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1.
Skorokhod's representation theorem states that if on a Polish space,there is a weakly convergent sequence of probability measures μnw→μ0,as n →∞,then there exist a probability space(Ω,F,P) and a sequence of random elements Xnsuch that Xn→ X almost surely and Xnhas the distribution function μn,n = 0,1,2,... We shall extend the Skorokhod representation theorem to the case where if there are a sequence of separable metric spaces Sn,a sequence of probability measures μnand a sequence of measurable mappings n such that μnn-1w→μ0,then there exist a probability space(Ω,F,P) and Sn-valued random elements Xndefined on Ω,with distribution μnand such that n(Xn) → X0 almost surely. In addition,we present several applications of our result including some results in random matrix theory,while the original Skorokhod representation theorem is not applicable.  相似文献   

2.
Let X 1,X 2,… be a sequence of i.i.d. mean zero random variables and let S n denote the sum of the first n random variables. We show that whenever we have with probability one, lim?sup? n→∞|S n |/c n =α 0<∞ for a regular normalizing sequence {c n }, the corresponding normalized partial sum process sequence is relatively compact in C[0,1] with canonical cluster set. Combining this result with some LIL type results in the infinite variance case, we obtain Strassen type results in this setting.  相似文献   

3.
Let be a sequence of i.i.d. random variables taking values in a real separable Hilbert space (H,‖⋅‖) with covariance operator Σ, and set Sn=X1+?+Xn, n?1. Let . We prove that, for any 1<r<3/2 and a>−d/2,
  相似文献   

4.
The Hartman–Wintner–Strassen law of the iterated logarithm states that if X 1, X 2,… are independent identically distributed random variables and S n =X 1+???+X n , then
$\limsup_{n}S_{n}/\sqrt{2n\log \log n}=1\quad \text{a.s.},\qquad \liminf_{n}S_{n}/\sqrt{2n\log \log n}=-1\quad \text{a.s.}$
if and only if EX 1 2 =1 and EX 1=0. We extend this to the case where the X n are no longer identically distributed, but rather their distributions come from a finite set of distributions.
  相似文献   

5.
For a double array of blockwise M-dependent random variables {X mn ,m ?? 1, n ?? 1}, strong laws of large numbers are established for double sums ?? i=1 m ?? j=1 n X ij , m ?? 1, n ?? 1. The main results are obtained for (i) random variables {X mn ,m ?? 1, n ?? 1} being non-identically distributed but satisfy a condition on the summability condition for the moments and (ii) random variables {X mn ,m ?? 1, n ?? 1} being stochastically dominated. The result in Case (i) generalizes the main result of Móricz et al. [J. Theoret. Probab., 21, 660?C671 (2008)] from dyadic to arbitrary blocks, whereas the result in Case (ii) extends a result of Gut [Ann. Probab., 6, 469?C482 (1978)] to the bockwise M-dependent setting. The sharpness of the results is illustrated by some examples.  相似文献   

6.
Let X 1,X 2,… be a sequence of random variables. Let S k =X 1+???+X k and assume that S k /b k converges in distribution for some numerical sequence (b k ). We study the weak convergence of the random processes {Λ n (z), z∈?}, where
$\Lambda_{n}(z)=\frac{1}{n}\sum_{k=1}^{n}I\left\{\frac{S_{k}}{b_{k}}\leq z\right\}.$
We consider the same problem when the normalized partial sums S k /b k are replaced by other functionals of the sequence (X n ). In particular, we investigate the case of sample extremes in detail.
  相似文献   

7.
Let {X, X_n; n ≥ 0} be a sequence of independent and identically distributed random variables with EX=0, and assume that EX~2I(|X| ≤ x) is slowly varying as x →∞, i.e., X is in the domain of attraction of the normal law. In this paper, a self-normalized law of the iterated logarithm for the geometrically weighted random series Σ~∞_(n=0)β~nX_n(0 β 1) is obtained, under some minimal conditions.  相似文献   

