In this paper, we consider functional limit theorems for Poisson cluster processes. We first present a maximal inequality for Poisson cluster processes. Then we establish a functional central limit theorem under the second moment and a functional moderate deviation principle under the Cramér condition for Poisson cluster processes. We apply these results to obtain a functional moderate deviation principle for linear Hawkes processes.

Let $A\,:=\,-\,\left( \nabla \,-\,i\overrightarrow{a} \right)\,.\,\left( \nabla \,-\,i\overrightarrow{a} \right)\,+\,V$ be a magnetic Schrödinger operator on ${{\mathbb{R}}^{n}}$, where

$$\vec{a}:=\left( {{a}_{1}},...,{{a}_{n}} \right)\in L_{\text{loc}}^{2}\left( {{\mathbb{R}}^{n}},{{\mathbb{R}}^{n}} \right)\operatorname{and}0\le V\in L_{\text{loc}}^{1}\left( {{\mathbb{R}}^{n}} \right)$$

satisfy some reverse Hölder conditions. Let $\phi :\,{{\mathbb{R}}^{n}}\,\times \,[0,\,\infty )\,\to \,[0,\,\infty )$ be such that $\phi \left( x,\,. \right)$ for any given $x\,\in \,{{\mathbb{R}}^{n}}$ is an Orlicz function, $\phi \left( ^{.}\,,\,t \right)\,\in \,{{\mathbb{A}}_{\infty }}\left( {{\mathbb{R}}^{n}} \right)$ for all $t\,\in \,\left( 0,\,\infty \right)$ (the class of uniformly Muckenhoupt weights) and its uniformly critical upper type index $I\left( \phi \right)\,\in \,(0,\,1]$. In this article, the authors prove that second-order Riesz transforms $V{{A}^{-1}}$ and ${{\left( \nabla \,-\,i\overrightarrow{a} \right)}^{2}}{{A}^{-1}}$ are bounded from the Musielak–Orlicz–Hardy space ${{H}_{\phi ,\,A}}\left( {{\mathbb{R}}^{n}} \right)$, associated with $A$, to theMusielak–Orlicz space ${{L}^{\phi }}\left( {{\mathbb{R}}^{n}} \right)$. Moreover, we establish the boundedness of $V{{A}^{-1}}$ on ${{H}_{\phi ,\,A}}\left( {{\mathbb{R}}^{n}} \right)$. As applications, some maximal inequalities associated with $A$ in the scale of ${{H}_{\phi ,\,A}}\left( {{\mathbb{R}}^{n}} \right)$ are obtained.

We prove that the Return Times Theoremholds true for pairs of ${{L}^{p}}\,-\,{{L}^{q}}$ functions, whenever $\frac{1}{p}\,+\,\frac{1}{q}\,<\,\frac{3}{2}$.

In this paper we study the functional central limit theorem (CLT) for stationary Markov chains with a self-adjoint operator and general state space. We investigate the case when the variance of the partial sum is not asymptotically linear in n, and establish that conditional convergence in distribution of partial sums implies the functional CLT. The main tools are maximal inequalities that are further exploited to derive conditions for tightness and convergence to the Brownian motion.

In this article, we consider the stochastic heat equation $du=(\Delta u+f(t,x)){\rm d}t+ \sum_{k=1}^{\infty} g^{k}(t,x) \delta \beta_t^k, t \in [0,T]$, with random coefficients f and gk,driven by a sequence (βk)k of i.i.d. fractional Brownianmotions of index H>1/2. Using the Malliavin calculus techniquesand a p-th moment maximal inequality for the infinite sum ofSkorohod integrals with respect to (βk)k, we prove that theequation has a unique solution (in a Banach space of summabilityexponent p ≥ 2), and this solution is Hölder continuous inboth time and space.

We consider a stochastic process in a modified Ehrenfest urn model. The modification prescribes there to be a minimum number of balls in each urn, and the process records the differences between treatment assignments under a sampling scheme implemented with this modified Ehrenfest urn model. In contrast to the result that the difference process forms a Markov chain and converges to a stationary distribution under the Ehrenfest urn model, the corresponding process under this modified Ehrenfest urn design satisfies the central limit property. We prove this asymptotic normality property using a central limit theorem for dependent random variables, renewal theory, and two Kolmogorov-type maximal inequalities.

In this short note we show that $\sup\{\|M_\nu\|:\nu\text{ is a measure on }\mathbb{R}^n\}$, where $\|M_\nu\|$ denotes the centred Hardy–Littlewood maximal operator, depends exponentially on $n$.

AMS 2000 Mathematics subject classification: Primary 42B25