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1.
The paper obtains a functional limit theorem for the empirical process of a stationary moving average process Xt with i.i.d. innovations belonging to the domain of attraction of a symmetric -stable law, 1<<2, with weights bj decaying as j−β, 1<β<2/. We show that the empirical process (normalized by N1/β) weakly converges, as the sample size N increases, to the process cx+L++cxL, where L+,L are independent totally skewed β-stable random variables, and cx+,cx are some deterministic functions. We also show that, for any bounded function H, the weak limit of suitably normalized partial sums of H(Xs) is an β-stable Lévy process with independent increments. This limiting behavior is quite different from the behavior of the corresponding empirical processes in the parameter regions 1/<β<1 and 2/<β studied in Koul and Surgailis (Stochastic Process. Appl. 91 (2001) 309) and Hsing (Ann. Probab. 27 (1999) 1579), respectively.  相似文献   

2.
In this paper we exhibit a triangular map F of the square with the following properties: (i) F is of type 2 but has positive topological entropy; we recall that similar example was given by Kolyada in 1992, but our argument is much simpler. (ii) F is distributionally chaotic in the wider sense, but not distributionally chaotic in the sense introduced by Schweizer and Smítal [Trans. Amer. Math. Soc. 344 (1994) 737]. In other words, there are lower and upper distribution functions Φxy and Φxy* generated by F such that Φxy*≡1 and Φxy(0+)<1, and no distribution functions Φuv, and Φuv* such that Φuv*≡1 and Φuv(t)=0 whenever 0<t<ε, for some ε>0. We also show that the two notions of distributional chaos used in the paper, for continuous maps of a compact metric space, are invariants of topological conjugacy.  相似文献   

3.
Let A be a matrix in r×r such that Re(z) > −1/2 for all the eigenvalues of A and let {πn(A,1/2) (x)} be the normalized sequence of Laguerre matrix polynomials associated with A. In this paper, it is proved that πn(A,1/2) (x) = O(n(A)/2lnr−1(n)) and πn+1(A,1/2) (x) − πn(A,1/2) (x) = O(n((A)−1)/2lnr−1(n)) uniformly on bounded intervals, where (A) = max{Re(z); z eigenvalue of A}.  相似文献   

4.
Consider the following Itô stochastic differential equation dX(t) = ƒ(θ0, X(t)) dt + dW(t), where (W(t), t 0), is a standard Wiener process in RN. On the basis of discrete data 0 = t0 < t1 < …<tn = T; X(t1),...,X(tn) we would like to estimate the parameter θ0. We shall define the least squares estimator and show that under some regularity conditions, is strongly consistent.  相似文献   

5.
Only photons are needed to explain the masses of the π0, η, Λ, Σ0, Ξ0, Ω, Λc+, Σc0, Ξc0 and Ωc0 mesons and baryons with the sum of the energies contained in the frequencies of standing electromagnetic waves in a cubic black body. Only neutrinos are needed to explain the mass of the π± mesons with the sum of the energies of standing oscillations of muon and electron neutrinos in a cubic lattice plus the energies contained in the rest masses of the neutrinos. Neutrinos and photons are needed to explain the masses of the K± mesons. Surprisingly the mass of the μ± mesons can also be explained without an additional assumption by the oscillation energies and rest masses of a neutrino lattice. From the difference of the masses of the π± mesons and μ± mesons we find that the rest mass of the muon neutrino is 47.5 meV/c2. From the difference of the masses of the neutron and proton we find that the rest mass of the electron neutrino is 0.55 meV/c2. The potential of the weak force between the lattice points can be determined from Born’s lattice theory. From the weak force between the lattice points follows automatically the existence of a strong force between the sides of two lattices. The strong nuclear force is the sum of the unsaturated weak forces at the sides of each lattice and is therefore about 106 times stronger than the weak force.  相似文献   

6.
For a positive integer k2, the k-Fibonacci sequence {gn(k)} is defined as: g1(k)==gk−2(k)=0, gk−1(k)=gk(k)=1 and for n>k2, gn(k)=gn−1(k)+gn−2(k)++gnk(k). Moreover, the k-Lucas sequence {ln(k)} is defined as ln(k)=gn−1(k)+gn+k−1(k) for n1. In this paper, we consider the relationship between gn(k) and ln(k) and 1-factors of a bipartite graph.  相似文献   

