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1.
Consider a standard row-column-exchangeable array X = (Xij : i,j ≥ 1), i.e., Xij = f(a, ξi, ηj, λij) is a function of i.i.d. random variables. It is shown that there is a canonical version of X, X′, such that X′, and α′, ξ1, ξ2,…, η1, η2,…, are conditionally independent given ∩n ≥ 1σ(Xij : max(i,j) ≥ n). This result is quite a bit simpler to prove than the analogous result for the original array X, which is due to Aldous.  相似文献   

2.
Let Xj = (X1j ,…, Xpj), j = 1,…, n be n independent random vectors. For x = (x1 ,…, xp) in Rp and for α in [0, 1], let Fj1(x) = αI(X1j < x1 ,…, Xpj < xp) + (1 ? α) I(X1jx1 ,…, Xpjxp), where I(A) is the indicator random variable of the event A. Let Fj(x) = E(Fj1(x)) and Dn = supx, α max1 ≤ Nn0n(Fj1(x) ? Fj(x))|. It is shown that P[DnL] < 4pL exp{?2(L2n?1 ? 1)} for each positive integer n and for all L2n; and, as n → ∞, Dn = 0((nlogn)12) with probability one.  相似文献   

3.
Let X1, X2, X3, … be i.i.d. r.v. with E|X1| < ∞, E X1 = μ. Given a realization X = (X1,X2,…) and integers n and m, construct Yn,i, i = 1, 2, …, m as i.i.d. r.v. with conditional distribution P1(Yn,i = Xj) = 1n for 1 ? j ? n. (P1 denotes conditional distribution given X). Conditions relating the growth rate of m with n and the moments of X1 are given to ensure the almost sure convergence of (1mmi=1 Yn,i toμ. This equation is of some relevance in the theory of Bootstrap as developed by Efron (1979) and Bickel and Freedman (1981).  相似文献   

4.
Motivated by problems occurring in the empirical identification and modelling of a n-dimensional ARMA time series X(t) we study the possibility of obtaining a factorization (I + a1B + … + apBp) X(t) = [Πi=1p (I ? αiB)] X(t), where B is the backward shift operator. Using a result in [3] we conclude that as in the univariate case such a factorization always exists, but unlike the univariate case in general the factorization is not unique for given a1, a2,…, ap. In fact the number of possibilities is limited upwards by (np)!(n!)p, there being cases, however, where this maximum is not reached. Implications for the existence and possible use of transformations which removes nonstationarity (or almost nonstationarity) of X(t) are mentioned.  相似文献   

5.
A complete comparison is made between the value V(X1,…, Xn) = sup{EXt: t is a stop rule for X1,…,Xn} and E(maxjnXj) for all uniformly bounded sequences of i.i.d. random variables X1, …, Xn. Specifically, the set of ordered pairs {(x,y): x = V(X1, …, Xn) and y = E(maxjnXj) for some i.i.d.r.v.'s X1,…, Xn taking values in [0, 1]} is precisely the set {(x, y): xyΓn(x); 0 ≤x≤1}, where the upper boundary function Γn is given in terms of recursively defined functions. The result yields families of inequalities for the “prophet” problem, relating the motal's value of a game V(X1, …, Xn) to the prophet's value of the game E(maxjnXj). The proofs utilize conjugate duality theory, probabilistic convexity arguments, and functional equation analysis. Asymptotic analysis of the “prophet” regions and inequalities is also given.  相似文献   

6.
It is known that the sums of the components of two random vectors (X 1,X 2,…,X n ) and (Y 1,Y 2,…,Y n ) ordered in the multivariate (s 1,s 2,…,s n )-increasing convex order are ordered in the univariate (s 1+s 2+?+s n )-increasing convex order. More generally, real-valued functions of (X 1,X 2,…,X n ) and (Y 1,Y 2,…,Y n ) are ordered in the same sense as long as these functions possess some specified non-negative cross-derivatives. This note extends these results to multivariate functions. In particular, we consider vectors of partial sums (S 1,S 2,…,S n ) and (T 1,T 2,…,T n ) where S j =X 1+?+X j and T j =Y 1+?+Y j and we show that these random vectors are ordered in the multivariate (s 1,s 1+s 2,…,s 1+?+s n )-increasing convex order. The consequences of these general results for the upper orthant order and the orthant convex order are discussed.  相似文献   

