共查询到20条相似文献,搜索用时 591 毫秒
1.
Let {Xn} be a stationary Gaussian sequence with E{X0} = 0, {X20} = 1 and E{X0Xn} = rnn Let cn = (2ln n), bn = cn? c-1n ln(4π ln n), and set Mn = max0 ?k?nXk. A classical result for independent normal random variables is that Berman has shown that (1) applies as well to dependent sequences provided rnlnn = o(1). Suppose now that {rn} is a convex correlation sequence satisfying rn = o(1), (rnlnn)-1 is monotone for large n and o(1). Then for all x, where Ф is the normal distribution function. While the normal can thus be viewed as a second natural limit distribution for {Mn}, there are others. In particular, the limit distribution is given below when rn is (sufficiently close to) γ/ln n. We further exhibit a collection of limit distributions which can arise when rn decays to zero in a nonsmooth manner. Continuous parameter Gaussian processes are also considered. A modified version of (1) has been given by Pickands for some continuous processes which possess sufficient asymptotic independence properties. Under a weaker form of asymptotic independence, we obtain a version of (2). 相似文献
2.
A.Larry Wright 《Journal of multivariate analysis》1982,12(2):178-185
Two related almost sure limit theorems are obtained in connection with a stochastic process {ξ(t), ?∞ < t < ∞} with independent increments. The first result deals with the existence of a simultaneous stabilizing function H(t) such that for almost all sample functions of the process. The second result deals with a wide-sense stationary process whose random spectral distributions is ξ. It addresses the question: Under what conditions does converge as T → ∞ for all τ for almost all sample functions? 相似文献
3.
V. V. Nekrutkin 《Vestnik St. Petersburg University: Mathematics》2007,40(3):243-249
It is proved that a mixed Poisson process ξt is a Pólya process if and only if there exists a nondegenerate linear transform ξt → ηt = a(t)ξt + b(t) such that ηt is a martingale. A similar result is valid for Pólya sequences. 相似文献
4.
Jürg Hüsler 《Journal of multivariate analysis》1981,11(2):273-279
Let {Xn, n ≥ 1} be a real-valued stationary Gaussian sequence with mean zero and variance one. Let Mn = max{Xt, i ≤ n} and Hn(t) = (M[nt] ? bn)an?1 be the maximum resp. the properly normalised maximum process, where , and . We characterize the almost sure limit functions of (Hn)n≥3 in the set of non-negative, non-decreasing, right-continuous, real-valued functions on (0, ∞), if r(n) (log n)3?Δ = O(1) for all Δ > 0 or if r(n) (log n)2?Δ = O(1) for all Δ > 0 and r(n) convex and fulfills another regularity condition, where r(n) is the correlation function of the Gaussian sequence. 相似文献
5.
Georg Lindgren 《Journal of multivariate analysis》1980,10(2):181-206
Let ζ(t), η(t) be continuously differentiable Gaussian processes with mean zero, unit variance, and common covariance function r(t), and such that ζ(t) and η(t) are independent for all t, and consider the movements of a particle with time-varying coordinates (ζ(t), η(t)). The time and location of the exists of the particle across a circle with radius u defines a point process in R3 with its points located on the cylinder {(t, u cos θ, u sin θ); t ≥ 0, 0 ≤ θ < 2π}. It is shown that if r(t) log t → 0 as t → ∞, the time and space-normalized point process of exits converges in distribution to a Poisson process on the unit cylinder. As a consequence one obtains the asymptotic distribution of the maximum of a χ2-process, χ2(t) = ζ2(t) + η2(t), P{sup0≤t≤Tχ2(t) ≤ u2} → e?τ if as T, u → ∞. Furthermore, it is shown that the points in R3 generated by the local ?-maxima of χ2(t) converges to a Poisson process in R3 with intensity measure (in cylindrical polar coordinates) (2πr2)?1dtdθdr. As a consequence one obtains the asymptotic extremal distribution for any function g(ζ(t), η(t)) which is “almost quadratic” in the sense that has a limit as r → ∞. Then if as T, u → ∞. 相似文献
6.
