Zones of attraction of self-similar multiple integrals, 1982
ZONES OF ATTRACTION OF SELF-SIMILAR MULTIPLE INTEGRALS D. Surgailis UDC 519.21 The goal of this p... more ZONES OF ATTRACTION OF SELF-SIMILAR MULTIPLE INTEGRALS D. Surgailis UDC 519.21 The goal of this paper is the proof of the theorem announced in [20]. Here we consider the "multidimensional case," i.e., convergence to self-similar fields. We give a short survey of the separate sections. In Sec. 1 the concepts needed are formulated as well as the basic result of the paper (Theorem i). The connection of this theorem with the result of Dobrushin-Major [8] is discussed as well as some similar questions. In Sec. 2 the term making the basic contribution to the distribution of the sums considered is isolated. Here we explain the idea of the following proof, which is broken up into several lemmas, and their formulations are given. The proofs of these lemmas (except for Lemma 5) are carried out to section 3. In section 4 there is proved a lemma (Lemma 1 of [20]) on the convergence with respect to distribution of "discrete multiple integrals" to "continuous" integr...
It is well-known that the aggregated time series might have very different properties from those ... more It is well-known that the aggregated time series might have very different properties from those of the individual series, in particular, long memory. At the present time, aggregation has become one of the main tools for modelling of long memory processes. We review recent work on contemporaneous aggregation of random-coefficient AR(1) and related models, with particular focus on various long memory properties of the aggregated process.
The abstracts (in two languages) can be found in the pdf file of the article. Original author nam... more The abstracts (in two languages) can be found in the pdf file of the article. Original author name(s) and title in Russian and Lithuanian: Д. Сургайлис. О единственности решения стохастического уравнения К. Ито D. Surgailis. Apie K. Ito stochastinės lygties sprendinio vienatinumą
The IMA Volumes in Mathematics and its Applications, 1993
Shot noise processes and random fields with long memory are discussed. Convergence of the distrib... more Shot noise processes and random fields with long memory are discussed. Convergence of the distribution of integrated polynomials of shot noise process with long memory to a self-similar process expressed as multiple Wiener-Ito integral is proved. Asymptotical normality of solutions of the Burgers equation in R 3 with random initial data as the gradient of a shot noise field with long memory is established, extending recent results by Bulinskii and Molchanov.
We discuss the covariance structure and long-memory properties of stationary solutions of the bil... more We discuss the covariance structure and long-memory properties of stationary solutions of the bilinear equation Xt = t At + Bt; (?), where t ; t ∈ Z are standard i.i.d. r.v.'s, and At; Bt are moving averages in Xs; s ¡ t. Stationary solution of (?) is obtained as an orthogonal Volterra expansion. In the case At ≡ 1; Xt is the classical AR(∞) process, while Bt ≡ 0 gives the LARCH model studied by Giraitis et al. (Ann. Appl. Probab. 10 (2000) 1002). In the general case, Xt may exhibit long memory both in conditional mean and in conditional variance, with arbitrary fractional parameters 0 ¡ d1 ¡ 1 2 and 0 ¡ d2 ¡ 1 2 , respectively. We also discuss the hyperbolic decay of auto-and=or cross-covariances of Xt and X 2 t and the asymptotic distribution of the corresponding partial sums' processes.
A (noncentral) limit theorem is proved for sums SN(m) ~-ZP,,(X,) 1 als of degree m > 1 of a linea... more A (noncentral) limit theorem is proved for sums SN(m) ~-ZP,,(X,) 1 als of degree m > 1 of a linear (or moving average) process of polynomi-X,= ~ a(t-s)~s-co with slowly decreasing coefficients a(t). Conditions assuring the convergence of the distributions SN(m)/B N are expressed in terms of the asymptotics of the N variances BN 2 = Var SN(m) and A~=VarZX ,. The limit distribution of the sums 1 SN(m)/B N is given by an m-fold stochastic Ito-Wiener integral. The theorem proved develops the results of [I-7], obtained under the hypothesis that the process X t is Gaussian and (or) of the regularity of the asymptotics of the coefficients a(t).
The abstracts (in two languages) can be found in the pdf file of the article. Original author nam... more The abstracts (in two languages) can be found in the pdf file of the article. Original author name(s) and title in Russian and Lithuanian: Д. Сургайлис. О стохастических уравнениях D. Surgailis. Apie stochastines lygtis
The abstracts (in two languages) can be found in the pdf file of the article. Original author nam... more The abstracts (in two languages) can be found in the pdf file of the article. Original author name(s) and title in Russian and Lithuanian: Д. Сургайлис. О стохастических уравнениях фильтрации марковских процессов D. Surgailis. Apie Markovo procesų filtracijos stochastines lygtis
Burgers turbulence is an accepted formalism for the adhesion model of the large-scale distributio... more Burgers turbulence is an accepted formalism for the adhesion model of the large-scale distribution of matter in the universe. The paper uses variational methods to establish evolution of quasi-Voronoi (curved boundaries) tessellation structure of shock fronts for solutions of the inviscid nonhomogeneous Burgers equation in R d in the presence of random forcing due to a degenerate potential. The mean rate of growth of the quasi-Voronoi cells is calculated and a scaled limit random tessellation structure is found. Time evolution of the probability that a cell contains a ball of a given radius is also determined.
