Randomly fractionally integrated processes

Philippe et al. [9], [10] introduced two distinct time-varying mutually invertible fractionally integrated filters A(d), B(d) depending on an arbitrary sequence d = (d _t ) _t∈ℤ of real numbers; if the parameter sequence is constant d _t ≡ d, then both filters A(d) and B(d) reduce to the usual fractional integration operator (1 − L)^−d . They also studied partial sums limits of filtered white noise nonstationary processes A(d)ε _t and B(d)ε _t for certain classes of deterministic sequences d. The present paper discusses the randomly fractionally integrated stationary processes X _t ^A = A(d)ε _t and X _t ^B = B(d)ε _t by assuming that d = (d _t , t ∈ ℤ) is a random iid sequence, independent of the noise (ε _t ). In the case where the mean , we show that large sample properties of X ^A and X ^B are similar to FARIMA(0, , 0) process; in particular, their partial sums converge to a fractional Brownian motion with parameter . The most technical part of the paper is the study and characterization of limit distributions of partial sums for nonlinear functions h(X _t ^A ) of a randomly fractionally integrated process X _t ^A with Gaussian noise. We prove that the limit distribution of those sums is determined by a conditional Hermite rank of h. For the special case of a constant deterministic sequence d _t , this reduces to the standard Hermite rank used in Dobrushin and Major [2]. Published in Lietuvos Matematikos Rinkinys, Vol. 47, No. 1, pp. 3–28, January–March, 2007.     

Belt-shaped π-systems: relating geometry to electronic structure in a six-porphyrin nanoring

Linear π-conjugated oligomers have been widely investigated, but the behavior of the corresponding cyclic oligomers is poorly understood, despite the recent synthesis of π-conjugated macrocycles such as [n]cycloparaphenylenes and cyclo[n]thiophenes. Here we present an efficient template-directed synthesis of a π-conjugated butadiyne-linked cyclic porphyrin hexamer directly from the monomer. Small-angle X-ray scattering data show that this nanoring is shape-persistent in solution, even without its template, whereas the linear porphyrin hexamer is relatively flexible. The crystal structure of the nanoring-template complex shows that most of the strain is localized in the acetylenes; the porphyrin units are slightly curved, but the zinc coordination sphere is undistorted. The electrochemistry, absorption, and fluorescence spectra indicate that the HOMO-LUMO gap of the nanoring is less than that of the linear hexamer and less than that of the corresponding polymer. The nanoring exhibits six one-electron reductions and six one-electron oxidations, most of which are well resolved. Ultrafast fluorescence anisotropy measurements show that absorption of light generates an excited state that is delocalized over the whole π-system within a time of less than 0.5 ps. The fluorescence spectrum is amazingly structured and red-shifted. A similar, but less dramatic, red-shift has been reported in the fluorescence spectra of cycloparaphenylenes and was attributed to a high exciton binding energy; however the exciton binding energy of the porphyrin nanoring is similar to those of linear oligomers. Quantum-chemical excited state calculations show that the fluorescence spectrum of the nanoring can be fully explained in terms of vibronic Herzberg-Teller (HT) intensity borrowing.     

Asymptotics of partial sums of linear processes with changing memory parameter*

We study the limit distribution of partial sums of nonstationary truncated linear process {X _t , t = 1,…, n} with long memory and changing memory parameter d _t,n ∈ (0,∞). Two classes of linear processes are investigated, namely, (i) the class of FARIMA-type truncated moving averages with time-varying fractional integration parameter and (ii) the class of time-varying fractionally integrated processes introduced in [A. Philippe, D. Surgailis, and M.-C. Viano, Invariance principle for a class of nonstationary processes with long memory, C. R. Acad. Sci., Paris, Sér. I, 342:269–274, 2006; A. Philippe, D. Surgailis, and M.-C. Viano, Time-varying fractionally integrated processes with nonstationary long memory, Theory Probab. Appl., 52:651–673, 2008]. The cases of fast-changing memory parameter (d _t,n = d _t does not depend on n) and slowly changing memory parameter (d _t,n = d(t/n) for some function d(τ), τ ∈ [0, 1]) are discussed. In the case of fast-changing memory, the limit partial sums process is a type II fractional Brownian motion (fBm) with the Hurst parameter equal to the global maximum of (d _t ) for class (i) processes and to the mean value of (d _t ) for class (ii) processes. In the case of slowly changing memory, the limit of partial sums for both classes (i) and (ii) is degenerate and “localized” at the global maximum of the memory function d(?); however, a nondegenerate limit of the partial sums process is shown to exist when time is suitably rescaled in the vicinity of the maximum point.     

On infinitely divisible self-similar random fields

Summary A class of self-similar stationary random fields in ^d , d1 with finite variance is constructed by means of multiple stochastic integrals with respect to the Poisson random measure in ^d+1. Various topics associated with these fields such as subordination, ergodicity, existence of higher order moments, uniqueness of stochastic integral representation, renormalized powers of linear generalized fields and some limit theorems are studied. A Lévy-Hinin type formula for the characteristic functional of general infinitely divisible self-similar random fields with finite variance is obtained.     

The thermodynamic limit of polygonal models

We discuss the basic models of polygonal Markov fields with a two-dimensional continuous parameter % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEaiabgI% GioNWaajaadsfaoiabgkOimlabl2riHoaaCaaabeqaaiaaikdaaaaa% aa!3DEB!\[x \in T \subset \mathbb{R}^2 \], which were introduced by Arak and studied later by Arak and surgailis. There are two types of polygonal models, either with a given initial distribution of the lines or of the points (vertices) of a random polygonal graph. The main result of this paper is proof of the existence of a thermodynamic limit (as % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbiqaaW2ajmaqca% WGubacciqceaQae8xKH0QdcqWIDesOdaahaaqabeaacaaIYaaaaaaa% !3C7D!\[T \uparrow \mathbb{R}^2 \]) for a class of polygonal models with a small contour length'. It is based on the study of Kirkwood-Salzburg-type equations for the correlation functions. We also discuss some examples of consistent polygonal models for which the existence of the thermodynamic limit is trivial.