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1.
Let (X, Xn; n ≥1) be a sequence of i.i.d, random variables taking values in a real separable Hilbert space (H, ||·||) with covariance operator ∑. Set Sn = X1 + X2 + ... + Xn, n≥ 1. We prove that, for b 〉 -1,
lim ε→0 ε^2(b+1) ∞ ∑n=1 (logn)^b/n^3/2 E{||Sn||-σε√nlogn}=σ^-2(b+1)/(2b+3)(b+1) B||Y|^2b+3
holds if EX=0,and E||X||^2(log||x||)^3bv(b+4)〈∞ where Y is a Gaussian random variable taking value in a real separable Hilbert space with mean zero and covariance operator ∑, and σ^2 denotes the largest eigenvalue of ∑. 相似文献
lim ε→0 ε^2(b+1) ∞ ∑n=1 (logn)^b/n^3/2 E{||Sn||-σε√nlogn}=σ^-2(b+1)/(2b+3)(b+1) B||Y|^2b+3
holds if EX=0,and E||X||^2(log||x||)^3bv(b+4)〈∞ where Y is a Gaussian random variable taking value in a real separable Hilbert space with mean zero and covariance operator ∑, and σ^2 denotes the largest eigenvalue of ∑. 相似文献
2.
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
3.
L. V. Rozovsky 《Journal of Mathematical Sciences》2009,159(3):341-349
Let Sn = X1 + · · · + X
n
, n ≥ 1, and S
0 = 0, where X
1, X
2, . . . are independent, identically distributed random variables such that the distribution of S
n/B
n converges weakly to a nondeoenerate distribution F
α
as n → ∞ for some positive B
n
. We study asymptotic behavior of sums of the form
where
a function d(t) is continuous at [0,1] and has power decay at zero,
Bibliography: 13 titles.
Translated from Zapiski Nauchnylch Serninarov POMI, Vol. 361, 2008, pp. 109–122. 相似文献
4.
Siegfried Hörmann 《Journal of Theoretical Probability》2007,20(3):613-636
Let X
1,X
2,… be i.i.d. random variables with EX
1=0, EX
12=1 and let S
k
=X
1+⋅⋅⋅+X
k
. We study the a.s. convergence of the weighted averages
where (d
k
) is a positive sequence with D
N
=∑
k=1
N
d
k
→∞. By the a.s. central limit theorem, the above averages converge a.s. to Φ(x) if d
k
=1/k (logarithmic averages) but diverge if d
k
=1 (ordinary averages). Under regularity conditions, we give a fairly complete solution of the problem for what sequences
(d
k
) the weighted averages above converge, resp. the corresponding LIL and CLT hold. Our results show that logarithmic averaging,
despite its prominent role in a.s. central limit theory, is far from optimal and considerably stronger results can be obtained
using summation methods near ordinary (Cesàro) summation. 相似文献
5.
Fix any n≥1. Let X
1,…,X
n
be independent random variables such that S
n
=X
1+⋅⋅⋅+X
n
, and let
S*n=sup1 £ k £ nSkS^{*}_{n}=\sup_{1\le k\le n}S_{k}
. We construct upper and lower bounds for s
y
and
sy*s_{y}^{*}
, the upper
\frac1y\frac{1}{y}
th quantiles of S
n
and
S*nS^{*}_{n}
, respectively. Our approximations rely on a computable quantity Q
y
and an explicit universal constant γ
y
, the latter depending only on y, for which we prove that
${l}\displaystyle s_y\le s_y^*\le Q_y\quad\mbox{for }y>1,\\[4pt]\displaystyle \gamma_{3y/16}Q_{3y/16}-Q_1\le s_y^*\quad\mbox{for }y>\frac{32}{3},$\begin{array}{l}\displaystyle s_y\le s_y^*\le Q_y\quad\mbox{for }y>1,\\[4pt]\displaystyle \gamma_{3y/16}Q_{3y/16}-Q_1\le s_y^*\quad\mbox{for }y>\frac{32}{3},\end{array} 相似文献
6.