We discuss a class of conditionally heteroscedastic time series models satisfying the equation r ... more We discuss a class of conditionally heteroscedastic time series models satisfying the equation r t = ζ t σ t , where ζ t are standardized i.i.d. r.v.'s and the conditional standard deviation σ t is a nonlinear function Q of inhomogeneous linear combination of past values r s , s < t with coefficients b j. The existence of stationary solution r t with finite pth moment, 0 < p < ∞ is obtained under some conditions on Q, b j and pth moment of ζ 0. Weak dependence properties of r t are studied, including the invariance principle for partial sums of Lipschitz functions of r t. In the case of quadratic Q 2 , we prove that r t can exhibit a leverage effect and long memory, in the sense that the squared process r 2 t has long memory autocorrelation and its normalized partial sums process converges to a fractional Brownian motion.
Zeitschrift f�r Wahrscheinlichkeitstheorie und Verwandte Gebiete, 1981
A class of self-similar stationary random fields in IR d, d > 1 with finite variance is construct... more A class of self-similar stationary random fields in IR d, d > 1 with finite variance is constructed by means of multiple stochastic integrals with respect to the Poisson random measure in IR 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 L6vy-Hin6in type formula for the characteristic functional of general infinitely divisible self-similar random fields with finite variance is obtained.
A central limit theorem for quadratic forms in strongly dependent linear (or moving average) vari... more A central limit theorem for quadratic forms in strongly dependent linear (or moving average) variables is proved, generalizing the results of Avram [1] and Fox and Taqqu [3] for Gaussian variables. The theorem is applied to prove asymptotical normality of Whittle's estimate of the parameter of strongly dependent linear sequences.
Philippe et al. [9], [10] introduced two distinct time-varying mutually invertible fractionally i... more 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 A t = A(d)ε t and X B t = 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 meand = Ed 0 ∈ (0, 1/2), we show that large sample properties of X A and X B are similar to FARIMA(0,d, 0) process; in particular, their partial sums converge to a fractional Brownian motion with parameterd + (1/2). The most technical part of the paper is the study and characterization of limit distributions of partial sums for nonlinear functions h(X A t) of a randomly fractionally integrated process X A t 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].
Zones of attraction of self-similar multiple integrals, 1982
ZONES OF ATTRACTION OF SELF-SIMILAR MULTIPLE INTEGRALS D. Surgailis UDC 519.21 The goal of this p... more ZONES OF ATTRACTION OF SELF-SIMILAR MULTIPLE INTEGRALS D. Surgailis UDC 519.21 The goal of this paper is the proof of the theorem announced in [20]. Here we consider the "multidimensional case," i.e., convergence to self-similar fields. We give a short survey of the separate sections. In Sec. 1 the concepts needed are formulated as well as the basic result of the paper (Theorem i). The connection of this theorem with the result of Dobrushin-Major [8] is discussed as well as some similar questions. In Sec. 2 the term making the basic contribution to the distribution of the sums considered is isolated. Here we explain the idea of the following proof, which is broken up into several lemmas, and their formulations are given. The proofs of these lemmas (except for Lemma 5) are carried out to section 3. In section 4 there is proved a lemma (Lemma 1 of [20]) on the convergence with respect to distribution of "discrete multiple integrals" to "continuous" integr...
It is well-known that the aggregated time series might have very different properties from those ... more It is well-known that the aggregated time series might have very different properties from those of the individual series, in particular, long memory. At the present time, aggregation has become one of the main tools for modelling of long memory processes. We review recent work on contemporaneous aggregation of random-coefficient AR(1) and related models, with particular focus on various long memory properties of the aggregated process.
The abstracts (in two languages) can be found in the pdf file of the article. Original author nam... more The abstracts (in two languages) can be found in the pdf file of the article. Original author name(s) and title in Russian and Lithuanian: Д. Сургайлис. О единственности решения стохастического уравнения К. Ито D. Surgailis. Apie K. Ito stochastinės lygties sprendinio vienatinumą
The IMA Volumes in Mathematics and its Applications, 1993
Shot noise processes and random fields with long memory are discussed. Convergence of the distrib... more Shot noise processes and random fields with long memory are discussed. Convergence of the distribution of integrated polynomials of shot noise process with long memory to a self-similar process expressed as multiple Wiener-Ito integral is proved. Asymptotical normality of solutions of the Burgers equation in R 3 with random initial data as the gradient of a shot noise field with long memory is established, extending recent results by Bulinskii and Molchanov.
