共查询到20条相似文献,搜索用时 328 毫秒
1.
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
2.
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
3.
Jack W Silverstein 《Journal of multivariate analysis》1984,15(3):295-324
Let {vij} i,j = 1, 2,…, be i.i.d. standardized random variables. For each n, let Vn = (vij) i = 1, 2,…, n; j = 1, 2,…, s = s(n), where as n → ∞, and let . Previous results [7, 8] have shown the eigenvectors of Mn to display behavior, for n large, similar to those of the corresponding Wishart matrix. A certain stochastic process Xn on [0, 1], constructed from the eigenvectors of Mn, is known to converge weakly, as n → ∞, on D[0, 1] to Brownian bridge when v11 is N(0, 1), but it is not known whether this property holds for any other distribution. The present paper provides evidence that this property may hold in the non-Wishart case in the form of limit theorems on the convergence in distribution of random variables constructed from integrating analytic function w.r.t. Xn(Fn(x)), where Fn is the empirical distribution function of the eigenvalues of Mn. The theorems assume certain conditions on the moments of v11 including E(v114) = 3, the latter being necessary for the theorems to hold. 相似文献
4.
D.J Hartfiel 《Journal of Mathematical Analysis and Applications》1985,108(1):230-240
Let Pij and qij be positive numbers for i ≠ j, i, j = 1, …, n, and consider the set of matrix differential equations x′(t) = A(t) x(t) over all A(t), where aij(t) is piecewise continuous, aij(t) = ?∑i ≠ jaij(t), and pij ? aij(t) ? qij all t. A solution x is also to satisfy ∑i = 1nxi(0) = 1. Let Ct denote the set of all solutions, evaluated at t to equations described above. It is shown that , the topological closure of Ct, is a compact convex set for each t. Further, the set valued function , of t is continuous and . 相似文献
5.
Thomas G. Kurtz 《Stochastic Processes and their Applications》1978,6(3):223-240
A variety of continuous parameter Markov chains arising in applied probability (e.g. epidemic and chemical reaction models) can be obtained as solutions of equations of the form where , the Y1 are independent Poisson processes, and N is a parameter with a natural interpretation (e.g. total population size or volume of a reacting solution).The corresponding deterministic model, satisfies Under very general conditions limN→∞XN(t)=X(t) a.s. The process XN(t) is compared to the diffusion processes given by and Under conditions satisfied by most of the applied probability models, it is shown that XN,ZN and V can be constructed on the same sample space in such a way that and 相似文献
6.
Charles Rennolet 《Journal of Mathematical Analysis and Applications》1979,70(1):42-60
Existence and boundedness theorems are given for solutions of nonlinear integrodifferential equations of type , (1.1) u(0) = u0, Here A and B are nonlinear, possibly multivalued, operators on a Banach space W and a Hilbert space H, where W ? H. The function f (0, ∞) → H and the kernel a(t, s): × → are known functions. The results of this paper extend the results of Crandall, Londen, and Nohel [4] for equation (1.1). They assumed the kernel to be of the type a(t, s) = a(t ? s). We relax this assumption and obtain similar results. Examples of kernels satisfying the conditions we require are given in section 4. 相似文献
7.
Eng-Bin Lim 《Journal of Differential Equations》1978,30(1):49-53
The author discusses the best approximate solution of the functional differential equation x′(t) = F(t, x(t), x(h(t))), 0 < t < l satisfying the initial condition x(0) = x0, where x(t) is an n-dimensional real vector. He shows that, under certain conditions, the above initial value problem has a unique solution y(t) and a unique best approximate solution of degree k (cf. [1]) for a given positive integer k. Furthermore, , where ¦ · ¦ is any norm in Rn. 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.
Let Ωm be the set of partitions, ω, of a finite m-element set; induce a uniform probability distribution on Ωm, and define Xms(ω) as the number of s-element subsets in ω. We alow the existence of an integer-valued function n=n(m)(t), t?[0, 1], and centering constants bms, 0?s? m, such that converges to the ‘Brownian Bridge’ process in terms of its finite-dimensional distributions. 相似文献
10.
