Large deviations for sums of independent random variables with dominatedly varying tails

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.     

Moderate Deviations for Random Sums of Heavy-Tailed Random Variables

Let {Xn;n≥ 1} be a sequence of independent non-negative random variables with common distribution function F having extended regularly varying tail and finite mean μ = E（X1） and let {N（t）; t ≥0} be a random process taking non-negative integer values with finite mean λ（t） = E（N（t）） and independent of {Xn; n ≥1}. In this paper, asymptotic expressions of P（（X1 ＋… ＋XN（t）） -λ（t）μ 〉 x） uniformly for x ∈[γb（t）, ∞） are obtained, where γ〉 0 and b（t） can be taken to be a positive function with limt→∞ b（t）/λ（t） = 0.     

A contribution to large deviations for heavy-tailed random sums

In this paper we consider the large deviations for random sums , whereX _n,n⩾1 are independent, identically distributed and non-negative random variables with a common heavy-tailed distribution function F, andN(t), t⩾0 is a process of non-negative integer-valued random variables, independent ofX _n,n⩾1. Under the assumption that the tail of F is of Pareto’s type (regularly or extended regularly varying), we investigate what reasonable condition can be given onN(t), t⩾0 under which precise large deviation for S( t) holds. In particular, the condition we obtain is satisfied for renewal counting processes.     

Large deviation results for generalized compound negative binomial risk models

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.     

Joint asymptotic distribution of exceedances point process and partial sum of stationary Gaussian sequence

Let {X _i } _i=1^∞ be a standardized stationary Gaussian sequence with covariance function r(n) = EX ₁ X _n+1, S _n = Σ _i=1 ⁿ X _i , and $\bar X_n = \tfrac{{S_n }} {n} $\bar X_n = \tfrac{{S_n }} {n} . And let N _n be the point process formed by the exceedances of random level $(\tfrac{x} {{\sqrt {2\log n} }} + \sqrt {2\log n} - \tfrac{{\log (4\pi \log n)}} {{2\sqrt {2\log n} }})\sqrt {1 - r(n)} + \bar X_n $(\tfrac{x} {{\sqrt {2\log n} }} + \sqrt {2\log n} - \tfrac{{\log (4\pi \log n)}} {{2\sqrt {2\log n} }})\sqrt {1 - r(n)} + \bar X_n by X ₁,X ₂,…, X _n . Under some mild conditions, N _n and S _n are asymptotically independent, and N _n converges weakly to a Poisson process on (0,1].     

Complete moment and integral convergence for sums of negatively associated random variables

For a sequence of identically distributed negatively associated random variables {Xn; n ≥ 1} with partial sums Sn = ∑i=1^n Xi, n ≥ 1, refinements are presented of the classical Baum-Katz and Lai complete convergence theorems. More specifically, necessary and sufficient moment conditions are provided for complete moment convergence of the form ∑n≥n0 n^r-2-1/pq anE（max1≤k≤n｜Sk｜^1/q-∈bn^1/qp）^＋〈∞to hold where r 〉 1, q 〉 0 and either n0 = 1,0 〈 p 〈 2, an = 1,bn = n or n0 = 3,p = 2, an = 1 （log n） ^1/2q, bn=n log n. These results extend results of Chow and of Li and Spataru from the indepen- dent and identically distributed case to the identically distributed negatively associated setting. The complete moment convergence is also shown to be equivalent to a form of complete integral convergence.     

A limit theorem for the moments of sums of independent random variables

Forn≧1, letS _n=ΣX _n,i (1≦i≦r _n <∞), where the summands ofS _n are independent random variables having medians bounded in absolute value by a finite number which is independent ofn. Letf be a nonnegative function on (− ∞, ∞) which vanishes and is continuous at the origin, and which satisfies, for some for allt≧1 and all values ofx. Theorem.For centering constants c _n,let S _n − c _n converge in distribution to a random variable S. (A)In order that Ef(S_n − c_n) converge to a limit L, it is necessary and sufficient that there exist a common limit (B)If L exists, then L<∞ if and only if R<∞, and when L is finite, L=Ef(S)+R. Applications are given to infinite series of independent random variables, and to normed sums of independent, identically distributed random variables.     

