On the law of the iterated logarithm for weighted sums of independent random variables in a Banach space

Assume that (X _n) are independent random variables in a Banach space, (b _n) is a sequence of real numbers, S_n= ₁ ⁿ b_iX_i, and B_n= ₁ ⁿ b _i ² . Under certain moment restrictions imposed on the variablesX _n, the conditions for the growth of the sequence (b_n) are established, which are sufficient for the almost sure boundedness and precompactness of the sequence (S_n/B _n ln ln B_n)^1/2).Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 45, No. 9, pp. 1225–1231, September, 1993.     

Estimation of the variance for strongly mixing sequences

Abstract. Let {Xn,n≥1} be a stationary strongly mixing random sequence satisfying EX1=u,     

Asymptotic distribution of sequences with random indices

A general form is determined for the limit distribution function of a sequence of random vectors with random indices (S₁(N _n ⁽¹⁾ ) ..., Sr(N_n ^(r)) in the case when sequences (S₁(n), ..., S_r(n)) and (N _n ⁽¹⁾ , ..., N _n ^(r) ) for appropriate normalization have a nonsingular joint limit distribution.Translated from Matematicheskie Zametki, Vol. 6, No. 6, pp. 705–712, December, 1969.     

On the almost sure central limit theorem for randomly indexed sums

Let {S_n, n ≥ 1} be partial sums of independent identically distributed random variables. The almost sure version of CLT is generalized on the case of randomly indexed sums {S_Nn, n ≥ 1}, where {N_n, n ≥ 1} is a sequence of positive integer‐valued random variables independent of {S_n, n ≥ 1}. The affects of nonrandom centering and norming are considered too (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Strong Approximation Theorems for Sums of Random Variables when Extreme Terms are Excluded

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     

Moments of the maximum of normed partial sums of ρ^--mixing random variables

Let {Xn,n ≥ 1} be a sequence of identically distributed ρ^--mixing random variables and set Sn =∑i^n=1 Xi,n ≥ 1,the suffcient and necessary conditions for the existence of moments of supn≥1 ｜Sn/n^1/r｜^p（0 〈 r 〈 2,p 〉 0） are given,which are the same as that in the independent case.     

On the strong law of large numbers for dependent random variables

Let {X _n} _n ^=1/∞ be a sequence of random variables with partial sumsS _n, and let {ie241-1} be the σ-algebra generated byX ₁,…,X _n. Letf be a function fromR toR and suppose {ie241-2}. Under conditions off and moment conditions on theX' _ns, we show thatS _n/n converges a.e. (almost everywhere). We give several applications of this result. Research supported by N.S.F. Grant MCS 77-26809     

On extremal sort sequences

A sort sequence S_n is a sequence of all unordered pairs of indices inI _n = {1, 2, ..., n}. With a sort sequenceS_n = (s₁ ,s₂ ,...,s_{( 2ⁿ )} )S_n = (s_1 ,s_2 ,...,s_{\left( {_2^n } \right)} ), one can associate a predictive sorting algorithm A(S_n). An execution of the algorithm performs pairwise comparisons of elements in the input setX in the order defined by the sort sequence S_n 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(S_n) — 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(S_n) — the minimum number of active predictions in S_n over all input orderings.     

The Weak Convergence for Functions of Negatively Associated Random Variables

Let {X_n, n1} be a sequence of stationary negatively associated random variables, S_j(l)=∑^l_i=1 X_j+i, S_n=∑ⁿ_i=1 X_i. Suppose that f(x) is a real function. Under some suitable conditions, the central limit theorem and the weak convergence for sums are investigated. Applications to limiting distributions of estimators of Var S_n are also discussed.     

A local limit theorem for random walks conditioned to stay positive

We consider a real random walk S_n=X₁+...+X_n attracted (without centering) to the normal law: this means that for a suitable norming sequence a_n we have the weak convergence S_n/a_n⇒ϕ(x)dx, ϕ(x) being the standard normal density. A local refinement of this convergence is provided by Gnedenko's and Stone's Local Limit Theorems, in the lattice and nonlattice case respectively. Now let denote the event (S₁>0,...,S_n>0) and let S_n⁺ denote the random variable S_n conditioned on : it is known that S_n⁺/a_n ↠ ϕ⁺(x) dx, where ϕ⁺(x):=x exp (−x²/2)1_(x≥0). What we establish in this paper is an equivalent of Gnedenko's and Stone's Local Limit Theorems for this weak convergence. We also consider the particular case when X₁ has an absolutely continuous law: in this case the uniform convergence of the density of S_n⁺/a_n towards ϕ⁺(x) holds under a standard additional hypothesis, in analogy to the classical case. We finally discuss an application of our main results to the asymptotic behavior of the joint renewal measure of the ladder variables process. Unlike the classical proofs of the LLT, we make no use of characteristic functions: our techniques are rather taken from the so–called Fluctuation Theory for random walks.     

A class of limit distributions for maximum cumulative sum

Let X₁, X₂, ... be a sequence of independent identically distributed random variables with zero mathematical expectation and finite variances. S_o=0 and S_n=∑ _i=1 ⁿ X_i. 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).     

