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1.
Let be an i.i.d. sequence of -valued random vectors belonging to the generalized domain of semistable attraction of some nonnormal law. Assume further
that is a sequence of positive integer valued random variables such that for some
for some discrete positive random variable D, where we do not assume that and are independent. Let . Then various laws of the iterated logarithm for the norm of as well as the radial projection onto a unit vector θ are presented.
(Received 31 January 2000; in revised form 5 April 2000) 相似文献
2.
Let X
1, X
2,... be independent, but not necessarily identically distributed random variables in the domain of attraction of a normal law or a stable law with index 0 < α < 2. Using suitable self-normalizing (or Studentizing) factors, laws of the iterated logarithm for self-normalized Hanson–Russo type increments are discussed. Also, some analogous results for self-normalized weighted sums of i.i.d. random variables are given. 相似文献
3.
Wen Jiwei Yan Yunliang 《高校应用数学学报(英文版)》2006,21(1):87-95
Let X,X1,X2 be i. i. d. random variables with EX^2+δ〈∞ (for some δ〉0). Consider a one dimensional random walk S={Sn}n≥0, starting from S0 =0. Let ζ* (n)=supx∈zζ(x,n),ζ(x,n) =#{0≤k≤n:[Sk]=x}. A strong approximation of ζ(n) by the local time for Wiener process is presented and the limsup type and liminf-type laws of iterated logarithm of the maximum local time ζ*(n) are obtained. Furthermore,the precise asymptoties in the law of iterated logarithm of ζ*(n) is proved. 相似文献
4.
Li Xin Zhang 《数学学报(英文版)》2002,18(3):605-614
Let be a field of i.i.d. random variables indexed by d-tuples of positive integers and taking values in a Banach space ℬ and let if is the r-th maximum of . Let and . We approximate the trimmed sums by a Brownian sheet and obtain sufficient and necessary conditions for to satisfy the compact and functional laws of the iterated logarithm. These results improve the previous works by Morrow
(1981), Li and Wu (1989) and Ledoux and Talagrand (1990).
Received March 22, 1999, Accepted October 13, 2000 相似文献
5.
LuCHUANRONG 《高校应用数学学报(英文版)》1997,12(4):375-380
Let(x1,j≥1)be a sequence of negatively associated random variables with ex1=o,ex^21<∞.in this paper a functional central limit theorem for negatively associated random variables under some conditions withbout stationarity is proved which is the same as the results for positively associated random variables. 相似文献
6.
George Davie 《Archive for Mathematical Logic》2001,40(8):629-638
Let ω be a Kolmogorov–Chaitin random sequence with ω1:
n
denoting the first n digits of ω. Let P be a recursive predicate defined on all finite binary strings such that the Lebesgue measure of the set {ω|∃nP(ω1:
n
)} is a computable real α. Roughly, P holds with computable probability for a random infinite sequence. Then there is an algorithm which on input indices for any
such P and α finds an n such that P holds within the first n digits of ω or not in ω at all. We apply the result to the halting probability Ω and show that various generalizations of
the result fail.
Received: 1 December 1998 / Published online: 3 October 2001 相似文献
7.
Let {Y
i
;−∞<i<∞} be a doubly infinite sequence of independent random elements taking values in a separable real Banach space and stochastically
dominated by a random variable X. Let {a
i
;−∞<i<∞} be an absolutely summable sequence of real numbers and set V
i
=∑
k=−∞∞
a
i+k
Y
i
,i≥1. In this paper, we derive that if
and E|X|
μ
log
ρ
|X|<0, for some μ (0<μ<2, μ≠1) and ρ>0 then
for all ε>0.
This work was partially supported by the Korean Research Foundation Grant funded by the Korean Government (KRF-2006-353-C00006,
KRF-2006-251-C00026). 相似文献
8.
