共查询到20条相似文献,搜索用时 31 毫秒
1.
Sh. A. Mirakhmedov 《Journal of Mathematical Sciences》1987,38(6):2346-2357
Assume that there are given random vectors ¯v1, ..., ¯vs, independent in their totality, where vj=(v1j, ... vNj) has the multinomial distribution M(nj, p1j, ..., PNj,. Assume further that
jm
(N)
(x1, ..., xs), j=1–k, k1 are random functions of s nonnegative integer arguments x1, ..., xs. One considers the multidimensional randomized separable statistic SN=(SN1, ..., SNk), where. One obtains estimates for the rate of convergence in the central limit theorem for SN.Translated from Veroyatnostnye Raspredeleniya i Matematicheskaya Statistika, pp. 315–337, 1986. 相似文献
2.
Paul Deheuvels 《Probability Theory and Related Fields》1981,58(1):1-6
Summary If X
1, X
2, ..., are i.i.d. random variables and Y
n
=Max(X
1, ..., X
n
); if for some sequences A
n
, Bn, n=1, 2, ..., E
n
(t)=AnY[nt]+Bn is such that E
n
(1) weakly converges to a non degenerate limit distribution, then we prove that it is possible to construct a sequence of replicates of extremal processes E
(n)(t) on the same probability space, such that d(E
n
(.), E
(n)(.))0 a.s., with the Levy metric. We give the rates of consistency of the approximations. 相似文献
3.
Li Xin Zhang 《数学学报(英文版)》2002,18(2):311-326
Let {X
n
;n≥1} be a sequence of i.i.d. random variables and let X
(r)
n
= X
j
if |X
j
| is the r-th maximum of |X
1|, ..., |X
n
|. Let S
n
= X
1+⋯+X
n
and
(r)
S
n
= S
n
−(X
(1)
n
+⋯+X
(r)
n
). Sufficient and necessary conditions for
(r)
S
n
approximating to sums of independent normal random variables are obtained. Via approximation results, the convergence rates
of the strong law of large numbers for
(r)
S
n
are studied.
Received March 22, 1999, Revised November 6, 2000, Accepted March 16, 2001 相似文献
4.
金敬森 《数学物理学报(A辑)》2009,29(4):1138-1143
设d是一个正整数, N d是d -维正整数格点.设{Xn , n∈N d} 是一同分布的负相伴随机场, 记Sn =∑k≤ n Xk, Sn(k)=Sn-Xk, 如果r >2, EX1 = 0 和σ2= Var(X1}, 则存在一个正数M:=100√(r-2)(1+σ2)使得下列条件等价
(I)E |X1|r (log|X1|)d-1-r/2 <∞;
(II)∑n∈ Nd |n|r/2-2P(max1≤ k≤ n |Sn(k)|≥ (2d+1 )ε√|n| log |n |) <∞,∨ε > M;
(III)∑n∈N d |n|r/2-2P(max1≤ k≤n |Sk |≥ε√| n} log| n |) <∞,∨ε > M.
(III)\ \ $\sum\limits_{{{\bf n}}\in {{\cal N}}^{d}} |n|^{r/2-2}
P(\max\limits_{{\bf 1}\leq{\bf k}\leq{\bf n}}|S_{{\bf k}}|\geq
\varepsilon \sqrt{|{\bf n}|\log |{\bf n}|})<\infty$,
$\forall\varepsilon>M$. 相似文献
5.
V. M. Zolotarev 《Journal of Mathematical Sciences》1987,38(5):2262-2272
Let {Q(n)(x1,...,xn)} be a sequence of symmetric polynomials having a fixed degree equal to k. Let {Xn1,...,Xnn}, n k, be some sequence of series of random variables (r.v.). We form the sequence of r.v. Yn=Q(n)(Xn1, ... Xnn), n k One obtains limit theorems for the sequence Yn, under very general assumptions.Translated from Veroyatnostnye Raspredeleniya i Matematicheskaya Statistika, pp. 170–188, 1986. 相似文献
6.
Letα r denote the number of cycles of length r in a random permutation, taking its values with equal probability from among the set Sn of all permutations of length n. In this paper we study the limiting behavior of linear combinations of random permutationsα 1, ...,α r having the form $$\zeta _{n, r} = c_{r1^{a_1 } } + ... + c_{rr} a_r $$ in the case when n, r→∞. We shall show that the class of limit distributions forξ n,r as n, r→∞ and r In r/h→0 coincides with the class of unbounded divisible distributions. For the random variables ηn, r=α 1+2α 2+... rα r, equal to the number of elements in the permutation contained in cycles of length not exceeding r, we find' limit distributions of the form r In r/n→0 and r=γ n, 0<γ<1. 相似文献
7.
