共查询到20条相似文献,搜索用时 15 毫秒
1.
S. Ihara 《Journal of multivariate analysis》1974,4(1):74-87
The message m = {m(t)} is a Gaussian process that is to be transmitted through the white Gaussian channel with feedback: . Under the average power constraint, , we construct causally the optimal coding, in the sense that the mutual information It(m, Y) between the message m and the channel output Y (up to t) is maximized. The optimal coding is presented by , where and A(s) is a positive function such that . 相似文献
2.
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
3.
Let K1 and K2 be number fields and . Suppose and are of prime degree p but are not necessarily normal. Let N1 and N2 be the normal closures of K1 and K2 over F, respectively, L = K1K2, N = N1N2, and be a prime divisor of N which divides p and is totally ramified in and . Let be the ramification index of in , be the total ramification number of in , and . Then (K1, K2) is exactly divisible by M, where . 相似文献
4.
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
5.
Let (Wt) = (W1t,W2t,…,Wdt), d ? 2, be a d-dimensional standard Brownian motion and let A(t) be a bounded measurable function from + into the space of d × d skew-symmetric matrices and x(t) such a function into d. A class of stochastic processes (LtA,x), a particular example of which is Levy's “stochastic area” , is dealt with.The joint characteristic function of Wt and L1A,x is calculated and based on this result a formula for fundamental solutions for the hypoelliptic operators which generate the diffusions (Wt,LtA,x) is given. 相似文献
6.
Let {} denote the N-parameter Wiener process on . For multiple sequences of certain independent random variables the authors find lower bounds for the distributions of maximum of partial sums of these random variables, and as a consequence a useful upper bound for the yet unknown function , c ≥ 0, is obtained where DN = Πk = 1N [0, Tk]. The latter bound is used to give three different varieties of N-parameter generalization of the classical law of iterated logarithm for the standard Brownian motion process. 相似文献
7.
Milton Rosenberg 《Journal of multivariate analysis》1978,8(2):295-316
Let p, q be arbitrary parameter sets, and let be a Hilbert space. We say that x = (xi)i?q, xi ? , is a bounded operator-forming vector (?Fq) if the Gram matrix 〈x, x〉 = [(xi, xj)]i?q,j?q is the matrix of a bounded (necessarily ≥ 0) operator on , the Hilbert space of square-summable complex-valued functions on q. Let A be p × q, i.e., let A be a linear operator from to . Then exists a linear operator ǎ from (the Banach space) Fq to Fp on (A) = {x:x ? Fq, is p × q bounded on } such that y = ǎx satisfies yj?σ(x) = {space spanned by the xi}, 〈y, x〉 = A〈x, x〉 and . This is a generalization of our earlier [J. Multivariate Anal.4 (1974), 166–209; 6 (1976), 538–571] results for the case of a spectral measure concentrated on one point. We apply these tools to investigate q-variate wide-sense Markov processes. 相似文献
8.
Arthur Lubin 《Journal of Functional Analysis》1974,17(4):388-394
Let m and vt, 0 ? t ? 2π be measures on T = [0, 2π] with m smooth. Consider the direct integral = ⊕L2(vt) dm(t) and the operator on , where e(s, t) = exp ∫st ∫Tdvλ(θ) dm(λ). Let μt be the measure defined by for all continuous ?, and let ?t(z) = exp[?∫ (eiθ + z)(eiθ ? z)?1dμt(gq)]. Call {vt} regular iff for all for 1 a.e. 相似文献
9.
D.C Doss 《Journal of multivariate analysis》1979,9(3):460-464
The probability generating function (pgf) of an n-variate negative binomial distribution is defined to be [β(s1,…,sn)]?k where β is a polynomial of degree n being linear in each si and k > 0. This definition gives rise to two characterizations of negative binomial distributions. An n-variate linear exponential distribution with the probability function h(x1,…,xn) is negative binomial if and only if its univariate marginals are negative binomial. Let St, t = 1,…, m, be subsets of {s1,…, sn} with empty ∩t=1mSt. Then an n-variate pgf is of a negative binomial if and only if for all s in St being fixed the function is of the form of the pgf of a negative binomial in other s's and this is true for all t. 相似文献
10.
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
11.
Loren D. Pitt 《Journal of multivariate analysis》1978,8(1):45-54
For Gaussian vector fields {X(t) ∈ Rn:t ∈ Rd} we describe the covariance functions of all scaling limits Y(t) = limα↓0 B?1(α) X(αt) which can occur when B(α) is a d × d matrix function with B(α) → 0. These matrix covariance functions are found to be homogeneous in the sense that for some matrix L and each α > 0, . Processes with stationary increments satisfying (1) are further analysed and are found to be natural generalizations of Lévy's multiparameter Brownian motion. 相似文献
12.
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
13.
Klaus D. Schmidt 《Journal of multivariate analysis》1980,10(1):123-134
For certain types of stochastic processes {Xn | n ∈ }, which are integrable and adapted to a nondecreasing sequence of σ-algebras n on a probability space (Ω, , P), several authors have studied the following problems: IfSdenotes the class of all stopping times for the stochastic basis {n | n ∈ }, when isfinite, and when is there a stopping time for which this supremum is attained? In the present paper we set the problem in a measure theoretic framework. This approach turns out to be fruitful since it reveals the root of the problem: It avoids the use of such notions as probability, null set, integral, and even σ-additivity. It thus allows a considerable generalization of known results, simplifies proofs, and opens the door to further research. 相似文献
14.