Thomas G Kurtz 《Journal of Functional Analysis》1976,23(2):135-144
For each t ? 0, let A(t) generate a contraction semigroup on a Banach space L. Suppose the solution of ut = ?A(t)u is given by an evolution operator V?(t, s). Conditions are given under which converges strongly as ? → 0 to a semigroup T(t) generated by the closure of .This result is applied to the following situation: Let B generate a contraction group S(t) and the closure of ?A + B generate a contraction semigroup S?(t). Conditions are given under which converges strongly to a semigroup generated by the closure of . This work was motivated by and generalizes a result of Pinsky and Ellis for the linearized Boltzmann Equation. 相似文献
11.
Consider the renewal equation in the form (1) , where is a probability density on [0, ∞) and limt → ∞g(t) = g0. Asymptotic solutions of (1) are given in the case when f(t) has no expectation, i.e., . These results complement the classical theorem of Feller under the assumption that f(t) possesses finite expectation. 相似文献
12.
L.R. Haff 《Journal of multivariate analysis》1977,7(3):374-385
Let Sp×p ~ Wishart (Σ, k), Σ unknown, k > p + 1. Minimax estimators of Σ?1 are given for L1, an Empirical Bayes loss function; and L2, a standard loss function (Ri ≡ E(Li ∣ Σ), i = 1, 2). The estimators are , a, b ≥ 0, r(·) a functional on . Stein, Efron, and Morris studied the special cases and , for certain, a, b. From their work , a = k ? p ? 1, b = p2 + p ? 2; whereas, we prove . The reversal is surprising because a.e. (for a particular L2). Assume (compact) ? , the set of p × p p.s.d. matrices. A “divergence theorem” on functions Fp×p : → implies identities for Ri, i = 1, 2. Then, conditions are given for , i = 1, 2. Most of our results concern estimators with r(S) = t(U)/tr(S), U = p ∣S∣1/p/tr(S). 相似文献
13.
Hermann König 《Journal of Functional Analysis》1977,24(1):32-51
For an open set Ω ? N, 1 ? p ? ∞ and λ ∈ +, let denote the Sobolev-Slobodetzkij space obtained by completing in the usual Sobolev-Slobodetzkij norm (cf. A. Pietsch, “r-nukleare Sobol. Einbett. Oper., Ellipt. Dgln. II,” Akademie-Verlag, Berlin, 1971, pp. 203–215). Choose a Banach ideal of operators , 1 ? p, q ? ∞ and a quasibounded domain Ω ? N. Theorem 1 of the note gives sufficient conditions on λ such that the Sobolev-imbedding map exists and belongs to the given Banach ideal : Assume the quasibounded domain fulfills condition Ckl for some l > 0 and 1 ? k ? N. Roughly this means that the distance of any to the boundary ?Ω tends to zero as for , and that the boundary consists of sufficiently smooth ?(N ? k)-dimensional manifolds. Take, furthermore, 1 ? p, q ? ∞, p > k. Then, if μ, ν are real positive numbers with λ = μ + v ∈ , μ > λ S(; p,q:N) and v > N/l · λD(;p,q), one has that belongs to the Banach ideal . Here λD(;p,q;N)∈+ and λS(;p,q;N)∈+ are the D-limit order and S-limit order of the ideal , introduced by Pietsch in the above mentioned paper. These limit orders may be computed by estimating the ideal norms of the identity mappings lpn → lqn for n → ∞. Theorem 1 in this way generalizes results of R. A. Adams and C. Clark for the ideals of compact resp. Hilbert-Schmidt operators (p = q = 2) as well as results on imbeddings over bounded domains.Similar results over general unbounded domains are indicated for weighted Sobolev spaces.As an application, in Theorem 2 an estimate is given for the rate of growth of the eigenvalues of formally selfadjoint, uniformly strongly elliptic differential operators with Dirichlet boundary conditions in , where Ω fulfills condition C1l.For an open set Ω in N, let denote the Sobolev-Slobodetzkij space obtained by completing in the usual Sobolev-Slobodetzkij norm, see below. Taking a fixed Banach ideal of operators and 1 ? p, q ? ∞, we consider quasibounded domains Ω in N and give sufficient conditions on λ such that the Sobolev imbedding operator exists and belongs to the Banach ideal. This generalizes results of C. Clark and R. A. Adams for compact, respectively, Hilbert-Schmidt operators (p = q = 2) to general Banach ideals of operators, as well as results on imbeddings over bounded domains. Similar results over general unbounded domains may be proved for weighted Sobolev spaces. As an application, we give an estimate for the rate of growth of the eigenvalues of formally selfadjoint, uniformly strongly elliptic differential operators with Dirichlet boundary conditions in , where Ω is a quasibounded open set in N. 相似文献
14.
