共查询到20条相似文献,搜索用时 46 毫秒
1.
E. Bolthausen 《Stochastic Processes and their Applications》1979,9(2):217-222
Let Xn be an irreducible aperiodic recurrent Markov chain with countable state space I and with the mean recurrence times having second moments. There is proved a global central limit theorem for the properly normalized sojourn times. More precisely, if , then the probability measures induced by {t(n)i/√n?√nπi}i?I(πi being the ergotic distribution) on the Hilbert-space of square summable I-sequences converge weakly in this space to a Gaussian measure determined by a certain weak potential operator. 相似文献
2.
P. Révész 《Stochastic Processes and their Applications》1983,15(2):169-179
Let U1, U2,… be a sequence of independent, uniform (0, 1) r.v.'s and let R1, R2,… be the lengths of increasing runs of {Ui}, i.e., X1=R1=inf{i:Ui+1<Ui},…, Xn=R1+R2+?+Rn=inf{i:i>Xn?1,Ui+1<Ui}. The first theorem states that the sequence can be approximated by a Wiener process in strong sense.Let τ(n) be the largest integer for which R1+R2+?+Rτ(n)?n, and . Here Mn is the length of the longest increasing block. A strong theorem is given to characterize the limit behaviour of Mn.The limit distribution of the lengths of increasing runs is our third problem. 相似文献
3.
F. Götze 《Journal of multivariate analysis》1985,16(1):1-20
Asymptotic expansions for a class of functional limit theorems are investigated. It is shown that the expansions in this class fit into a common scheme, defined by a sequence of functions hn (ε1,…, εn), n ≥ 1, of “weights” (for n observations), which are smooth, symmetric, compatible and have vanishing first derivatives at zero. Then admits an asymptotic expansion in powers of . Applications to quadratic von Mises functionals, the C.L.T. in Banach spaces, and the invariance principle are discussed. 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.
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
6.
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
7.
Let (μt)∞t=0 be a k-variate (k?1) normal random walk process with successive increments being independently distributed as normal N(δ, R), and μ0 being distributed as normal N(0, V0). Let Xt have normal distribution N(μt, Σ) when μt is given, t = 1, 2,….Then the conditional distribution of μt given X1, X2,…, Xt is shown to be normal N(Ut, Vt) where Ut's and Vt's satisfy some recursive relations. It is found that there exists a positive definite matrix V and a constant θ, 0 < θ < 1, such that, for all t?1, where the norm |·| means that |A| is the largest eigenvalue of a positive definite matrix A. Thus, Vt approaches to V as t approaches to infinity. Under the quadratic loss, the Bayesian estimate of μt is Ut and the process {Ut}∞t=0, U0=0, is proved to have independent successive increments with normal N(θ, Vt?Vt+1+R) distribution. In particular, when V0 =V then Vt = V for all t and {Ut}∞t=0 is the same as {μt}∞t=0 except that U0 = 0 and μ0 is random. 相似文献
8.
David L Russell 《Journal of Mathematical Analysis and Applications》1982,87(2):528-550
We suppose that K is a countable index set and that is a sequence of distinct complex numbers such that forms a Riesz (strong) basis for L2[a, b], a < b. Let Σ = {σ1, σ2,…, σm} consist of m complex numbers not in Λ. Then, with p(λ) = Πk = 1m (λ ? σk), forms a Riesz (strong) bas Sobolev space Hm[a, b]. If we take σ1, σ2,…, σm to be complex numbers already in Λ, then, defining p(λ) as before, forms a Riesz (strong) basis for the space H?m[a, b]. We also discuss the extension of these results to “generalized exponentials” tneλkt. 相似文献
9.
Let , 0<aT?T<∞, and {W(t);0?t<∞} be a standard Wiener process. This exposition studies the almost sure behaviour of , under varying conditions on aT and T/aT. The following analogue of Lévy's modulus of continuity of a Wiener Process is also given: and this may be viewed as the exact “modulus of non-differentiability” of a Wiener Process. 相似文献
10.
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
11.
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
12.
Let Fn(x) be the empirical distribution function based on n independent random variables X1,…,Xn from a common distribution function F(x), and let be the sample mean. We derive the rate of convergence of to normality (for the regular as well as nonregular cases), a law of iterated logarithm, and an invariance principle for . 相似文献
13.
