Functional Limit Theorem for the Empirical Process of a Class of Bernoulli Shifts with Long Memory

We prove a functional central limit theorem for the empirical process of a stationary process X_t=Y_t+V_t, where Y_t is a long memory moving average in i.i.d. r.v.s _s, s t, and V_t=V (_t, _t-1,...) is a weakly dependent nonlinear Bernoulli shift. Conditions of weak dependence of V_t are written in terms of L²-norms of shift-cut differences V (_t, _t-n, 0,...,) – V(_t,...,_t-n+1, 0,...). Examples of Bernoulli shifts are discussed. The limit empirical process is a degenerated process of the form f(x)Z, where f is the marginal p.d.f. of X₀ and Z is a standard normal r.v. The proof is based on a uniform reduction principle for the empirical process.     

Stable mixed moving averages

Summary The class of (non-Gaussian) stable moving average processes is extended by introducing an appropriate joint randomization of the filter function and of the stable noise, leading to stable mixed moving averages. Their distribution determines a certain combination of the filter function and the mixing measure, leading to a generalization of a theorem of Kanter (1973) for usual moving averages. Stable mixed moving averages contain sums of independent stable moving averages, are ergodic and are not harmonizable. Also a class of stable mixed moving averages is constructed with the reflection positivity property.Research supported by AFSOR Contract 91-0030Research also supported by ARO DAAL-91-G-0176Research also supported by AFOSR 90-0168Research also supported by ONR N00014-91-J-0277     

Self-similar random fields

The present paper is devoted to a survey of results of recent years in the theory of self-similar fields and convergence of renormalized transformations to such fields (limit theorems) and also applications in the theory of critical phenomena of statistical physics and quantum field theory.Translated from Itogi Nauki i Tekhniki, Seriya Teoriya Veroyatnostei, Matematicheskaya Statistika, Teoreticheskaya Kibernetika, Vol. 20, pp. 3–52, 1983.     

CLT and other limit theorems for functionals of Gaussian processes

Summary Conditions for the CLT for non-linear functionals of stationary Gaussian sequences are discussed, with special references to the borderline between the CLT and the non-CLT. Examples of the non-CLT for such functionals with the norming factor are given.     

Consistent polygonal fields

Summary We extend our results [3] on the construction of polygonal Markov fields on the plane taking finite number of values and having a given symmetric Markov process as the marginal process on each line, to the case when is reversible.     

Stratified structure of the Universe and Burgers' equation—a probabilistic approach

Summary The model of the potential turbulence described by the 3-dimensional Burgers' equation with random initial data was developped by Zeldovich and Shandarin, in order to explain the existing Large Scale Structure of the Universe. Most of the recent probabilistic investigations of large time asymptotics of the solution deal with the central limit type results (the Gaussian scenario), under suitable moment assumptions on the initial velocity field. These results and some open questions are discussed in Sect. 2, where we concentrate on the Gaussian model and the shot-noise model. In Sect. 3 we construct a probabilistic model of strong initial fluctuations (a zero-range shot-noise field with high amplitudes) which reveals an intermittent large time behaviour, with the velocity determined by the position of the largest initial fluctuation (discounted by the heat kernelg(t,x·)) in a neighborhood ofx. The asymptoties of such local maximum ast can be analyzed with the help of the theory of records (Sect. 4). Finally, in Sect. 5 we introduce a global definition of a point process oft-local maxima, and show the weak convergence of the suitably rescaled process to a non-trivial limit ast.     

Kinetic modeling of charge-transfer quenching in the CP29 minor complex

We performed transient absorption (TA) measurements on CP29 minor light-harvesting complexes that were reconstituted in vitro with either violaxanthin (Vio) or zeaxanthin (Zea) and demonstrate that the Zea-bound CP29 complexes exhibit charge-transfer (CT) quenching that has been correlated with the energy-dependent quenching (qE) in higher plants. Simulations of the difference TA kinetics reveal two-phase kinetics for intracomplex energy transfer to the CT quenching site in CP29 complexes, with a fast <500 fs component and a approximately 6 ps component. Specific chlorophyll sites within CP29 are identified as likely locations for CT quenching. We also construct a kinetic model for CT quenching during qE in an intact system that incorporates CP29 as a CT trap and show that the model is consistent with previous in vivo measurements on spinach thylakoid membranes. Finally, we compare simulations of CT quenching in thylakoids with those of the individual CP29 complexes and propose that CP29 rather than LHCII is a site of CT quenching.     

Time-Varying Fractionally Integrated Processes with Finite or Infinite Variance and Nonstationary Long Memory

Recently, Philippe et al. (C.R. Acad. Sci. Paris. Ser. I 342, 269–274, 2006; Theory Probab. Appl., 2007, to appear) introduced a new class of time-varying fractionally integrated filters A(d)x _t =∑ _j=0 ^∞ a _j (t)x _t?j , B(d)x _t =∑ _j=0 ^∞ b _j (t)x _t?j depending on arbitrary given sequence d=(d _t ,t∈?) of real numbers, such that A(d)^?1=B(?d), B(d)^?1=A(?d) and such that when d _t ≡d is a constant, A(d)=B(d)=(1?L) ^d is the usual fractional differencing operator. Philippe et al. studied partial sums limits of (nonstationary) filtered white noise processes X _t =B(d)ε _t and Y _t =A(d)ε _t in the case when (1) d is almost periodic having a mean value $\bar{d}\in (0,1/2)$ , or (2) d admits limits d _±=lim? _t→±∞ d _t ∈(0,1/2) at t=±∞. The present paper extends the above mentioned results of Philippe et al. into two directions. Firstly, we consider the class of time-varying processes with infinite variance, by assuming that ε _t ,t∈? are iid rv’s in the domain of attraction of α-stable law (1<α≤2). Secondly, we combine the classes (1) and (2) of sequences d=(d _t ,t∈?) into a single class of sequences d=(d _t ,t∈?) admitting possibly different Cesaro limits $\bar{d}_{\pm}\in(0,1-(1/\alpha))$ at ±∞. We show that partial sums of X _t and Y _t converge to some α-stable self-similar processes depending on the asymptotic parameters $\bar{d}_{\pm}$ and having asymptotically stationary or asymptotically vanishing increments.     

Anisotropic scaling limits of long-range dependent random fields

Lithuanian Mathematical Journal - We review recent results on anisotropic scaling limits and the scaling transition for linear and their subordinated nonlinear long-range dependent stationary...