7.
We discuss the covariance structure and long-memory properties of stationary solutions of the bilinear equation XttAt+Bt,(), where 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 and , respectively. We also discuss the hyperbolic decay of auto- and/or cross-covariances of Xt and Xt2 and the asymptotic distribution of the corresponding partial sums’ processes.  相似文献   

8.
For the pth-order linear ARCH model,
, where 0 > 0, i 0, I = 1, 2, …, p, {t} is an i.i.d. normal white noise with Et = 0, Et2 = 1, and t is independent of {Xs, s < t}, Engle (1982) obtained the necessary and sufficient condition for the second-order stationarity, that is, 1 + 2 + ··· + p < 1. In this note, we assume that t has the probability density function p(t) which is positive and lower-semicontinuous over the real line, but not necessarily Gaussian, then the geometric ergodicity of the ARCH(p) process is proved under Et2 = 1. When t has only the first-order absolute moment, a sufficient condition for the geometric ergodicity is also given.  相似文献   

9.
For a 1-dependent stationary sequence {Xn} we first show that if u satisfies p1=p1(u)=P(X1>u)0.025 and n>3 is such that 88np131, then
P{max(X1,…,Xn)u}=ν·μn+O{p13(88n(1+124np13)+561)}, n>3,
where
ν=1−p2+2p3−3p4+p12+6p22−6p1p2,μ=(1+p1p2+p3p4+2p12+3p22−5p1p2)−1
with
pk=pk(u)=P{min(X1,…,Xk)>u}, k1
and
|O(x)||x|.
From this result we deduce, for a stationary T-dependent process with a.s. continuous path {Ys}, a similar, in terms of P{max0skTYs<u}, k=1,2 formula for P{max0stYsu}, t>3T and apply this formula to the process Ys=W(s+1)−W(s), s0, where {W(s)} is the Wiener process. We then obtain numerical estimations of the above probabilities.  相似文献   

10.
From GCH and Pm(κ)-hypermeasurable (1 <m<gw), we construct a model satisfying 2n = a(n) and 2ω = ω+m for a monotone a:ω→ω satisfying a(n)>n.  相似文献   

11.
We are concerned with the behavior of the minimum (maximum) eigenvalue λ0(n) (λn(n)) of an (n + 1) × (n + 1) Hermitian Toeplitz matrix Tn(ƒ) where ƒ is an integrable real-valued function. Kac, Murdoch, and Szegö, Widom, Parter, and R. H. Chan obtained that λ0(n) — min ƒ = O(1/n2k) in the case where ƒ C2k, at least locally, and ƒ — inf ƒ has a zero of order 2k. We obtain the same result under the second hypothesis alone. Moreover we develop a new tool in order to estimate the extreme eigenvalues of the mentioned matrices, proving that the rate of convergence of λ0(n) to inf ƒ depends only on the order ρ (not necessarily even or integer or finite) of the zero of ƒ — inf ƒ. With the help of this tool, we derive an absolute lower bound for the minimal eigenvalues of Toeplitz matrices generated by nonnegative L1 functions and also an upper bound for the associated Euclidean condition numbers. Finally, these results are extended to the case of Hermitian block Toeplitz matrices with Toeplitz blocks generated by a bivariate integrable function ƒ.  相似文献   

12.
In this note we characterize doubly stochastic matrices A whose powers A,A2,A3,… eventually stop, i.e., Ap=Ap+1= for some positive integer p. The characterization enables us to determine the set of all such matrices.  相似文献   

13.
14.
For a double array {V_(m,n), m ≥ 1, n ≥ 1} of independent, mean 0 random elements in a real separable Rademacher type p(1 ≤ p ≤ 2) Banach space and an increasing double array {b_(m,n), m ≥1, n ≥ 1} of positive constants, the limit law ■ and in L_p as m∨n→∞ is shown to hold if ■ This strong law of large numbers provides a complete characterization of Rademacher type p Banach spaces. Results of this form are also established when 0 p ≤ 1 where no independence or mean 0 conditions are placed on the random elements and without any geometric conditions placed on the underlying Banach space.  相似文献   

15.
A target, whose initial position is unknown, is performing a random walk on the integers. A searcher, starting at the origin, wants to follow a search plan for which E[τk] is finite, where k ≥ 1 and τ is the time to capture. The searcher, who has a prior distribution over the target's initial position, can move only to adjacent positions, and cannot travel faster than the target. Necessary and sufficient conditions are given for the existence of search plans for which E[τk] is finite and a minimum.  相似文献   