Stable Laws and Products of Positive Random Matrices

Let S be the multiplicative semigroup of q×q matrices with positive entries such that every row and every column contains a strictly positive element. Denote by (X _n ) _n≥1 a sequence of independent identically distributed random variables in S and by X ⁽ⁿ⁾=X _n ⋅⋅⋅ X ₁,  n≥1, the associated left random walk on S. We assume that (X _n ) _n≥1 satisfies the contraction property

负相依随机变量的自正则化中心极限定理与部分和的方差估计

§ 1　 IntroductionWe firstintroduce some concepts.Random variables X and Y are called negative dependent ( ND) if for any pair ofmonotonically non-decresing functions f and g,Cov{ f( X) ,g( Y) }≤ 0 .Clearly itis equivalenttoP( X≤ x,Y≤ y)≤ P( X≤ x) P( Y≤ y)for all x,y∈R.A random sequence{ Xi,i≥ 1 } is said to be negative quadrant dependent( NQD) if any pairof variables Xi,Xj( i≠j) are ND.A sequence of random variables{ Xi,i≥ 1 } is said to be linear negative quadrand depend…     

An almost sure invariance principle for trimmed sums of random vectors

Let {X _n ; n ≥ 1} be a sequence of independent and identically distributed random vectors in ℜ ^p with Euclidean norm |·|, and let X _n ^(r) = X _m if |X _m | is the r-th maximum of {|X _k |; k ≤ n}. Define S _n = Σ _k≤n X _k and ^(r) S _n − (X _n ⁽¹⁾ + ... + X _n ^(r)). In this paper a generalized strong invariance principle for the trimmed sums ^(r) S _n is derived.     

The law of the iterated logarithm for subsequences and characterization of the cluster set ofS n k/(2n k log logn k)1/2 in Banach spaces

LetS _n=X ₁+...+X _n, where {X _n;n=1, 2,...} is a sequence of i.i.d. random vectors with values in a Banach space and let be any infinite set of positive integers. In this paper we obtain the bounded and the compact laws of the iterated logarithm for {S _n;n}. We characterize the cluster set of {S _n/(2n log logn)^1/2;n}, for random vectors with mean zero, weak second moment, and satisfying certain additional conditions. Special situations, like type 2 Banach spaces and the real-valued case are also considered.     

Uniformly Conditioned Bases of Spectral Subspaces

A condition number of an ordered basis of a finite-dimensional normed space is defined in an intrinsic manner. This concept is extended to a sequence of bases of finite-dimensional normed spaces, and is used to determine uniform conditioning of such a sequence. We address the problem of finding a sequence of uniformly conditioned bases of spectral subspaces of operators of the form T _n  = S _n  + U _n , where S _n is a finite-rank operator on a Banach space and U _n is an operator which satisfies an invariance condition with respect to S _n . This problem is reduced to constructing a sequence of uniformly conditioned bases of spectral subspaces of operators on ? ^n×1. The applicability of these considerations in practical as well as theoretical aspects of spectral approximation is pointed out.     

On the strong law of large numbers for sequences of random variables without the independence condition

New sufficient conditions for the applicability of the strong law of large numbers to a sequence of dependent random variables X ₁, X ₂, …, with finite variances are established. No particular type of dependence between the random variables in the sequence is assumed. The statement of the theorem involves the classical condition Σ _n ^∞ (log₂ n)²/n ² < ∞, which appears in various theorems on the strong law of large numbers for sequences of random variables without the independence condition.     

Convergence of Point Processes with Weakly Dependent Points

For each n≥1, let {X _j,n }_1≤j≤n be a sequence of strictly stationary random variables. In this article, we give some asymptotic weak dependence conditions for the convergence in distribution of the point process $N_{n}=\sum_{j=1}^{n}\delta_{X_{j,n}}For each n≥1, let {X _j,n }_1≤j≤n be a sequence of strictly stationary random variables. In this article, we give some asymptotic weak dependence conditions for the convergence in distribution of the point process N_n=?_j=1ⁿd_Xj,nN_{n}=\sum_{j=1}^{n}\delta_{X_{j,n}} to an infinitely divisible point process. From the point process convergence we obtain the convergence in distribution of the partial sum sequence S _n =∑ _j=1 ⁿ X _j,n to an infinitely divisible random variable whose Lévy measure is related to the canonical measure of the limiting point process. As examples, we discuss the case of triangular arrays which possess known (row-wise) dependence structures, like the strong mixing property, the association, or the dependence structure of a stochastic volatility model.     

Ein Null-Zwei-Gesetz auf lokalkompakten abelschen Halbgruppen

Let λ, μ be regular probability measures on a locally compact abelian semigroup S, λ * μ the convolution of λ and μ, λⁿ the n^th iterated convolution of λ, δ_x the point measure of x?S. We study the totalvariation of λⁿ–δ_x * λⁿ for n → ∞. We shall see that for a certain class of semigroups the limit of this sequence is either 0 or 2.