On the chromatic number of random geometric graphs

Given independent random points X ₁,...,X _n ∈ℝ ^d with common probability distribution ν, and a positive distance r=r(n)>0, we construct a random geometric graph G _n with vertex set {1,..., n} where distinct i and j are adjacent when ‖X _i −X _j ‖≤r. Here ‖·‖ may be any norm on ℝ ^d , and ν may be any probability distribution on ℝ ^d with a bounded density function. We consider the chromatic number χ(G _n ) of G _n and its relation to the clique number ω(G _n ) as n→∞. Both McDiarmid [11] and Penrose [15] considered the range of r when $r \ll \left( {\tfrac{{\ln n}} {n}} \right)^{1/d}$r \ll \left( {\tfrac{{\ln n}} {n}} \right)^{1/d} and the range when $r \gg \left( {\tfrac{{\ln n}} {n}} \right)^{1/d}$r \gg \left( {\tfrac{{\ln n}} {n}} \right)^{1/d}, and their results showed a dramatic difference between these two cases. Here we sharpen and extend the earlier results, and in particular we consider the ‘phase change’ range when $r \sim \left( {\tfrac{{t\ln n}} {n}} \right)^{1/d}$r \sim \left( {\tfrac{{t\ln n}} {n}} \right)^{1/d} with t>0 a fixed constant. Both [11] and [15] asked for the behaviour of the chromatic number in this range. We determine constants c(t) such that $\tfrac{{\chi (G_n )}} {{nr^d }} \to c(t)$\tfrac{{\chi (G_n )}} {{nr^d }} \to c(t) almost surely. Further, we find a “sharp threshold” (except for less interesting choices of the norm when the unit ball tiles d-space): there is a constant t ₀>0 such that if t≤t ₀ then $\tfrac{{\chi (G_n )}} {{\omega (G_n )}}$\tfrac{{\chi (G_n )}} {{\omega (G_n )}} tends to 1 almost surely, but if t>t ₀ then $\tfrac{{\chi (G_n )}} {{\omega (G_n )}}$\tfrac{{\chi (G_n )}} {{\omega (G_n )}} tends to a limit >1 almost surely.     

Asymptotics of random sums of negatively dependent random variables in the presence of dominatedly varying tails

Let X ₁ , X ₂ , . . . be a sequence of negatively dependent and identically distributed random variables, and let N be a counting random variable independent of X _i ’s. In this paper, we study the asymptotics for the tail probability of the random sum S_N = ?_{k = 1}^N X_k {S_N} = \sum\nolimits_{k = 1}^N {{X_k}} in the presence of heavy tails. We consider the following three cases: (i) P(N > x) = o(P(X ₁  > x)), and the distribution function (d.f.) of X ₁ is dominatedly varying; (ii) P(X ₁  > x) = o(P(N > x)), and the d.f. of N is dominatedly varying; (iii) the tails of X ₁ and N are asymptotically comparable and dominatedly varying.     

Onp-absolutely summing constants of banach spaces

Given 1≦p<∞ and a real Banach spaceX, we define thep-absolutely summing constantμ _p(X) as inf{Σ _i ^=1/m |x*(x _i)|^p p Σ _i ^=1/m ‖x _i‖^p p]¹ p}, where the supremum ranges over {x*∈X*; ‖x*‖≤1} and the infimum is taken over all sets {x ₁,x ₂, …,x _m} ⊂X such that Σ _i ^=1/m ‖x _i‖>0. It follows immediately from [2] thatμ _p(X)>0 if and only ifX is finite dimensional. In this paper we find the exact values ofμ _p(X) for various spaces, and obtain some asymptotic estimates ofμ _p(X) for general finite dimensional Banach spaces. This is a part of the author’s Ph.D. Thesis prepared at the Hebrew University of Jerusalem, under the supervision of Prof. A. Dvoretzky and Prof. J. Lindenstrauss.     

The ratio of the extreme to the sum in a random sequence

For X ₁ , X ₂ , ..., X _n a sequence of non-negative independent random variables with common distribution function F(t), X _(n) denotes the maximum and S _n denotes the sum. The ratio variate R _n  = X _(n) / S _n is a quantity arising in the analysis of process speedup and the performance of scheduling. O’Brien (J. Appl. Prob. 17:539–545, 1980) showed that as n → ∞, R _n →0 almost surely iff is finite. Here we show that, provided either (1) is finite, or (2) 1 − F (t) is a regularly varying function with index ρ < − 1, then . An integral representation for the expected ratio is derived, and lower and upper asymptotic bounds are developed to obtain the result. Since is often known or estimated asymptotically, this result quantifies the rate of convergence of the ratio’s expected value. The result is applied to the performance of multiprocessor scheduling.      

Precise rates in the law of the logarithm for the moment convergence in Hilbert spaces

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 ∑.     

Laws of large numbers for weighted sums of random variables and random elements

Consider the weighted sums of a sequence {X _n} of independent random variables or random elements inD [0,1]. For convergence ofS _n in probability and with probability one, in [2],[3] etc., the following stronger condition is required: {X _n} is uniformly bounded by a random variableX,i.e.P(¦X _n¦x)P(¦X¦x) for allx>0. Our paper aims at trying to drop this restriction.The Project supported by National Natural Science Foundation of China     

On normal approximation of discounted and strongly mixing random variables

We estimate the difference for bounded functions h: ℝ → ℝ satisfying the Lipschitz condition, where Z _v = B _v ⁻¹ ∑ _i=0 ^∞ v ⁱ X _i and with discount factor ν such that 0 < ν < 1. Here {X _n , n ≥ 0} is a sequence of strongly mixing random variables with , and N is a standard normal random variable. In a particular case, the obtained upper bounds are of order O((1 − ν)^1/2). Published in Lietuvos Matematikos Rinkinys, Vol. 47, No. 3, pp. 399–409, July–September, 2007. The research was partially supported by the Lithuanian State Science and Studies Foundation, grant No. T-15/07.     

