Testing diffusion processes for non-stationarity

Financial data are often assumed to be generated by diffusions. Using recent results of Fan et al. (J Am Stat Assoc, 102:618–631, 2007; J Financ Econometer, 5:321–357, 2007) and a multiple comparisons procedure created by Benjamini and Hochberg (J R Stat Soc Ser B, 59:289–300, 1995), we develop a test for non-stationarity of a one-dimensional diffusion based on the time inhomogeneity of the diffusion function. The procedure uses a single sample path of the diffusion and involves two estimators, one temporal and one spatial. We first apply the test to simulated data generated from a variety of one-dimensional diffusions. We then apply our test to interest rate data and real exchange rate data. The application to real exchange rate data is of particular interest, since a consequence of the law of one price (or the theory of purchasing power parity) is that real exchange rates should be stationary. With the exception of the GBP/USD real exchange rate, we find evidence that interest rates and real exchange rates are generally non-stationary. The software used to implement the estimation and testing procedure is available on demand and we describe its use in the paper.     

Experiments with low-energy muons

Two experiments with low-energy muons are described: the determination of the stopping power of C, Si, Ti and Au for muons at energies down to 2 keV and the measurement of the diffusion times for pµ and dµ atoms in low-pressure (0.25–12 hPa) hydrogen gas. A pronounced Barkas effect was found for muons at the Bragg peak (about 10 keV): the stopping power for µ^– in C, e.g., is about 30% lower than that for µ⁺. The mean kinetic energy of pµ atoms at the end of the cascade in 1 hPa hydrogen gas was determined to be (2.6 ± 0.6) eV (preliminary value).     

Stable Convergence of Multiple Wiener-Itô Integrals

We prove sufficient conditions ensuring that a sequence of multiple Wiener-Itô integrals (with respect to a general Gaussian process) converges stably to a mixture of normal distributions. Note that stable convergence is stronger than convergence in distribution. Our key tool is an asymptotic decomposition of contraction kernels, realized by means of increasing families of projection operators. We also use an infinite-dimensional Clark-Ocone formula, as well as a version of the correspondence between “abstract” and “concrete” filtered Wiener spaces, in a spirit similar to that of Üstünel and Zakai (J. Funct. Anal. 143, 10–32, [1997]).     

Observation of long-lived muonic hydrogen in the 2S state

The kinetic energy distribution of ground state muonic hydrogen atoms mup(1S) is determined from time-of-flight spectra measured at 4, 16, and 64 hPa H2 room-temperature gas. A 0.9 keV component is discovered and attributed to radiationless deexcitation of long-lived mu p(2S) atoms in collisions with H2 molecules. The analysis reveals a relative population of about 1%, and a pressure-dependent lifetime (e.g., 30.4 +21.4/-9.7 ns at 64 hPa) of the long-lived mu p(2S) population, equivalent to a 2S quench rate in mu p(2S)+H2 collisions of 4.4 +2.1/-1.8 x 10(11) s(-1) at liquid-hydrogen density.     

Rate Optimality of Wavelet Series Approximations of Fractional Brownian Motion

Consider the fractional Brownian motion process $B_H(t), t\in [0,T]$, with parameter $H\in (0,1)$. Meyer, Sellan and Taqqu have developed several random wavelet representations for $B_H(t)$, of the form $\sum_{k=0}^\infty U_k(t)\epsilon_k$ where $\epsilon_k$ are Gaussian random variables and where the functions $U_k$ are not random. Based on the results of Kühn and Linde, we say that the approximation $\sum_{k=0}^n U_k(t)\epsilon_k$ of $B_H(t)$ is optimal if $$ \displaystyle \left( E \sup_{t\in [0,T]} \left| \sum_{k=n}^\infty U_k(t) \epsilon_k\right|^2 \right)^{1/2} =O \left( n^{-H} (1+\log n)^{1/2} \right), $$ as $n\rightarrow\infty$. We show that the random wavelet representations given in Meyer, Sellan and Taqqu are optimal.     

Laser spectroscopy of the Lamb shift in muonic hydrogen

The muonic hydrogen atom in the 2s state provides the possibility of achieving high precision laser spectroscopy experiments from which a high precision value of the proton radius can be deduced. This will ultimately allow an increased precision in the test of QED in bound systems. Important progress has been made in recent years in the ability to stop muons in a low pressure gas target and in the understanding of the 2s-metastability in muonic hydrogen. As a consequence the 2s–2p laser spectroscopy experiment is now feasible and we present here the basic experimental concept considered by our collaboration. This revised version was published online in August 2006 with corrections to the Cover Date.     

Frictional cooling: Experimental results

The classical methods used in beam cooling are hard to be adapted for a beam of short-lived elementary particles. A novel method, the so-called frictional cooling – that is cooling a beam of low-energy charged particles by moderation in matter and acceleration in an electrostatic field – has been shown to be feasible. In our experiments performed in 1994/1995 a beam of short-lived particles was cooled for the first time ever. Utilizing frictional cooling on a beam of slow negative muons we observed increase in phase space density by about one order of magnitude. This revised version was published online in August 2006 with corrections to the Cover Date.     

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