Integration questions related to fractional Brownian motion

Let {B _H (u)} _{u ∈ℝ} be a fractional Brownian motion (fBm) with index H∈(0, 1) and (B _H ) be the closure in L ²(Ω) of the span Sp(B _H ) of the increments of fBm B _H . It is well-known that, when B _H = B _1/2 is the usual Brownian motion (Bm), an element X∈(B _1/2) can be characterized by a unique function f _X ∈L ²(ℝ), in which case one writes X in an integral form as X = ∫_ℝ f _X (u)dB _1/2(u). From a different, though equivalent, perspective, the space L ²(ℝ) forms a class of integrands for the integral on the real line with respect to Bm B _1/2. In this work we explore whether a similar characterization of elements of (B _H ) can be obtained when H∈ (0, 1/2) or H∈ (1/2, 1). Since it is natural to define the integral of an elementary function f = ∑ _{k =1} ⁿ f _k 1 _[uk,uk+1) by ∑ _{k =1} ⁿ f _k (B _H (u _{k +1}) −B _H (u _k )), we want the spaces of integrands to contain elementary functions. These classes of integrands are inner product spaces. If the space of integrands is not complete, then it characterizes only a strict subset of (B _H ). When 0<H<1/2, by using the moving average representation of fBm B _H , we construct a complete space of integrands. When 1/2<H<1, however, an analogous construction leads to a space of integrands which is not complete. When 0<H<1/2 or 1/2<H<1, we also consider a number of other spaces of integrands. While smaller and henceincomplete, they form a natural choice and are convenient to workwith. We compare these spaces of integrands to the reproducing kernel Hilbert space of fBm. Received: 9 August 1999 / Revised version: 10 January 2000 / Published online: 18 September 2000     

Limit theorems for bivariate Appell polynomials. Part II: Non-central limit theorems

Summary. Let (X _t ,t∈Z) be a linear sequence with non-Gaussian innovations and a spectral density which varies regularly at low frequencies. This includes situations, known as strong (or long-range) dependence, where the spectral density diverges at the origin. We study quadratic forms of bivariate Appell polynomials of the sequence (X _t ) and provide general conditions for these quadratic forms, adequately normalized, to converge to a non-Gaussian distribution. We consider, in particular, circumstances where strong and weak dependence interact. The limit is expressed in terms of multiple Wiener-It? integrals involving correlated Gaussian measures. Received: 22 August 1996 / In revised form: 30 August 1997     

Toward the 2s-2p spectroscopy of the µp atom

Very recent results relevant to the planning of laser spectroscopy of the 2s-2p splitting in the µp atom are presented. An experimental configuration providing a sufficiently small muon stopping volume at the lowest pressures has been tested. The first experimental indications for the existence of a long-lived 2s fraction at low pressure are discussed.     

Limit theorems for bivariate Appell polynomials. Part I: Central limit theorems

Summary. Consider the stationary linear process , , where is an i.i.d. finite variance sequence. The spectral density of may diverge at the origin (long-range dependence) or at any other frequency. Consider now the quadratic form , where denotes a non-linear function (Appell polynomial). We provide general conditions on the kernels and for to converge to a Gaussian distribution. We show that this convergence holds if and are not too badly behaved. However, the good behavior of one kernel may compensate for the bad behavior of the other. The conditions are formulated in the spectral domain. Received: 28 February 1996 / In revised form: 10 July 1996     

Low-energy μ− beam from the PSI anticyclotron

Negative muons of 28.6 Mev/c initial momentum were decelerated in and extracted from the PSI anticyclotron equipped with a Mylar foil in the median plane as moderator to provide a continuous μ⁻ beam of 3–30 keV energy. This technique can also be used to post-decelerate LEAR antiprotons in a continuous or pulsed mode to keV energies with an efficiency up to 10–20%.     

Asymptotic normality of quadratic forms of martingale differences

We establish the asymptotic normality of a quadratic form \(Q_n\) in martingale difference random variables \(\eta _t\) when the weight matrix A of the quadratic form has an asymptotically vanishing diagonal. Such a result has numerous potential applications in time series analysis. While for i.i.d. random variables \(\eta _t\), asymptotic normality holds under condition \(||A||_{sp}=o(||A||) \), where \(||A||_{sp}\) and ||A|| are the spectral and Euclidean norms of the matrix A, respectively, finding corresponding sufficient conditions in the case of martingale differences \(\eta _t\) has been an important open problem. We provide such sufficient conditions in this paper.     

A radial compression scheme for cooled antiprotons

A novel method is proposed for achieving a fast and efficient radial compression of antiproton clouds trapped and cooled in high magnetic field. The basic operation takes place in a low field solenoid during the transfer of the cooled antiprotons from one trap to the next. It makes use of a special device that imparts to the particles a transverse momentum which compensates exactly their initial generalized momentum. They can therefore be focused onto the solenoid axis and injected in the next high field region with a relatively small radial envelope. After standard trapping and cooling procedures a strongly radial extension is obtained. The operation can be repeated to lead to very small size of the final antiproton cloud. It can be estimated that an initial 1 cm radial extension can be reduced to 1 mm in a first stage and to less than 0.1 mm in a second stage. In this way highest densities (approaching the space charge limit) of trapped antiprotons can be obtained in a relatively short time.     

(1/α)-Self similar α-stable processes with stationary increments

In this note we settle a question posed by Kasahara, Maejima, and Vervaat. We show that the α-stable Lévy motion is the only (1/α)-self-similar α-stable process with stationary increments if 0 < α < 1. We also introduce new classes of (1/α)-self-similar α-stable processes with stationary increments for 1 < α < 2.