Let {T1, ..., TN} be a finite set of linear contraction mappings of a Hilbert space H into itself, and let r be a mapping from the natural numbersN to {1, ..., N} which assumes each value infinitely often. One can form Sn=Tr(n)...Tr(1) which could be described as a random product of the Ti's. If the contractions have the condition (W): Tx<x whenever Txx, then Sn converges weakly to the projection Q onto the subspace
. This theorem is due to Amemiya and Ando. We demonstrate a basic property of the algebraic semigroupS=S(T1, ..., TN) generated by N contractions, each having (W). We prove that if the semigroup of an infinite set of contractions is equipped with this property, and the maps satisfy a minor condition parallel to (W) on each of N maps, then random products still converge weakly. Our proof is different from Amemiya and Ando's. We illustrate our method with a new proof of the fact that if a contraction T is completely non-unitary, then Tn0 weakly. 相似文献
8.
Summary The asymptotic behaviour of elementary symmetric polynomials S
n
(k)
of order k, based on n independent and identically distributed random variables X
1,..., X
n,is investigated for the case that both k and n get large. If
, then the distribution function of a suitably normalised S
n
(k)
is shown to converge to a standard normal limit. The speed of this convergence to normality is of order
, provided
and certain natural moment assumptions are imposed. This order bound is sharp, and cannot be inferred from one of the existing Berry-Esseen bounds for U-statistics. If k at the rate n
1/2 then a non-normal weak limit appears, provided the X
i's are positive and S
n
(k)
is standardised appropriately. On the other hand, if k at a rate faster than n
1/2 then it is shown that for positive X
j'sthere exists no linear norming which causes S
n
(k)
to converge weakly to a nondegenerate weak limit. 相似文献
9.
Let (Xn)n? be a sequence of real, independent, not necessarily identically distributed random variables (r.v.) with distribution functions FXn, and Sn = Σi=1nXi. The authors present limit theorems together with convergence rates for the normalized sums ?(n)Sn, where ?: → +, ?(n) → 0, n → ∞, towards appropriate limiting r.v. X, the convergence being taken in the weak (star) sense. Thus higher order estimates are given for the expression ∝f(x) d[F?(n)Sn(x) ? FX(x)] which depend upon the normalizing function ?, decomposability properties of X and smoothness properties of the function f under consideration. The general theorems of this unified approach subsume O- and o-higher order error estimates based upon assumptions on associated moments. These results are also extended to multi-dimensional random vectors. 相似文献
10.
V. B. Nevzorov 《Journal of Mathematical Sciences》1983,23(3):2286-2290
Let X1, X2, ... be a sequence of independent identically distributed random variables with zero mathematical expectation and finite variances. So=0 and Sn=∑ i=1 n Xi. It is proved that is the limit distribution function of the normalized random variable a(k, n)} for some sequence of centering constants a (k,n). 相似文献
11.
LU Chuanrong QIU Jin & XU Jianjun School of Mathematics Statistics Zhejiang University of Finance Economics Hangzhou China Department of Mathematics Zhejiang University Hangzhou China 《中国科学A辑(英文版)》2006,49(12):1788-1799
Let {Xn,-∞< n <∞} be a sequence of independent identically distributed random variables with EX1 = 0, EX12 = 1 and let Sn =∑k=1∞Xk, and Tn = Tn(X1,…,Xn) be a random function such that Tn = ASn Rn, where supn E|Rn| <∞and Rn = o(n~(1/2)) a.s., or Rn = O(n1/2-2γ) a.s., 0 <γ< 1/8. In this paper, we prove the almost sure central limit theorem (ASCLT) and the function-typed almost sure central limit theorem (FASCLT) for the random function Tn. As a consequence, it can be shown that ASCLT and FASCLT also hold for U-statistics, Von-Mises statistics, linear processes, moving average processes, error variance estimates in linear models, power sums, product-limit estimators of a continuous distribution, product-limit estimators of a quantile function, etc. 相似文献
12.
A characteristic property of spheres 总被引:1,自引:1,他引:0
A. D. Alexandrov 《Annali di Matematica Pura ed Applicata》1962,58(1):303-315
Summary We prove: Let S be a closed n-dimensional surface in an(n+1)-space of constant curvature (n ≥ 2); k1 ≥ ... ≥ kn denote its principle curvatures. Let φ(ξ1, ..., ξn) be such that
. Then if φ(k1, ..., kn)=const on S and S is subject to some additional general conditions (those(II
0) or(II) no 1), S is a sphere.
To Enrico Bompiani on his scientific Jubilee 相似文献
13.
L. B. Klebanov 《Mathematical Notes》1973,13(6):531-532
Let (X1, ..., Xn) be a random vector with independent components. It is proven in this paper that, under certain restrictions, the distributions of the pairS
1=sup (a
1X1, ..., anXn) andS
2=sup (b1X1,...,bnXn) univocally define the distribution function of the components Xj.Translated from Matematicheskie Zametki, Vol. 13, No. 6, pp. 889–892, June, 1973. 相似文献
14.