Yashaswini Mittal 《Stochastic Processes and their Applications》1979,9(1):67-84
Let {Xi, i?0} be a sequence of independent identically distributed random variables with finite absolute third moment. Then Darling and Erdös have shown that for -∞<t<∞ where and . The result is extended to dependent sequences but assuming that {Xi} is a standard stationary Gaussian sequence with covariance function {ri}. When {Xi} is moderately dependent (e.g. when we get where Ha is a constant. In the strongly dependent case (e.g. when we get for-∞<t<∞. 相似文献
15.
Let Ω = {1, 0} and for each integer n ≥ 1 let (n-tuple) and for all k = 0,1,…,n. Let {Ym}m≥1 be a sequence of i.i.d. random variables such that . For each A in , let TA be the first occurrence time of A with respect to the stochastic process {Ym}m≥1. R. Chen and A.Zame (1979, J. Multivariate Anal. 9, 150–157) prove that if n ≥ 3, then for each element A in , there is an element B in such that the probability that TB is less than TA is greater than . This result is sharpened as follows: (I) for n ≥ 4 and 1 ≤ k ≤ n ? 1, each element A in , there is an element B also in such that the probability that TB is less than TA is greater than ; (II) for n ≥ 4 and 1 ≤ k ≤ n ? 1, each element A = (a1, a2,…,an) in , there is an element C also in such that the probability that TA is less than TC is greater than if n ≠ 2m or n = 2m but ai = ai + 1 for some 1 ≤ i ≤ n?1. These new results provide us with a better and deeper understanding of the fair coin tossing process. 相似文献
16.
James D Child 《Journal of Mathematical Analysis and Applications》1980,78(1):133-143
Let Ω denote a simply connected domain in the complex plane and let be the collection of all entire functions of exponential type whose Laplace transforms are analytic on Ω′, the complement of Ω with respect to the sphere. Define a sequence of functionals by , where F denotes the Laplace transform of f, Γ ? Ω is a simple closed contour chosen so that F is analytic outside and on Ω, and gn is analytic on Ω. The specific functionals considered by this paper are patterned after the Lidstone functions, L2n(f) = f(2n)(0) and L2n + 1(f) = f(2n)(1), in that their sequence of generating functions {gn} are “periodic.” Set gpn + k(ζ) = hk(ζ) ζpn, where p is a positive integer and each hk (k = 0, 1,…, p ? 1) is analytic on Ω. We find necessary and sufficient conditions for . DeMar previously was able to find necessary conditions [7]. Next, we generalize {Ln} in several ways and find corresponding necessary and sufficient conditions. 相似文献
17.
Douglas N. Clark 《Journal of Functional Analysis》1973,14(3):269-280
The operator acting on H=∝02π⊕L2(vt), where m and vt, 0 ? t ? 2π are measures on [0, 2π] with m smooth and e(s, t) = exp[?∝ts∝Tdvλ(θ) dm(λ)], satisfies . It is, therefore, unitarily equivalent to a scalar Sz.-Nagy-Foia? canonical model. The purpose of this paper is to determine the model explicitly and to give a formula for the unitary equivalence. 相似文献
18.
Alan F. Karr 《Stochastic Processes and their Applications》1985,20(1):115-131
Let Ni, i?1, be i.i.d. observable Cox processes on a compact metric space E, directed by unobservable random measures Mi. Assume that the probability law of the Mi is completely unknown. Techniques are developed for approximation of state estimators using data from the processes N1,…,Nn to estimate necessary attributes of the unknown probability law of the time Mi. The techniques are based on representation of the state estimators in terms of reduced Palm distributions of the Ni and on estimation of these Palm distributions. Estimators of Palm distributions are shown to be strongly consistent and asymptotically normal. The difference between the true and the pseudo-state estimators converges to zero in L2 at rate n? for each δ > 0. 相似文献
19.
Georg Neuhaus 《Journal of multivariate analysis》1980,10(1):19-25
Let X be a Gaussian rv with values in a separable Hilbert space H having a covariance operator R of the form , where L0, A are linear operators on H. A method is given for computing in terms of and A the distribution of |X|2, |·| being the norm in H. The result is applied to the evaluation of the asymptotic distribution of Cramér-von Mises statistics when parameters are present. L0 corresponds to the case where the true underlying parameter is known and A represents the effect of estimating the unknown parameter. 相似文献
20.
Kamal C Chanda 《Statistics & probability letters》1985,3(5):261-268
Let {Xt; t = 1, 2,…} be a linear process with a location parameter θ defined by Xt ? θ = Σ0∞grZt?r where {Zt; t = 0, ±1,…} is a sequence of independent and identically distributed random variables, with E∥Z1∥δ < ∞ for some δ > 0. If δ ? 1 we assume further than E(Z1) = 0. Let η = δ if 0 < δ < 2, and η = 2 if δ ? 2. Then assume that Σ0∞∥ gr ∥η < ∞. Consider the class of estimators given by is of the form cnt = Σp = 0sβnptp for some s ? 0. An attempt has been made to investigate the distributional properties of in large samples for various choices of βnp (0 ? p ? s), s, and the distribution of Z1 under the constraints Σ0∞rkgr = 0, 0 ? k ? q where q in an arbitrary integer, 0 ? q ? s. 相似文献