Robert Chen 《Journal of multivariate analysis》1978,8(2):328-333
Let {Xn}n≥1 be a sequence of independent and identically distributed random variables. For each integer n ≥ 1 and positive constants r, t, and ?, let Sn = Σj=1nXj and . In this paper, we prove that (1) lim?→0+?α(r?1)E{N∞(r, t, ?)} = K(r, t) if E(X1) = 0, Var(X1) = 1, and E(| X1 |t) < ∞, where 2 ≤ t < 2r ≤ 2t, , and ; (2) if 2 < t < 4, E(X1) = 0, Var(X1) > 0, and E(|X1|t) < ∞, where G(t, ?) = E{N∞(t, t, ?)} = Σn=1∞nt?2P{| Sn | > ?n} → ∞ as ? → 0+ and , i.e., H(t, ?) goes to infinity much faster than G(t, ?) as ? → 0+ if 2 < t < 4, E(X1) = 0, Var(X1) > 0, and E(| X1 |t) < ∞. Our results provide us with a much better and deeper understanding of the tail probability of a distribution. 相似文献
7.
Let Mt be the maximum of a recurrent one-dimensional diffusion up till time t. Under appropriate conditions, there exists a distribution function F such that |P(Mt?x) ? Ft(x)|→0as t and x go to infinity. This reduces the asymptotic behavior of the maximum to that of the maximum of independent and identically distributed random variables with distribution function F. A new proof of this fact is given which is based on a time change of the Ornstein-Uhlenbeck process. Using this technique, the asymptotic independence of the maximum and minimum is also established. Moreover, this method allows one to construct stationary processes in which the limiting behavior of Mt is essentially unaffected by the stationary distribution. That is, there may be no relationship between the distribution F above and the marginal distribution of the process. 相似文献
8.
E. Bolthausen 《Stochastic Processes and their Applications》1984,16(2):199-204
Let ξt, t ? 0, be a d-dimensional Brownian motion. The asymptotic behaviour of the random field ??∫t0?(ξs) ds is investigated, where ? belongs to a Sobolev space of periodic functions. Particularly a central limit theorem and a law of iterated logarithm are proved leading to a so-called universal law of iterated logarithm. 相似文献
9.
B. S. Pitskel' 《Mathematical Notes》1971,9(1):54-60
Let (X,μ, T) be an ergodic dynamic system and let ξ = (C1, C2, ...) be a discrete decomposition of X. Conditions are considered for the existence almost everywhere of $$\mathop {\lim }\limits_{n \to \infty } \frac{1}{n}\left| {\log \mu (C_{\xi ^n } (x))} \right|,$$ whereC ξn(x) is the element of the decomposition ξn = ξ V T ξ V ... < Tn-1ξ containing x. It is proved that the condition H(ξ) < ∞ is close to being necessary. If T is a Markov automorphism and ξ is the decomposition into states, then the limit exists, even if H(ξ) = ∞, and is equal to the entropy of the chain. 相似文献
10.
Robert Stelzer 《Journal of multivariate analysis》2008,99(6):1177-1190
The tail behaviour of stationary Rd-valued Markov-switching ARMA (MS-ARMA) processes driven by a regularly varying noise is analysed. It is shown that under appropriate summability conditions the MS-ARMA process is again regularly varying as a sequence. Moreover, it is established that these summability conditions are satisfied if the sum of the norms of the autoregressive parameters is less than one for all possible values of the parameter chain, which leads to feasible sufficient conditions.Our results complement in particular those of Saporta [Tail of the stationary solution of the stochastic equation Yn+1=anYn+bn with Markovian coefficients, Stochastic Process. Appl. 115 (2005) 1954-1978.] where regularly varying tails of one-dimensional MS-AR(1) processes coming from consecutive large values of the parameter chain were studied. 相似文献
11.
O.K Zakusilo 《Journal of multivariate analysis》1984,14(3):269-284
Let z(t) ∈ Rn be a generalized Poisson process with parameter λ and let A: Rn → Rn be a linear operator. The conditions of existence and limiting properties as λ → ∞ or as λ → 0 of the stationary distribution of the process x(t) ∈ Rn which satisfies the equation dx(t) = Ax(t)dt + dz(t) are investigated. 相似文献
12.
Yu. I. Alimov 《Mathematical Notes》1970,8(2):558-563
An investigation of measurable almost-everywhere finite functions ξ(t), -∞ $$\varphi _T^\xi (\tau _{(n)} , \lambda _{(n)} ) = \frac{1}{{2T}}\int_{ - T}^T {\exp i} \sum\nolimits_{k - 1}^n {\lambda _k \xi (t - \tau _k )dt} $$ tends to an asymptotic characteristic function? ∞ ξ (τ (n), λ(n)) when T → ∞. Here n is any positive integer and T(n)=(τ1; τ2, ..., τn) is arbitrary. It is proved that the class of such functions ξ(t) is larger than the class of Besicovich almost-periodic functions. 相似文献
13.
