Catalogue Search | MBRL
Are you sure you want to remove the book from the shelf?
{{itemTitle}}
78
result(s) for
"Surgailis, Donatas"
Sort by:
On Some Open Problems in Spatial Fractional Integration
Some open problems regarding fractional powers of the negative generator of a discrete-time random walk and a Markov process are discussed. The suggested approach combines analytic and probabilistic ideas and may be useful for developing fractional operators with multidimensional and/or abstract discrete arguments.
Journal Article
Consistent Markov Edge Processes and Random Graphs
We discuss Markov edge processes Ye;e∈E defined on edges of a directed acyclic graph (V,E) with the consistency property PE′(Ye;e∈E′)=PE(Ye;e∈E′) for a large class of subgraphs (V′,E′) of (V,E) obtained through a mesh dismantling algorithm. The probability distribution PE of such edge process is a discrete version of consistent polygonal Markov graphs. The class of Markov edge processes is related to the class of Bayesian networks and may be of interest to causal inference and decision theory. On regular ν-dimensional lattices, consistent Markov edge processes have similar properties to Pickard random fields on Z2, representing a far-reaching extension of the latter class. A particular case of binary consistent edge process on Z3 was disclosed by Arak in a private communication. We prove that the symmetric binary Pickard model generates the Arak model on Z2 as a contour model.
Journal Article
Fractional Operators and Fractionally Integrated Random Fields on ν
We consider fractional integral operators (I−T)d,d∈(−1,1) acting on functions g: Z ν→ R ,ν≥1 , where T is the transition operator of a random walk on Z ν . We obtain the sufficient and necessary conditions for the existence, invertibility, and square summability of kernels τ( s ;d), s ∈ Z ν of (I−T)d . The asymptotic behavior of τ( s ;d) as | s |→∞ is identified following the local limit theorem for random walks. A class of fractionally integrated random fields X on Z ν solving the difference equation (I−T)dX=ε with white noise on the right-hand side is discussed and their scaling limits. Several examples, including fractional lattice Laplace and heat operators, are studied in detail.
Journal Article
Fractional Operators and Fractionally Integrated Random Fields on Zν
We consider fractional integral operators (I−T)d,d∈(−1,1) acting on functions g:Zν→R,ν≥1, where T is the transition operator of a random walk on Zν. We obtain the sufficient and necessary conditions for the existence, invertibility, and square summability of kernels τ(s;d),s∈Zν of (I−T)d. The asymptotic behavior of τ(s;d) as |s|→∞ is identified following the local limit theorem for random walks. A class of fractionally integrated random fields X on Zν solving the difference equation (I−T)dX=ε with white noise on the right-hand side is discussed and their scaling limits. Several examples, including fractional lattice Laplace and heat operators, are studied in detail.
Journal Article
Aggregation of autoregressive random fields and anisotropic long-range dependence
We introduce the notions of scaling transition and distributional long-range dependence for stationary random fields Y on ℤ² whose normalized partial sums on rectangles with sides growing at rates O(n) and O(nγ) tend to an operator scaling random field Vγ on ℝ² , for any γ > 0. The scaling transition is characterized by the fact that there exists a unique γ₀ > 0 such that the scaling limits Vγ are different and do not depend on γ for γ > γ₀ and γ < γ₀. The existence of scaling transition together with anisotropic and isotropic distributional long-range dependence properties is demonstrated for a class of α-stable (1 < α ≤ 2) aggregated nearest-neighbor autoregressive random fields on ℤ² with a scalar random coefficient A having a regularly varying probability density near the \"unit root\" A = 1.
Journal Article
JOINT TEMPORAL AND CONTEMPORANEOUS AGGREGATION OF RANDOM-COEFFICIENT AR(1) PROCESSES WITH INFINITE VARIANCE
We discuss the joint temporal and contemporaneous aggregation of N independent copies of random-coefficient AR(1) processes driven by independent and identically distributed innovations in the domain of normal attraction of an α-stable distribution, 0 < α ≥ 2, as both N and the time scale n tend to infinity, possibly at different rates. Assuming that the tail distribution function of the random autoregressive coefficient regularly varies at the unit root with exponent β > 0, we show that, for β < max (α, 1), the joint aggregate displays a variety of stable and non-stable limit behaviors with stability index depending on α, β and the mutual increase rate of N and n. The paper extends the results of Pilipauskaite and Surgailis (2014) from α = 2 to 0 < α < 2.
Journal Article
Anisotropic scaling of the random grain model with application to network traffic
We obtain a complete description of anisotropic scaling limits of the random grain model on the plane with heavy-tailed grain area distribution. The scaling limits have either independent or completely dependent increments along one or both coordinate axes and include stable, Gaussian, and ‘intermediate’ infinitely divisible random fields. The asymptotic form of the covariance function of the random grain model is obtained. Application to superimposed network traffic is included.
Journal Article
Projective Stochastic Equations and Nonlinear Long Memory
A projective moving average X t , t ∈ ℤ is a Bernoulli shift written as a backward martingale transform of the innovation sequence. We introduce a new class of nonlinear stochastic equations for projective moving averages, termed projective equations, involving a (nonlinear) kernel Q and a linear combination of projections of X t on ‘intermediate’ lagged innovation subspaces with given coefficients α i and β i , j . The class of such equations includes usual moving average processes and the Volterra series of the LARCH model. Solvability of projective equations is obtained using a recursive equality for projections of the solution X t . We show that, under certain conditions on Q , α i , and β i , j , this solution exhibits covariance and distributional long memory, with fractional Brownian motion as the limit of the corresponding partial sums process.
Journal Article
STATIONARY INTEGRATED ARCH(∞) AND AR(∞) PROCESSES WITH FINITE VARIANCE
We prove the long standing conjecture of Ding and Granger (1996) about the existence of a stationary Long Memory ARCH model with finite fourth moment. This result follows from the necessary and sufficient conditions for the existence of covariance stationary integrated AR(∞), ARCH(∞), and FIGARCH models obtained in the present article. We also prove that such processes always have long memory.
Journal Article