Li Xin Zhang 《数学学报(英文版)》2008,24(4):631-646
Let X, X1, X2,... be i.i.d, random variables with mean zero and positive, finite variance σ^2, and set Sn = X1 +... + Xn, n≥1. The author proves that, if EX^2I{|X|≥t} = 0((log log t)^-1) as t→∞, then for any a〉-1 and b〉 -1,lim ε↑1/√1+a(1/√1+a-ε)b+1 ∑n=1^∞(logn)^a(loglogn)^b/nP{max κ≤n|Sκ|≤√σ^2π^2n/8loglogn(ε+an)}=4/π(1/2(1+a)^3/2)^b+1 Г(b+1),whenever an = o(1/log log n). The author obtains the sufficient and necessary conditions for this kind of results to hold. 相似文献
7.
Precise Rates in the Law of Iterated Logarithm for the Moment of I.I.D. Random Variables 总被引:1,自引:0,他引:1
Ye JIANG Li Xin ZHANG 《数学学报(英文版)》2006,22(3):781-792
Let{X,Xn;n≥1} be a sequence of i,i.d, random variables, E X = 0, E X^2 = σ^2 〈 ∞.Set Sn=X1+X2+…+Xn,Mn=max k≤n│Sk│,n≥1.Let an=O(1/loglogn).In this paper,we prove that,for b〉-1,lim ε→0 →^2(b+1)∑n=1^∞ (loglogn)^b/nlogn n^1/2 E{Mn-σ(ε+an)√2nloglogn}+σ2^-b/(b+1)(2b+3)E│N│^2b+3∑k=0^∞ (-1)k/(2k+1)^2b+3 holds if and only if EX=0 and EX^2=σ^2〈∞. 相似文献
8.
Wel Dong LIU Zheng Yan LIN 《数学学报(英文版)》2008,24(1):59-74
Let {X, X1, X2,...} be a strictly stationaryφ-mixing sequence which satisfies EX = 0,EX^2(log2{X})^2〈∞and φ(n)=O(1/log n)^Tfor some T〉2.Let Sn=∑k=1^nXk and an=O(√n/(log2n)^γ for some γ〉1/2.We prove that limε→√2√ε^2-2∑n=3^∞1/nP(|Sn|≥ε√ESn^2log2n+an)=√2.The results of Gut and Spataru (2000) are special cases of ours. 相似文献
9.
Let β
0=0.308443… denote the Littlewood-Salem-Izumi number, i.e., the unique solution of the equation
10.
T. V. Malovichko 《Ukrainian Mathematical Journal》2008,60(11):1789-1802
We consider the solution x
ε of the equation
11.
Kong Fanchao Zhang Ying 《高校应用数学学报(英文版)》2007,22(1):78-86
In this paper the large deviation results for partial and random sums Sn-ESn=n∑i=1Xi-n∑i=1EXi,n≥1;S(t)-ES(t)=N(t)∑i=1Xi-E(N(t)∑i=1Xi),t≥0are proved, where {N(t); t≥ 0} is a counting process of non-negative integer-valued random variables, and {Xn; n ≥ 1} are a sequence of independent non-negative random variables independent of {N(t); t ≥ 0}. These results extend and improve some known conclusions. 相似文献
12.
Euler considered sums of the form
13.
V. Bentkus 《Lithuanian Mathematical Journal》2008,48(2):137-157
Let S = X
1 + ⋯ + X
n
be a sum of independent random variables such that 0 ⩽ X
k
⩽ 1 for all k. Write p = E S/n and q = 1 − p. Let 0 < t < q. In this paper, we extend the Hoeffding inequality [16, Theorem 1]
14.