A. A. Mogul’skii 《Siberian Mathematical Journal》2008,49(4):669-683
We obtain an integro-local limit theorem for the sum S(n) = ξ(1)+?+ξ(n) of independent identically distributed random variables with distribution whose right tail varies regularly; i.e., it has the form P(ξ≥t) = t ?β L(t) with β > 2 and some slowly varying function L(t). The theorem describes the asymptotic behavior on the whole positive half-axis of the probabilities P(S(n) ∈ [x, x + Δ)) as x → ∞ for a fixed Δ > 0; i.e., in the domain where the normal approximation applies, in the domain where S(n) is approximated by the distribution of its maximum term, as well as at the “junction” of these two domains. 相似文献
14.
15.
We consider a diffusion (ξ t ) t≥0 whose drift contains some deterministic periodic signal. Its shape being fixed and known, up to scaling in time, the periodicity of the signal is the unknown parameter ? of interest. We consider sequences of local models at ? corresponding to continuous observation of the process ξ on the time interval [0, n] as n → ∞, with suitable choice of local scale at ?. Our tools - under an ergodicity condition — are path segments of ξ corresponding to the period ?, and limit theorems for certain functionals of the process ξ, which are not additive functionals. When the signal is smooth, with local scale n ?3/2 at ?, we have local asymptotic normality (LAN) in the sense of Le Cam [21]. When the signal has a finite number of discontinuities, with local scale n ?2 at ?, we obtain a limit experiment of different type, studied by Ibragimov and Khasminskii [14], where smoothness of the parametrization (in the sense of Hellinger distance) is Hölder 1/2. 相似文献
16.
Pierrette Cassou-Noguès 《Journal of Number Theory》1982,14(1):32-64
In this paper, we are studying Dirichlet series Z(P,ξ,s) = Σn?1rP(n)?s ξn, where P ∈ + [X1,…,Xr] and ξn = ξ1n1 … ξrnr, with ξi ∈ , such that |ξi| = 1 and ξi ≠ 1, 1 ≦ i ≦ r. We show that Z(P, ξ,·) can be continued holomorphically to the whole complex plane, and that the values Z(P, ξ, ?k) for all non negative integers, belong to the field generated over by the ξi and the coefficients of P. If, there exists a number field K, containing the ξi, 1 ≦ i ≦ r, and the coefficients of P, then we study the denominators of Z(P, ξ, ?k) and we define a -adic function Z(P, ξ,·) which is equal, on class of negative integers, to Z(P, ξ, ?k). 相似文献
17.
O. Gulinsky 《Linear algebra and its applications》2011,435(7):1575-1584
A probability model ∫Rexp(ι[nP(x)])dΦn(x) with Φn(x) the distribution function of random variable (ξk is i.i.d. sequence of r.v.’s with zero expectation and unit variance), being in a framework of stationary phase method is analyzed. The asymptotic expansion in CLT and Hörmander’s theorem play crucial role in asymptotic analysis of the model. 相似文献
18.
For the Gaussian channel Y(t) = Φ(ξ(s), Y(s); s ≦ t) + X(t), the mutual information I(ξ, Y) between the message ξ(·) and the output Y(·) is evaluated, where X(·) is a Gaussian noise. Furthermore, the optimal coding under average power constraints is constructed. 相似文献
19.
If ξ∈ (0,1) and A=an, n?? is a sequence of real numbers define Sn(ξ,A)∶=Σ{ak∶:k=[nξ]+1 to n}, n??, where [x] is the greatest integer less than or equal to x. In the theory of regularly varying sequences the problem arose to conclude from the convergence of the sequence Sn (ξ,A), n??, for all ξ in an appropriate set K of real numbers, that the sequence an, n??, converges to zero. It was shown that such a conclusion is possible if K={ξ,1?ξ} with ξ∈ (0,1) irrational. Then the following three questions were posed and will be answered in this paper:
- does the convergence of Sn (ξ,A), n??, for a single irrational number ξ imply an→0.
- does the convergence of Sn(ξ,A), n??, for finitely many rational numbers ξ∈ (0, 1) imply an→0.
- does the convergence of Sn (ξ,A), n??, for all rational numbers ξ∈ (0,1) imply an→0?
20.
Consider the system, of linear equations Ax = b where A is an n × n real symmetric, positive definite matrix and b is a known vector. Suppose we are given an approximation to x, ξ, and we wish to determine upper and lower bounds for ∥ x − ξ ∥ where ∥ ··· ∥ indicates the euclidean norm. Given the sequence of vectors {ri}ik = 0, where ri = Ari − 1 and r0 = b − Aξ, it is shown how to construct a sequence of upper and lower bounds for ∥ x − ξ ∥ using the theory of moments. 相似文献