8.
Consider independent identically distributed random variables (Xi) valued in [0,1]. Let B(n) be the optimal (minimum) number of unit size bins needed to pack n items of size X1, X2,…,Xn. We prove that there exists a numerical constant C such that for t > 0,
Pr(∣B(n)?E(B(n))∣>tn)≤ C exp(? t).
The constant C does not depend on the distribution of X.  相似文献   

9.
In this paper, Let X, Y be two real Banach spaces and ε ≥ 0. A mapping f: XY is said to be a standard ε-isometry provided f(0) = 0 and
$$\parallel f\left( x \right) - f\left( y \right)\parallel - \parallel x - y\parallel | \leqslant \varepsilon $$
(1)
for all x, yX. If ε = 0, then it is simply called a standard isometry. We prove a sufficient and necessary condition for which {f(xn)}n≥1 is a basic sequence of Y equivalent to {xn}n≥1 whenever {xn}n≥1 is a basic sequence in X and f: XY is a nonlinear standard isometry. As a corollary we obtain the stability of basic sequences under the perturbation by nonlinear and non-surjective standard ε-isometries.
  相似文献   

10.
Let be a sequence of i.i.d. random variables with EX=0 and EX2=σ2<∞. Set , Mn=maxk?n|Sk|, n?1. Let r>1, then we obtain
  相似文献   

11.
Let X,X1,X2,… be a sequence of nondegenerate i.i.d. random variables with zero means. Set Sn=X1+?+Xn and . In the present paper we examine the precise asymptotic behavior for the general deviation probabilities of self-normalized sums, Sn/Wn. For positive functions g(x), ?(x), α(x) and κ(x), we obtain the precise asymptotics for the following deviation probabilities of self-normalized sums:
  相似文献   

12.
Let {Xni, 1 ≤ n,i <∞} be an array of rowwise NA random variables and {an, n ≥ 1} a sequence of constants with 0 < an ↑∞. The limiting behavior of maximum partial sums 1/an max 1≤k≤n| kΣi=1 Xni| is investigated and some new results are obtained. The results extend and improve the corresponding theorems of rowwise independent random variable arrays by Hu and Taylor [1] and Hu and Chang [2].  相似文献   

13.
Let {X n ; n≥1} be a sequence of independent copies of a real-valued random variable X and set S n =X 1+???+X n , n≥1. This paper is devoted to a refinement of the classical Kolmogorov–Marcinkiewicz–Zygmund strong law of large numbers. We show that for 0<p<2,
$\sum_{n=1}^{\infty}\frac{1}{n}\biggl(\frac{|S_{n}|}{n^{1/p}}\biggr)<\infty\quad \mbox{almost surely}$
if and only if
$\begin{cases}\mathbb{E}|X|^{p}<\infty &; \mbox{if }0 < p < 1,\\\mathbb{E}X=0,\ \sum_{n=1}^{\infty}\frac{|\mathbb{E}XI\{|X|\leq n\}|}{n}<\infty,\mbox{ and }\\\sum_{n=1}^{\infty}\frac{\int_{\min\{u_{n},n\}}^{n}\mathbb{P}(|X|>t)\,dt}{n}<\infty &; \mbox{if }p = 1,\\\mathbb{E}X=0\mbox{ and }\int_{0}^{\infty}\mathbb{P}^{1/p}(|X|>t)\,dt<\infty,&;\mbox{if }1 < p < 2,\end{cases}$
where \(u_{n}=\inf \{t:~\mathbb{P}(|X|>t)<\frac{1}{n}\}\), n≥1. Versions of the above result in a Banach space setting are also presented. To establish these results, we invoke the remarkable Hoffmann-Jørgensen (Stud. Math. 52:159–186, 1974) inequality to obtain some general results for sums of the form \(\sum_{n=1}^{\infty}a_{n}\|\sum_{i=1}^{n}V_{i}\|\) (where {V n ; n≥1} is a sequence of independent Banach-space-valued random variables, and a n ≥0, n≥1), which may be of independent interest, but which we apply to \(\sum_{n=1}^{\infty}\frac{1}{n}(\frac{|S_{n}|}{n^{1/p}})\).
  相似文献   