We discuss the covariance structure and long-memory properties of stationary solutions of the bil... more We discuss the covariance structure and long-memory properties of stationary solutions of the bilinear equation Xt = t At + Bt; (?), where t ; t ∈ Z are standard i.i.d. r.v.'s, and At; Bt are moving averages in Xs; s ¡ t. Stationary solution of (?) is obtained as an orthogonal Volterra expansion. In the case At ≡ 1; Xt is the classical AR(∞) process, while Bt ≡ 0 gives the LARCH model studied by Giraitis et al. (Ann. Appl. Probab. 10 (2000) 1002). In the general case, Xt may exhibit long memory both in conditional mean and in conditional variance, with arbitrary fractional parameters 0 ¡ d1 ¡ 1 2 and 0 ¡ d2 ¡ 1 2 , respectively. We also discuss the hyperbolic decay of auto-and=or cross-covariances of Xt and X 2 t and the asymptotic distribution of the corresponding partial sums' processes.
A (noncentral) limit theorem is proved for sums SN(m) ~-ZP,,(X,) 1 als of degree m > 1 of a linea... more A (noncentral) limit theorem is proved for sums SN(m) ~-ZP,,(X,) 1 als of degree m > 1 of a linear (or moving average) process of polynomi-X,= ~ a(t-s)~s-co with slowly decreasing coefficients a(t). Conditions assuring the convergence of the distributions SN(m)/B N are expressed in terms of the asymptotics of the N variances BN 2 = Var SN(m) and A~=VarZX ,. The limit distribution of the sums 1 SN(m)/B N is given by an m-fold stochastic Ito-Wiener integral. The theorem proved develops the results of [I-7], obtained under the hypothesis that the process X t is Gaussian and (or) of the regularity of the asymptotics of the coefficients a(t).
The abstracts (in two languages) can be found in the pdf file of the article. Original author nam... more The abstracts (in two languages) can be found in the pdf file of the article. Original author name(s) and title in Russian and Lithuanian: Д. Сургайлис. О стохастических уравнениях D. Surgailis. Apie stochastines lygtis
The abstracts (in two languages) can be found in the pdf file of the article. Original author nam... more The abstracts (in two languages) can be found in the pdf file of the article. Original author name(s) and title in Russian and Lithuanian: Д. Сургайлис. О стохастических уравнениях фильтрации марковских процессов D. Surgailis. Apie Markovo procesų filtracijos stochastines lygtis
Burgers turbulence is an accepted formalism for the adhesion model of the large-scale distributio... more Burgers turbulence is an accepted formalism for the adhesion model of the large-scale distribution of matter in the universe. The paper uses variational methods to establish evolution of quasi-Voronoi (curved boundaries) tessellation structure of shock fronts for solutions of the inviscid nonhomogeneous Burgers equation in R d in the presence of random forcing due to a degenerate potential. The mean rate of growth of the quasi-Voronoi cells is calculated and a scaled limit random tessellation structure is found. Time evolution of the probability that a cell contains a ball of a given radius is also determined.
We discuss a class of conditionally heteroscedastic time series models satisfying the equation r ... more We discuss a class of conditionally heteroscedastic time series models satisfying the equation r t = ζ t σ t , where ζ t are standardized i.i.d. r.v.'s and the conditional standard deviation σ t is a nonlinear function Q of inhomogeneous linear combination of past values r s , s < t with coefficients b j. The existence of stationary solution r t with finite pth moment, 0 < p < ∞ is obtained under some conditions on Q, b j and pth moment of ζ 0. Weak dependence properties of r t are studied, including the invariance principle for partial sums of Lipschitz functions of r t. In the case of quadratic Q 2 , we prove that r t can exhibit a leverage effect and long memory, in the sense that the squared process r 2 t has long memory autocorrelation and its normalized partial sums process converges to a fractional Brownian motion.
Zeitschrift f�r Wahrscheinlichkeitstheorie und Verwandte Gebiete, 1981
A class of self-similar stationary random fields in IR d, d > 1 with finite variance is construct... more A class of self-similar stationary random fields in IR d, d > 1 with finite variance is constructed by means of multiple stochastic integrals with respect to the Poisson random measure in IR 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 L6vy-Hin6in type formula for the characteristic functional of general infinitely divisible self-similar random fields with finite variance is obtained.
A central limit theorem for quadratic forms in strongly dependent linear (or moving average) vari... more A central limit theorem for quadratic forms in strongly dependent linear (or moving average) variables is proved, generalizing the results of Avram [1] and Fox and Taqqu [3] for Gaussian variables. The theorem is applied to prove asymptotical normality of Whittle's estimate of the parameter of strongly dependent linear sequences.
Philippe et al. [9], [10] introduced two distinct time-varying mutually invertible fractionally i... more 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 A t = A(d)ε t and X B t = 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 meand = Ed 0 ∈ (0, 1/2), we show that large sample properties of X A and X B are similar to FARIMA(0,d, 0) process; in particular, their partial sums converge to a fractional Brownian motion with parameterd + (1/2). The most technical part of the paper is the study and characterization of limit distributions of partial sums for nonlinear functions h(X A t) of a randomly fractionally integrated process X A t 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].
Uploads
Papers by D. Surgailis