16.
In this paper we investigate the quasi-shadowing property for C~1 random dynamical systems on their random partially hyperbolic sets. It is shown that for any pseudo orbit {xk}_(-∞)~(+∞)on a random partially hyperbolic set there exists a "center" pseudo orbit {yk}_(-∞)~(+∞)shadowing it in the sense that yk+1 is obtained from the image of yk by a motion along the center direction. Moreover, when the random partially hyperbolic set has a local product structure, the above "center" pseudo orbit {yk}_(-∞)~(+∞)can be chosen such that yk+1 and the image of yk lie in their common center leaf.  相似文献   

17.
The mixed volume optimization problem is to determine the point of duality Q for a given convex set K that minimizes the “mixed volume” of the associated polar set (K*;Q). In the plane, the mixed volumes translate as the area and length; in space, the mixed volumes include the volume, surface area, and mean width. In this paper, the geometric optimization problems associated with minimizing mixed volumes are examined from two perspectives: enumerative search and symbolic computation. The problem of minimizing the polar area through an enumerative search is first considered. The dual polygon (Pm*;Q) is constructed for an arbitrary point of duality QPm° by using an algebraic correspondence between the edges of Pm and the vertices of (Pm*;Q), and the area of (Pm*;Q), A(P*m;Q), is calculated and minimized using naive search techniques. A result due to Santaló is applied to verify the minimizing solution, and computational tests are described for various classes of randomly generated polygons. Statistical evidence indicates that a “good” approximation to the minimum area polar polygon occurs when the duality point is located at the center-of-gravity of Pm. The polar area problem is then investigated using symbolic procedures. Explicit symbolic expressions for the polar area and length functionals are computed and solved directly using the differential optimality conditions and Newton's iterative method of solution. The mixed volume and surface area functionals are formulated and solved using numerical products, and the mean width functional is described. Examples are used throughout to illustratethe methodology.  相似文献   

18.
We have considered the problem of the weak convergence, as tends to zero, of the multiple integral processes
in the space , where fL2([0,T]n) is a given function, and {η(t)}>0 is a family of stochastic processes with absolutely continuous paths that converges weakly to the Brownian motion. In view of the known results when n2 and f(t1,…,tn)=1{t1<t2<<tn}, we cannot expect that these multiple integrals converge to the multiple Itô–Wiener integral of f, because the quadratic variations of the η are null. We have obtained the existence of the limit for any {η}, when f is given by a multimeasure, and under some conditions on {η} when f is a continuous function and when f(t1,…,tn)=f1(t1)fn(tn)1{t1<t2<<tn}, with fiL2([0,T]) for any i=1,…,n. In all these cases the limit process is the multiple Stratonovich integral of the function f.  相似文献   

19.
Suppose {k, −∞ < k < ∞} is an independent, not necessarily identically distributed sequence of random variables, and {cj}j=0, {dj}j=0 are sequences of real numbers such that Σjc2j < ∞, Σjd2j < ∞. Then, under appropriate moment conditions on {k, −∞ < k < ∞}, yk Σj=0cjk-j, zk Σj=0djk-j exist almost surely and in 4 and the question of Gaussian approximation to S[t]Σ[t]k=1 (yk zkE{yk zk}) becomes of interest. Prior to this work several related central limit theorems and a weak invariance principle were proven under stationary assumptions. In this note, we demonstrate that an almost sure invariance principle for S[t], with error bound sharp enough to imply a weak invariance principle, a functional law of the iterated logarithm, and even upper and lower class results, also exists. Moreover, we remove virtually all constraints on k for “time” k ≤ 0, weaken the stationarity assumptions on {k, −∞ < k < ∞}, and improve the summability conditions on {cj}j=0, {dj}j=0 as compared to the existing weak invariance principle. Applications relevant to this work include normal approximation and almost sure fluctuation results in sample covariances (let dj = cj-m for jm and otherwise 0), quadratic forms, Whittle's and Hosoya's estimates, adaptive filtering and stochastic approximation.  相似文献   

20.
In this paper it is proved that the exponential generating function of the numbers, denoted by N(p, q), of irreducible coverings by edges of the vertices of complete bipartite graphs Kp,q equals exp(xey + yexxyxy) − 1.  相似文献   

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