Bao-huai Sheng 《应用数学学报(英文版)》2005,21(4):529-536
Let S^1-1,q≥2,be the surface of the unit sphere in the Euclidean space R^1,f(x)∈L^p(S^q-1),f(x)≥0,f absohutely unegual to 0,1≤p≤+∞,Then,it is proved in the present paper that there is a spherical harmonics PN(x) of order≤N and a constant C〉0 such that where ω(f,δ)L^p=sup 0〈t≤δ‖St(f)-f‖L^p is a kind of moduli of continuity and ^‖f-1/PN‖L^p≤Cω(f,N^-1)L^p,St(f,μ)=1/|S^q-2|Sin^2λt ∫-μμ’=t f(μ')dμ' is a translation operator. 相似文献
15.
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
16.
Hiroaki Taniguchi 《Graphs and Combinatorics》2007,23(4):455-465
Let q = 2l with l≥ 1 and d ≥ 2. We prove that any automorphism of the d-dimensional dual hyperoval
over GF(q), constructed in [3] for any (d + 1)-dimensional GF(q)-vector subspace V in GF(qn) with n≥ d + 1 and for any generator σ of the Galois group of GF(qn) over GF(q), always fixes the special member X(∞). Moreover, we prove that, in case V = GF(qd+1), two dual hyperovals
and
in PG(2d + 1,q), where σ and τ are generators of the Galois group of GF(qd+1) over GF(q), are isomorphic if and only if (1) σ = τ or (2) σ τ = id. Therefore, we have proved that, even in the case q > 2, there exist non isomorphic d-dimensional dual hyperovals in PG(2d + 1,q) for d ≥ 3. 相似文献
17.
I. K. Matsak 《Ukrainian Mathematical Journal》1998,50(9):1405-1415
We prove that
18.
Let G be a graph with n vertices, m edges and a vertex degree sequence (d
1, d
2,..., d
n
), where d
1 ≥ d
2 ≥ ... ≥ d
n
. The spectral radius and the largest Laplacian eigenvalue are denoted by ϱ(G) and μ(G), respectively. We determine the graphs with
19.
Mariko Hagita Makoto Matsumoto Fumio Natsu Yuki Ohtsuka 《Graphs and Combinatorics》2008,24(3):185-194
Let X be a finite set of q elements, and n, K, d be integers. A subset C ⊂ X
n
is an (n, K, d) error-correcting code, if #(C) = K and its minimum distance is d. We define an (n, K, d) error-correcting sequence over X as a periodic sequence {a
i
}
i=0,1,... (a
i
∈ X) with period K, such that the set of all consecutive n-tuples of this sequence form an (n, K, d) error-correcting code over X. Under a moderate conjecture on the existence of some type of primitive polynomials, we prove that there is a error correcting sequence, such that its code-set is the q-ary Hamming code with 0 removed, for q > 2 being a prime power. For the case q = 2, under a similar conjecture, we prove that there is a error-correcting sequence, such that its code-set supplemented with 0 is the subset of the binary Hamming code [2
m
− 1, 2
m
− 1 − m, 3] obtained by requiring one specified coordinate being 0.
Received: October 27, 2005. Final Version received: December 31, 2007 相似文献
20.
Henry Teicher 《Journal of Theoretical Probability》1995,8(4):779-793
Conditions are obtained for (*)E|S
T
|γ<∞, γ>2 whereT is a stopping time and {S
n=∑
1
n
,X
j
ℱ
n
,n⩾1} is a martingale and these ensure when (**)X
n
,n≥1 are independent, mean zero random variables that (*) holds wheneverET
γ/2<∞, sup
n≥1
E|X
n
|γ<∞. This, in turn, is applied to obtain conditions for the validity ofE|S
k,T
|γ<∞ and of the second moment equationES
k,T
2
=σ
2
EΣ
j=k
T
S
k−1,j−1
2
where
and {X
n
, n≥1} satisfies (**) and
,n≥1. The latter is utilized to elicit information about a moment of a stopping rule. It is also shown for i.i.d. {X
n
, n≥1} withEX=0,EX
2=1 that the a.s. limit set of {(n log logn)−k/2
S
k,n
,n≥k} is [0,2
k/2/k!] or [−2
k/2/k!] according ask is even or odd and this can readily be reformulated in terms of the corresponding (degenerate kernel)U-statistic
. 相似文献
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