Christer Borell 《Journal of Mathematical Analysis and Applications》1973,41(2):300-312
We study certain functionals and obtain an inverse Hölder inequality for n functions f1a1,…,fnan (fk concave, 1 dimension).We also prove a multidimensional inverse Hölder inequality for n functions f1,…,fn, where .Finally we give an inverse Minkowski inequality for concave functions. 相似文献
14.
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
15.
Robert Donaghey 《Journal of Combinatorial Theory, Series A》1980,28(1):111-114
For a formal power series with nonnegative integer coefficients, the compositional inverse of is shown to be the generating function for the colored planted plane trees in which each vertex of degree i + 1 is colored one of hi colors. Since the compositional inverse of the Euler transformation of f(t) is the star transformation [[g(t)]?1 ? 1]?1 of g(t), [2], it follows that the Euler transformation of f(t) is the generating function for the colored planted plane trees in which each internal vertex of degree i + 1 is colored one of hi colors for i > 1, and h1 ? 1 colors for i = 1. 相似文献
16.
Let Z = {Z0, Z1, Z2,…} be a martingale, with difference sequence X0 = Z0, Xi = Zi ? Zi ? 1, i ≥ 1. The principal purpose of this paper is to prove that the best constant in the inequality λP(supi |Xi| ≥ λ) ≤ C supiE |Zi|, for λ > 0, is C = (log 2)?1. If Z is finite of length n, it is proved that the best constant is . The analogous best constant Cn(z) when Z0 ≡ z is also determined. For these finite cases, examples of martingales attaining equality are constructed. The results follow from an explicit determination of the quantity Gn(z, E) = supzP(maxi=1,…,n |Xi| ≥ 1), the supremum being taken over all martingales Z with Z0 ≡ z and E|Zn| = E. The expression for Gn(z,E) is derived by induction, using methods from the theory of moments. 相似文献
17.
Ludwig Arnold 《Linear algebra and its applications》1976,13(3):185-199
It is proved that Wigner's semicircle law for the distribution of eigenvalues of random matrices, which is important in the statistical theory of energy levels of heavy nuclei, possesses the following completely deterministic version. Let An=(aij), 1?i, ?n, be the nth section of an infinite Hermitian matrix, {λ(n)}1?k?n its eigenvalues, and {uk(n)}1?k?n the corresponding (orthonormalized column) eigenvectors. Let , put (bookeeping function for the length of the projections of the new row v1n of An onto the eigenvectors of the preceding matrix An?1), and let finally (empirical distribution function of the eigenvalues of . Suppose (i) , (ii) limnXn(t)=Ct(0<C<∞,0?t?1). Then ,where W is absolutely continuous with (semicircle) density 相似文献
18.
R.N. Buttsworth 《Journal of Number Theory》1980,12(4):487-498
The polynomial functions f1, f2,…, fm are found to have highest common factor h for a set of values of the variables x1, x2,…,xm whose asymptotic density is For the special case f1(x) = f2(x) = … = fm(x) = x and h = 1 the above formula reduces to , the density if m-tuples with highest common factor 1. Necessary and sufficient conditions on the polynomials f1, f2,…, fm for the asymptotic density to be zero are found. In particular it is shown that either the polynomials may never have highest common factor h or else h is the highest common factor infinitely often and in fact with positive density. 相似文献
19.
Allan R Sampson 《Journal of multivariate analysis》1983,13(2):375-381
Let X1, …, Xp have p.d.f. g(x12 + … + xp2). It is shown that (a) X1, …, Xp are positively lower orthant dependent or positively upper orthant dependent if, and only if, X1,…, Xp are i.i.d. N(0, σ2); and (b) the p.d.f. of |X1|,…, |Xp| is TP2 in pairs if, and only if, In g(u) is convex. Let X1, X2 have p.d.f. . Necessary and sufficient conditions are given for f(x1, x2) to be TP2 for fixed correlation ?. It is shown that if f is TP2 for all ? >0. then (X1, X2)′ ~ N(0, Σ). Related positive dependence results and applications are also considered. 相似文献
20.
Let fk(n) denote the maximum of k-subsets of an n-set satisfying the condition in the title. It is proven that f2t ? 1(n) ? f2t(n + 1) ? with equalities holding iff there exists a Steiner-system S(t, 2t ? 1, n). The bounds are approximately best possile for k ? 6 and of correct order of magnitude for k >/ 7, as well, even if the corresponding Steiner-systems do not exist.Exponential lower and upper bounds are obtained for the case if we do not put size restrictions on the members of the family (i.e., the nonuniform case). 相似文献