14.
We consider the asymptotic behavior of the values P(S > x), E(S 1{S>x}), and E(S | S > x). Here S = θ1X1 + θ2X2 + · · · + θnXn is a randomly weighted sum of the basic random variables X1,X2, . . . , Xn, which follow some special dependence structure, and 1, θ2, . . . , θn} is a collection of nonnegative and arbitrarily dependent random weights; the collections {X1,X2, . . .,Xn} and 1, θ2, . . . , θn} are supposed to be independent. We derive asymptotic formulas in the case where the number of summands n is fixed and the distributions of the basic random variables are dominatedly varying.We apply them to some values related to the risk measures of certain weighted sums.  相似文献   

15.
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   

16.
Let X 1,…,X n be pairwise asymptotically independent or pairwise upper extended negatively dependent real-valued random variables. Under the condition that the distribution of the maximum of X 1,…,X n belongs to some subclass of heavy-tailed distributions, we investigate the asymptotic behavior of the partial sum and its maximum generated by dependent X 1,…,X n . As an application, we consider a discrete-time risk model with insurance and financial risks and derive the asymptotics for the finite-time ruin probability.  相似文献   

17.
Any (measurable) function K from Rn to R defines an operator K acting on random variables X by K(X) = K(X1,..., Xn), where the Xj are independent copies of X. The main result of this paper concerns continuous selectors H, continuous functions defined in Rn and such that H(x1, x2,..., xn) ∈ {x1, x2,..., xn}. For each such continuous selector H (except for projections onto a single coordinate) there is a unique point ωH in the interval (0, 1) so that, for any random variable X, the iterates H(N) acting on X converge in distribution as N → ∞ to the ωH-quantile of X.  相似文献   

18.
We provide an optimal Berry-Esseen type inequality for Zolotarev’s ideal ζ3-metric measuring the difference between expectations of sufficiently smooth functions, like |·|3, of a sum of independent random variables X 1,..., X n with finite third-order moments and a sum of independent symmetric two-point random variables, isoscedastic to the X i . In the homoscedastic case of equal variances, and in particular, in case of identically distributed X 1,..., X n the approximating law is a standardized symmetric binomial one. As a corollary, we improve an already optimal estimate of the accuracy of the normal approximation due to Tyurin (2009).  相似文献   

19.
Let X be a symmetric Banach function space on [0, 1] and let E be a symmetric (quasi)-Banach sequence space. Let f = {f k } k=1 n , n ≥ 1 be an arbitrary sequence of independent random variables in X and let {e k } k=1 ? E be the standard unit vector sequence in E. This paper presents a deterministic characterization of the quantity
$||||\sum\limits_{k = 1}^n {{f_k}{e_k}|{|_E}|{|_X}} $
in terms of the sum of disjoint copies of individual terms of f. We acknowledge key contributions by previous authors in detail in the introduction, however our approach is based on the important recent advances in the study of the Kruglov property of symmetric spaces made earlier by the authors. Authors acknowledge support from the ARC.
  相似文献   

20.
Let the independent random variables X1, X2, … have the same continuous distribution function. The upper record values X(1) = X1 < X(2) < … generated by this sequence of variables, as well as the lower record values x(1) = X1 > x(2) > …, are considered. It is known that in this situation, the mean value c(n) of the total number of the both types of records among the first n variables X is given by the equality c(n)=2(1+1/2+…+1/n), n = 1, 2, …. The problem considered here is following: how, sequentially obtaining the observed values x1, x2, … of variables X and selecting one of them as the initial point, to obtain the maximal mean value e(n) of the considered numbers of records among the rest random variables. It is not possible to come back to rejected elements of the sequence. Some procedures of the optimal choice of the initial element X r are discussed. The corresponding tables for the values e(n) and differences δ(n)= e(n)–c(n) are presented for different values of n. The value of δ= limn→∞δ(n)is also given. In some sense, the considered problem and optimization procedure presented in this paper are quite similar to the classical “secretary problem,” in which the probability of selecting the last record value in the set of independent identically distributed X is maximized.  相似文献   

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