The bijection between composition structures and random closed subsets of the unit interval implies that the composition structures
associated with S ⋂ [0, 1] for a self-similar random set S ⊂ ℝ+ are those that are consistent with respect to a simple truncation operation. Using the standard coding of compositions by
finite strings of binary digits starting with a 1, the random composition of n is defined by the first n terms of a random
binary sequence of infinite length. The locations of 1’s in the sequence are the positions visited by an increasing time-homogeneous
Markov chain on the positive integers if and only if S = exp(−W) for some stationary regenerative random subset W of the real line. Complementing our study presented in previous papers,
we identify self-similar Markov composition structures associated with the two-parameter family of partition structures. Bibliography:
19 titles.
Published in Zapiski Nauchnykh Seminarov POMI, Vol. 326, 2005, pp. 59–84. 相似文献
9.
Zhen-long Chen San-yang Liu 《应用数学学报(英文版)》2005,21(4):623-636
Let φ be a Hausdorff measure function and A be an infinite increasing sequence of positive integers. The Hausdorff-type measure φ - mA associated to φ and A is studied. Let X(t)(t ∈ R^N) be certain Gaussian random fields in R^d. We give the exact Hausdorff measure of the graph set GrX([0, 1]N), and evaluate the exact φ - mA measure of the image and graph set of X(t). A necessary and sufficient condition on the sequence A is given so that the usual Hausdorff measure function for X([0, 1] ^N) and GrX([0, 1]^N) are still the correct measure functions. If the sequence A increases faster, then some smaller measure functions will give positive and finite ( φ A)-Hausdorff measure for X([0, 1]^N) and GrX([0, 1]N). 相似文献
10.
Let {X
n
,n ≥ 1} be a sequence of i.i.d. random variables. Let M
n
and m
n
denote the first and the second largest maxima. Assume that there are normalizing sequences a
n
> 0, b
n
and a nondegenerate limit distribution G, such that . Assume also that {d
k
,k ≥ 1} are positive weights obeying some mild conditions. Then for x > y we have
when G(y) > 0 (and to zero when G(y) = 0).
相似文献
11.
Complete convergence for arrays 总被引:4,自引:0,他引:4
A. Gut 《Periodica Mathematica Hungarica》1992,25(1):51-75
Let {(X
nk
, 1≤k≤n),n≥1}, be an array of rowwise independent random variables. We extend and generalize some recent results due to Hu, Móricz and
Taylor concerning complete convergence, in the sense of Hsu and Robbins, of the sequence of rowwise arithmetic means. 相似文献
12.
In this paper we extend and improve some results of the large deviation for random sums of random variables. Let {Xn;n 〉 1} be a sequence of non-negative, independent and identically distributed random variables with common heavy-tailed distribution function F and finite mean μ ∈R^+, {N(n); n ≥0} be a sequence of negative binomial distributed random variables with a parameter p C (0, 1), n ≥ 0, let {M(n); n ≥ 0} be a Poisson process with intensity λ 〉 0. Suppose {N(n); n ≥ 0}, {Xn; n≥1} and {M(n); n ≥ 0} are mutually independent. Write S(n) =N(n)∑i=1 Xi-cM(n).Under the assumption F ∈ C, we prove some large deviation results. These results can be applied to certain problems in insurance and finance. 相似文献
13.
Let {Y i;∞ < i < ∞} be a doubly infinite sequence of identically distributed-mixing random variables and let {a i;∞ < i < ∞} be an absolutely summable sequence of real numbers.In this paper we study the moments of sup(1 ≤ r < 2,p > 0) under the conditions of some moments. 相似文献
14.
V. Bentkus 《Lithuanian Mathematical Journal》2008,48(3):237-255
Let S
n = X
1 + ⋯ + X
n be a sum of independent random variables such that 0 ⩽ X
k ⩽ 1 for all k. Write {ie237-01} and q = 1 − p. Let 0 < t < q. In our recent paper [3], we extended the inequality of Hoeffding ([6], Theorem 1) {fx237-01} to the case where X
k are unbounded positive random variables. It was assumed that the means {ie237-02} of individual summands are known. In this
addendum, we prove that the inequality still holds if only an upper bound for the mean {ie237-03} is known and that the i.i.d.
case where {ie237-04} dominates the general non-i.i.d. case. Furthermore, we provide upper bounds expressed in terms of certain
compound Poisson distributions. Such bounds can be more convenient in applications. Our inequalities reduce to the related
Hoeffding inequalities if 0 ⩽ X
k ⩽ 1. Our conditions are X
k ⩾ 0 and {ie237-05}. In particular, X
k can have fat tails. We provide as well improvements comparable with the inequalities in Bentkus [2]. The independence of
X
k can be replaced by super-martingale type assumptions. Our methods can be extended to prove counterparts of other inequalities
in Hoeffding [6] and Bentkus
The research was partially supported by the Lithuanian State Science and Studies Foundation, grant No T-25/08. 相似文献
15.