J.J.A.M Brands 《Journal of Mathematical Analysis and Applications》1978,63(1):54-64
This paper presents some comparison theorems on the oscillatory behavior of solutions of second-order functional differential equations. Here we state one of the main results in a simplified form: Let q, τ1, τ2 be nonnegative continuous functions on (0, ∞) such that τ1 ? τ2 is a bounded function on [1, ∞) and t ? τ1(t) → ∞ if t → ∞. Then is oscillatory if and only if is oscillatory. 相似文献
15.
Amitai Regev 《Advances in Mathematics》1982,46(2):230-240
An asymptotic formula, involving integrals, is given for certain combinatorial sums. By evaluating a multi-integral it is then found that as n → ∞, the codimensions cn(F2) and the trace codimensions tn(F2) of F2, the 2 × 2 matrices, are asymptotically equal: . 相似文献
16.
A process which has just one jump, and whose time parameter is the positive quadrant [0, ∞] × [0, ∞], is considered. Following Merzbach, related stopping lines are introduced, and the filtration {t1,t23} considered in this paper is such that, modulo completion, the σ-field t1,t23 is the Borel field on the region , together with the atom which is the complement in Ω = [0, ∞]2 of Lt1,t2. Optional and predictable projections of related processes are defined, together with their dual projections, and an integral representation for martingales is obtained. 相似文献
17.
C.G Khatri 《Journal of multivariate analysis》1978,8(3):453-467
Optimization problems are connected with maximization of three functions, namely, geometric mean, arithmetic mean and harmonic mean of the eigenvalues of (X′ΣX)?1ΣY(Y′ΣY)?1Y′ΣX, where Σ is positive definite, X and Y are p × r and p × s matrices of ranks r and s (≥r), respectively, and X′Y = 0. Some interpretations of these functions are given. It is shown that the maximum values of these functions are obtained at the same point given by X = (h1 + ?1hp, …, hr + ?rhp?r+1) and , where h1, …, hp are the eigenvectors of Σ corresponding to the eigenvalues λ1 ≥ λ2 ≥ … ≥ λp > 0, ?j = +1 or ?1 for j = 1,2,…, r and , are linear functions of hr+1,…, hp?r. These results are extended to intermediate stationary values. They are utilized in obtaining the inequalities for canonical correlations θ1,…,θr and they are given by expressions (3.8)–(3.10). Further, some new union-intersection test procedures for testing the sphericity hypothesis are given through test statistics (3.11)–(3.13). 相似文献
18.
David Terman 《Journal of Differential Equations》1983,47(3):406-443
We consider the pure initial value problem for the system of equations , the initial data being (ν(x, 0), w(x, 0)) = (?(x), 0). Here , where H is the Heaviside step function and . This system is of the FitzHugh-Nagumo type and has several applications including nerve conduction and distributed chemical/ biochemical systems. It is demonstrated that this system exhibits a threshold phenomenon. This is done by considering the curve s(t) defined by s(t) = sup{x: v(x, t) = a}. The initial datum, ?(x), is said to be superthreshold if limt→∞ s(t) = ∞. It is proven that the initial datum is superthreshold if ?(x) > a on a sufficiently long interval, ?(x) is sufficiently smooth, and ?(x) decays sufficiently fast to zero as . 相似文献
19.
Derek W Robinson 《Journal of Functional Analysis》1977,24(3):280-290
Let U, V be two strongly continuous one-parameter groups of bounded operators on a Banach space with corresponding infinitesimal generators S, T. We prove the following: ∥Ut, ? Vt ∥ = O(t), t → 0, if and only if U = V; ∥Ut ? Vt∥ = O(tα), t → 0; with 0 ? α ? 1, if and only if , where Ω, P, are bounded operators on such that if and only if has a bounded extension to 1. Further results of this nature are inferred for semigroups, reflexive spaces, Hilbert spaces, and von Neumann algebras. 相似文献
20.
Ming-Po Chen Cheh-Chih Yeh Cheng-Shu Yu 《Journal of Mathematical Analysis and Applications》1977,59(2):211-215
For nonlinear retarded differential equations and the sufficient conditions are given on fi, pi, Fi, and h under which every bounded nonoscillatory solution of () or () tends to zero as t → ∞. 相似文献