian motion

We derive some maximal inequalities for the sub-fractional Brownian motion using comparison theorems for Gaussian processes.

Functional CLT for martingale-like nonstationary dependent structures

In this paper, we develop non-stationary martingale techniques for dependent data. We shall stress the non-stationary version of the projective Maxwell–Woodroofe condition, which will be essential for obtaining maximal inequalities and functional central limit theorem for the following examples: nonstationary $\rho$-mixing sequences, functions of linear processes with non-stationary innovations, locally stationary processes, quenched version of the functional central limit theorem for a stationary sequence, evolutions in random media such as a process sampled by a shifted Markov chain.

Weak Hardy-Type Spaces Associated with Ball Quasi-Banach Function Spaces II: Littlewood–Paley Characterizations and Real Interpolation

Let X be a ball quasi-Banach function space on $${\mathbb R}^n$$ . In this article, assuming that the powered Hardy–Littlewood maximal operator satisfies some Fefferman–Stein vector-valued maximal inequality on X as well as it is bounded on both the weak ball quasi-Banach function space WX and the associated space, the authors establish various Littlewood–Paley function characterizations of $$WH_X({{\mathbb {R}}}^n)$$ under some weak assumptions on the Littlewood–Paley functions. The authors also prove that the real interpolation intermediate space $$(H_{X}({{\mathbb {R}}}^n),L^\infty ({{\mathbb {R}}}^n))_{\theta ,\infty }$$ , between the Hardy space associated with X, $$H_{X}({{\mathbb {R}}}^n)$$ , and the Lebesgue space $$L^\infty ({\mathbb R}^n)$$ , is $$WH_{X^{{1}/{(1-\theta )}}}({{\mathbb {R}}}^n)$$ , where $$\theta \in (0, 1)$$ . All these results are of wide applications. Particularly, when $$X:=M_q^p({{\mathbb {R}}}^n)$$ (the Morrey space), $$X:=L^{\vec {p}}({{\mathbb {R}}}^n)$$ (the mixed-norm Lebesgue space) and $$X:=(E_\Phi ^q)_t({{\mathbb {R}}}^n)$$ (the Orlicz-slice space), all these results are even new; when $$X:=L_\omega ^\Phi ({\mathbb R}^n)$$ (the weighted Orlicz space), the result on the real interpolation is new and, when $$X:=L^{p(\cdot )}({{\mathbb {R}}}^n)$$ (the variable Lebesgue space) and $$X:=L_\omega ^\Phi ({{\mathbb {R}}}^n)$$ , the Littlewood–Paley function characterizations of $$WH_X({{\mathbb {R}}}^n)$$ obtained in this article improves the existing results via weakening the assumptions on the Littlewood–Paley functions.