7.
Let {Xi, Yi}i=1,2,... be an i.i.d. sequence of bivariate random vectors with P(Y1 = y) = 0 for all y. Put Mn(j) = max0≤k≤n-j (Xk+1 + ... Xk+j)Ik,j, where Ik,k+j = I{Yk+1 < ⋯ < Yk+j} denotes the indicator function for the event in brackets, 1 ≤ j ≤ n. Let Ln be the largest index l ≤ n for which Ik,k+l = 1 for some k = 0, 1, ..., n - l. The strong law of large numbers for “the maximal gain over the longest increasing runs,” i.e., for Mn(Ln) has been recently derived for the case where X1 has a finite moment of order 3 + ε, ε > 0. Assuming that X1 has a finite mean, we prove for any a = 0, 1, ..., that the s.l.l.n. for M(Ln - a) is equivalent to EX 1 3+a I{X1 > 0} < ∞. We derive also some new results for the a.s. asymptotics of Ln. Bibliography: 5 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 311, 2004, pp. 179–189.  相似文献   

8.
For 1 ≦ lj, let al = ?h=1q(l){alh + Mv: v = 0, 1, 2,…}, where j, M, q(l) and the alh are positive integers such that j > 1, al1 < … < alq(2)M, and let al = al ∪ {0}. Let p(n : B) be the number of partitions of n = (n1,…,nj) where, for 1 ≦ lj, the lth component of each part belongs to Bl and let p1(n : B) be the number of partitions of n into different parts where again the lth component of each part belongs to Bl. Asymptotic formulas are obtained for p(n : a), p1(n : a) where all but one nl tend to infinity much more rapidly than that nl, and asymptotic formulas are also obtained for p(n : a′), p1(n ; a′), where one nl tends to infinity much more rapidly than every other nl. These formulas contrast with those of a recent paper (Robertson and Spencer, Trans. Amer. Math. Soc., to appear) in which all the nl tend to infinity at approximately the same rate.  相似文献   

9.
Suppose X and Y are n × 1 random vectors such that lX + f(l) and lY have the same marginal distribution for all n × 1 real vectors l and some real valued function f(l), and the existence of expectations of X and Y is not necessary. Under these conditions it is proven that there exists a vector M such that f(l) = lM and X + M and Y have the same joint distribution. This result is extended to Banach-space valued random vectors.  相似文献   

10.
On the basis of a random sample of size n on an m-dimensional random vector X, this note proposes a class of estimators fn(p) of f(p), where f is a density of X w.r.t. a σ-finite measure dominated by the Lebesgue measure on Rm, p = (p1,…,pm), pj ≥ 0, fixed integers, and for x = (x1,…,xm) in Rm, f(p)(x) = ?p1+…+pm f(x)/(?p1x1 … ?pmxm). Asymptotic unbiasedness as well as both almost sure and mean square consistencies of fn(p) are examined. Further, a necessary and sufficient condition for uniform asymptotic unbisedness or for uniform mean square consistency of fn(p) is given. Finally, applications of estimators of this note to certain statistical problems are pointed out.  相似文献   

11.
Let S = (1/n) Σt=1n X(t) X(t)′, where X(1), …, X(n) are p × 1 random vectors with mean zero. When X(t) (t = 1, …, n) are independently and identically distributed (i.i.d.) as multivariate normal with mean vector 0 and covariance matrix Σ, many authors have investigated the asymptotic expansions for the distributions of various functions of the eigenvalues of S. In this paper, we will extend the above results to the case when {X(t)} is a Gaussian stationary process. Also we shall derive the asymptotic expansions for certain functions of the sample canonical correlations in multivariate time series. Applications of some of the results in signal processing are also discussed.  相似文献   

12.
This note deals with the numerical solution of the matrix differential system Y′ = [B(t,Y), Y], Y(0) = Y0, t ⩾ 0, where Y0 is a real constant symmetric matrix, B maps symmetric into skew-symmetric matrices, and [B(t,Y),Y] is the Lie bracket commutator of B(t,Y) and Y, i.e. [B(t,Y),Y] = B(t,Y)YYB(t,Y). The unique solution of (1) is isospectral, that is the matrix Y(t) preserves the eigenvalues of Y0 and is symmetric for all t (see [1, 5]). Isospectral methods exploit the Flaschka formulation of (1) in which Y(t) is written as Y(t) = U(t)Y0UT(t), for t ⩾ 0, where U(t) is the orthogonal solution of the differential system U′ = B(t, UY0UT)U, U(0) = I, t ⩾ 0, (see [5]). Here a numerical procedure based on the Cayley transform is proposed and compared with known isospectral methods.  相似文献   

13.
Let G be a finitely presented group given by its pre-abelian presentation <X1,…,Xm; Xe11ζ1,…,Xemmζ,ζm+1,…>, where ei≥0 for i = 1,…, m and ζj?G′ for j≥1. Let N be the subgroup of G generated by the normal subgroups [xeii, G] for i = 1,…, m. Then Dn+2(G)≡γn+2(G) (modNG′) for all n≥0, where G” is the second commutator subgroup of Gn+2(G) is the (n+2)th term of the lower central series of G and Dn+2(G) = G∩(1+△n+2(G)) is the (n+2)th dimension subgroup of G.  相似文献   