Let {Z
t
,t≥1} be a sequence of trials taking values in a given setA={0, 1, 2,...,m}, where we regard the value 0 as failure and the remainingm values as successes. Let ε be a (single or compound) pattern. In this paper, we provide a unified approach for the study
of two joint distributions, i.e., the joint distribution of the numberX
n
of occurrences of ε, the numbers of successes and failures inn trials and the joint distribution of the waiting timeT
r
until ther-th occurrence of ε, the numbers of successes and failures appeared at that time. We also investigate some distributions as
by-products of the two joint distributions. Our methodology is based on two types of the random variablesX
n
(a Markov chain imbeddable variable of binomial type and a Markov chain imbeddable variable of returnable type). The present
work develops several variations of the Markov chain imbedding method and enables us to deal with the variety of applications
in different fields. Finally, we discuss several practical examples of our results.
This research was partially supported by the ISM Cooperative Research Program (2002-ISM·CRP-2007). 相似文献
16.
F. Luca I. E. Shparlinski 《Abhandlungen aus dem Mathematischen Seminar der Universit?t Hamburg》2006,76(1):143-156
Let (un)n≥0 be a non-degenerate linear recurrence sequence of integers. We show that the set of positive integersn such that either ω)(n) orΩ(n) dividesu
n
is of asymptotic density zero, where ω(n) and Ω(n) are the numbers of prime and prime power divisors ofn, respectively. The same also holds for the set of positive integersn such that τ(n)u
n
, where τ(n) is the number of the positive integer divisors of n, provided thatu
n
satisfies some mild technical conditions. 相似文献
17.
Let be a sequence of natural numbers > 1, and set . The sequence is called admissible if a i divides for all i. It is known that the admissible sequences are counted by the Catalan numbers. We present a proof of this fact which, in turn, leads to some interesting combinatorial and number-theoretic questions. 相似文献
18.
Simeon M. Berman 《Annals of the Institute of Statistical Mathematics》1984,36(1):301-321
Summary Let {X
n,j,−∞<j<∞∼,n≧1, be a sequence of stationary sequences on some probability space, with nonnegative random variables. Under appropriate
mixing conditions, it is shown thatS
n=Xn,1+…+X
n,n has a limiting distribution of a general infinitely divisible form. The result is applied to sequences of functions {f
n(x)∼ defined on a stationary sequence {X
j∼, whereX
n.f=fn(Xj). The results are illustrated by applications to Gaussian processes, Markov processes and some autoregressive processes of
a general type.
This paper represents results obtained at the Courant Institute of Mathematical Sciences, New York University, under the sponsorship
of the National Sciences Foundation, Grant MCS 82-01119. 相似文献
19.
A. I. Martikainen 《Journal of Mathematical Sciences》2006,133(3):1308-1313
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
20.
We show that an i.i.d. uniformly colored scenery on ℤ observed along a random walk path with bounded jumps can still be reconstructed
if there are some errors in the observations. We assume the random walk is recurrent and can reach every point with positive
probability. At time k, the random walker observes the color at her present location with probability 1−δ and an error Y
k
with probability δ. The errors Y
k
, k≥0, are assumed to be stationary and ergodic and independent of scenery and random walk. If the number of colors is strictly
larger than the number of possible jumps for the random walk and δ is sufficiently small, then almost all sceneries can be
almost surely reconstructed up to translations and reflections.
Received: 3 February 2002 / Revised version: 15 January 2003 Published online: 28 March 2003
Mathematics Subject Classification (2000): 60K37, 60G50
Key words or phrases:Scenery reconstruction – Random walk – Coin tossing problems 相似文献