Peter Schatte 《Mathematische Nachrichten》1988,137(1):249-256
Let Sn be the sum of n i.i.d.r.v. and let 1(-∞,x)(·) be the indicator function of the interval (-∞, x). Then the sequence 1(-∞, x)(Sn/√n) does not converge for any x. Likewise the arithmetic means of this sequence converge only with probability zero. But the logarithmic means converge with probability one to the standard normal distribution Ø(x). Then for any bounded and a.e. continuous function a(y) the logarithmic means of a(Sn/√n) converge a.s. to a = ∫a(y)dØ(y). The arithmetic means of a(Snk/√n) converge to the same limit a for all subsequences nk = [ck], c > 1. 相似文献
15.
Let {X
j} be independent, identically distributed random variables which are symmetric about the origin and have a continuous nondegenerate distributionF. Let {X
n(1),...,X
n(n)} denote the arrangement of {X
1,...,X
n} in decreasing order of magnitude, so that with probability one, |X
n(1)|>|X
n(2)|>...> |X
n(n)|. For initegersr
n such thatr
n/n0, define the self-normalized trimmed sumT
n=
i=rn
n
X
n(i)/{
i=rn
n
X
n
2
(i)}1/2. Hahn and Weiner(6) showed that under a probabilistically meaningful analytic condition generalizing the asymptotic normality criterion forT
n, various nonnormal limit laws forT
n arise which are represented by means of infinite random series. The analytic condition is now extended and the previous approach is refined to obtain limits which are mixtures of a normal, a Rademacher, and a law represented by a more general random series. Each such limit law actually arises as can be seen from the construction of a single distribution whose correspondingL(T
n
) generates all of the law along different subsequences, at least if {r
n} grows sufficiency fast. Another example clarifies the limitations of the basic approach. 相似文献
16.
Let X
1
, X
2
, ..., Xn be n independent identically distributed real random variables and Sn = Σ
n=1
n
Xi. We obtain precise asymptotics forP (Sn ∈ nA) for rather arbitrary Borel sets A1 in terms of the density of the dominating points in A. Our result extends classical theorems in the field of large deviations
for independent samples. We also obtain asymptotics forP (Sn ∈ γnA), with γn/n → ∞.
Proceedings of the Seminar on Stability Problems for Stochastic Models, Vologda, Russia, 1998, Part I. 相似文献
17.
Yuval Peres 《Israel Journal of Mathematics》1996,95(1):341-347
Say that a sequenceS
0, ..., Sn has a (global) point of increase atk ifS
k is maximal amongS
0, ..., Sk and minimal amongS
k, ..., Sn. We give an elementary proof that ann-step symmetric random walk on the line has a (global) point of increase with probability comparable to 1/logn. (No moment assumptions are needed.) This implies the classical fact, due to Dvoretzky, Erdős and Kakutani (1961), that Brownian
motion has no points of increase.
Research partially supported by NSF grant # DMS-9404391. 相似文献
18.
Amiram Braun 《Israel Journal of Mathematics》1986,55(3):345-349
An example is given of a ringR (with 1) satisfying the standard identityS
6[x
1, ...,x
6] butM
2(R), the 2 × 2 matrix ring overR, does not satisfyS
12[x
1, ...,x
12]. This is in contrast to the caseR=M
n (F),F a field, where by the Amitsur-Levitzki theoremR satisfiesS
2n [x
1, ...,x
2n] andM
2(R) satisfiesS
4n
[x
1, ...,x
n].
Part of this work was done while the author enjoyed the hospitality of the University of California at San Diego, the University
of Texas at Austin and the University of Washington at Seattle. 相似文献
19.
A. Kopotowski M G. Nadkarni K. P. S. Bhaskara Rao 《Proceedings Mathematical Sciences》2003,113(1):77-86
We discuss subsetsS of ℝn such that every real valued functionf onS is of the formf(x1, x2, ..., xn) =u
1(x1) +u
2(x2) +...+u
n(xn), and the related concepts and situations in analysis. 相似文献
20.
A sort sequence Sn is a sequence of all unordered pairs of indices inI n = {1, 2, ..., n}. With a sort sequenceSn = (s1 ,s2 ,...,s( 2n ) )S_n = (s_1 ,s_2 ,...,s_{\left( {_2^n } \right)} ), one can associate a predictive sorting algorithm A(Sn). An execution of the algorithm performs pairwise comparisons of elements in the input setX in the order defined by the sort sequence Sn except that the comparisons whose outcomes can be inferred from the results of the preceding comparisons are not performed. A sort sequence is said to be extremal if it maximizes a given objective function. First we consider the extremal sort sequences with respect to the objective function ω(Sn) — the expected number of active predictions inS n. We study ω-extremal sort sequences in terms of their prediction vectors. Then we consider the objective function Ω(Sn) — the minimum number of active predictions in Sn over all input orderings. 相似文献