14.
Let FX,Y(x,y) be a bivariate distribution function and Pn(x), Qm(y), n, m = 0, 1, 2,…, the orthonormal polynomials of the two marginal distributions FX(x) and FY(y), respectively. Some necessary conditions are derived for the co-efficients cn, n = 0, 1, 2,…, if the conditional expectation E[Pn(X) ∥ Y] = cnQn(Y) holds for n = 0, 1, 2,…. Several examples are given to show the application of these necessary conditions.  相似文献   

15.
We consider the problem of the identification of the time-varying matrix A(t) of a linear m-dimensional differential system y′ = A(t)y. We develop an approximation An,k = ∑nj ? 1cj{Y(tk + τj) Y?1(tk) ? I} to A(tk) for grid points tk = a + kh, k = 0,…, N using specified τj = θjh, 0 < θj < 1, j = 1, …, n, and show that for each tk, the L1 norm of the error matrix is O(hn). We demonstrate an efficient scheme for the evaluation of An,k and treat sample problems.  相似文献   

16.
Garsia-Haiman modules C[Xn,Yn]/Iγ are quotient rings in the variables Xn={x1,x2,…,xn} and Yn={y1,y2,…,yn} that generalize the quotient ring C[Xn]/I, where I is the ideal generated by the elementary symmetric polynomials ej(Xn) for 1?j?n. A bitableau basis for the Garsia-Haiman modules of hollow type is constructed. Applications of this basis to representation theory and other related polynomial spaces are considered.  相似文献   

17.
LetX 1,X 2, ...,X n be independent and identically distributed random vectors inR d , and letY=(Y 1,Y 2, ...,Y n )′ be a random coefficient vector inR n , independent ofX j /′ . We characterize the multivariate stable distributions by considering the independence of the random linear statistic $$U = Y_1 X_1 + Y_2 X_2 + \cdot \cdot \cdot + Y_n X_n $$ and the random coefficient vectorY.  相似文献   

18.
Given a polynomial P(X1,…,XN)∈R[X], we calculate a subspace Gp of the linear space 〈X〉 generated by the indeterminates which is minimal with respect to the property P∈R[Gp] (the algebra generated by Gp, and prove its uniqueness. Furthermore, we use this result to characterize the pairs (P,Q) of polynomials P(X1,…,Xn) and Q(X1,…,Xn) for which there exists an isomorphism T:X〉 →〈X〉 that “separates P from Q,” i.e., such that for some k(1<k<n) we can write P and Q as P1(Y1,…,Yk) and Q1(Yk+1,…,Yn) respectively, where Y=TX.  相似文献   

19.
《随机分析与应用》2013,31(3):491-509
Abstract

Let X 1, X 2… and B 1, B 2… be mutually independent [0, 1]-valued random variables, with EB j  = β > 0 for all j. Let Y j  = B 1 … sB j?1 X j for j ≥ 1. A complete comparison is made between the optimal stopping value V(Y 1,…,Y n ):=sup{EY τ:τ is a stopping rule for Y 1,…,Y n } and E(max 1≤jn Y j ). It is shown that the set of ordered pairs {(x, y):x = V(Y 1,…,Y n ), y = E(max 1≤jn Y j ) for some sequence Y 1,…,Y n obtained as described} is precisely the set {(x, y):0 ≤ x ≤ 1, x ≤ y ≤ Ψ n, β(x)}, where Ψ n, β(x) = [(1 ? β)n + 2β]x ? β?(n?2) x 2 if x ≤ β n?1, and Ψ n, β(x) = min j≥1{(1 ? β)jx + β j } otherwise. Sharp difference and ratio prophet inequalities are derived from this result, and an analogous comparison for infinite sequences is obtained.  相似文献   

20.
A recursive kernel estimate i = 1n YiK⧸(x − Xi)hi)⧸∑j = 1n K((x − Xj)⧸hj) of a regression m(x) = E{Y|X = x} calculated from independent observations (X1, Y1),…, (Xn, Yn) of a pair (X, Y) of random variables is examined. ForE|Y|1 + δ < ∞, δ > 0, the estimate is weakly pointwise consistent for almost all (μ) x ∈ Rd, μ is the probability measure of X, if and only if∑i−1n hid I{hi > ɛ } ⧸ ∑j = 1n hjd → 0 as n → ∞, all ɛ > 0, and∑i = 1 hid = ∞, d is the dimension of X. For E|Y|1 + δ < ∞, δ > 0, the estimate is strongly pointwise consistent for almost all (μ) x ∈ Rd, if and only if the same conditions hold. ForE|Y|1 + δ < ∞, δ > 0, weak and strong consistency are equivalent. Similar results are given for complete convergence.  相似文献   

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