Weak Convergence of a Nonlinear Functional of a Stationary Gaussian Process. Application to the Local Time

Let {X _t : 0 ≦ t ≦ 1} be a centered stationary Gaussian process, with correlation function satisfying the condition ρ(t) = 1 − t ^β L(t), 0 < β < 2, and let L be a slowly varying function at zero. Observing the process at points i/N, i = 0,1,..., N and considering |X _i/N − X _(i-1)/N | ^p with p > 0, we study the properties of the Donsker line associated with p-th order variations . We also study the relationship between the number of crossings of a regularization of the initial process and the local time of the initial process. The results depend on the values of β. This revised version was published online in June 2006 with corrections to the Cover Date.     

Uniform estimates for order statistics and Orlicz functions

We establish uniform estimates for order statistics: Given a sequence of independent identically distributed random variables ξ ₁, … , ξ _n and a vector of scalars x = (x ₁, … , x _n ), and 1 ≤ k ≤ n, we provide estimates for \mathbb E   k-min_{1 ￡ i ￡ n} |x_ix_i|{\mathbb E \, \, k-{\rm min}_{1\leq i\leq n} |x_{i}\xi _{i}|} and \mathbb E k-max_{1 ￡ i ￡ n}|x_ix_i|{\mathbb E\,k-{\rm max}_{1\leq i\leq n}|x_{i}\xi_{i}|} in terms of the values k and the Orlicz norm ||y_x||_M{\|y_x\|_M} of the vector y _x  = (1/x ₁, … , 1/x _n ). Here M(t) is the appropriate Orlicz function associated with the distribution function of the random variable |ξ ₁|, G(t) = \mathbb P ({ |x₁| ￡ t}){G(t) =\mathbb P \left(\left\{ |\xi_1| \leq t\right\}\right)}. For example, if ξ ₁ is the standard N(0, 1) Gaussian random variable, then G(t) = ?{\tfrac2p}ò₀^t e^-\fracs22ds {G(t)= \sqrt{\tfrac{2}{\pi}}\int_{0}^t e^{-\frac{s^{2}}{2}}ds }  and M(s)=?{\tfrac2p}ò₀^se^-\frac12t2dt{M(s)=\sqrt{\tfrac{2}{\pi}}\int_{0}^{s}e^{-\frac{1}{2t^{2}}}dt}. We would like to emphasize that our estimates do not depend on the length n of the sequence.     

Central limit theorem for integrated square error of kernel estimators of spherical density

LetX ₁,…,X _n be iid observations of a random variableX with probability density functionf(x) on the q-dimensional unit sphere Ω_q in R^q+1,q ⩾ 1. Let be a kernel estimator off(x). In this paper we establish a central limit theorem for integrated square error off _n under some mild conditions.     

Almost sure central limit theorems for random functions

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.     

On antipodal Euclidean tight (2e + 1)-designs

Neumaier and Seidel (1988) generalized the concept of spherical designs and defined Euclidean designs in ℝ ⁿ . For an integer t, a finite subset X of ℝ ⁿ given together with a weight function w is a Euclidean t-design if holds for any polynomial f(x) of deg(f)≤ t, where {S _i , 1≤ i ≤ p} is the set of all the concentric spheres centered at the origin that intersect with X, X _i = X∩ S _i , and w:X→ ℝ_{> 0}. (The case of X⊂ S ⁿ⁻¹ with w≡ 1 on X corresponds to a spherical t-design.) In this paper we study antipodal Euclidean (2e+1)-designs. We give some new examples of antipodal Euclidean tight 5-designs. We also give the classification of all antipodal Euclidean tight 3-designs, the classification of antipodal Euclidean tight 5-designs supported by 2 concentric spheres.     

On generalized shift bases for the wiener disc algebra

LetW(D) denote the set of functionsf(z)=Σ _n=0 ^∞ A _n Z ⁿ a _nzⁿ for which Σ_n=0 ^∞|a _n |<+∞. Given any finite set lcub;f _i (z)rcub; _i=1 ⁿ inW(D) the following are equivalent: (i) The generalized shift sequence lcub;f ₁(z)z ^kn ,f ₂(z)z ^kn+1, …,f _n (z)z ^(k+1)n−1rcub; _k=0 ^∞ is a basis forW(D) which is equivalent to the basis lcub;z ^m rcub; _m=0 ^∞ . (ii) The generalized shift sequence is complete inW(D), (iii) The function has no zero in |z|≦1, wherew=e ^2πiti /n.