Estimation of the variance for strongly mixing sequences

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

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…     

The average of the values of a function at random points

The behavior of (1/N) asN→∞ is considered, wheref is a bounded measurable function on (−∞, ∞) and (S _n) _n ^=1/∞ are the partial sums of a sequence of independent and identically distributed rondom variables.     

Convergence of weighted sums of independent random variables

Let {X_k} be a sequence of i.i.d. random variables with d.f. F(x). In the first part of the paper the weak convergence of the d.f.'s F_n(x) of sums is studied, where 0<α≤2, a_nk>0, 1≤k≤m_n, and, as n→∞, bothmax _1≤k≤mn_{a nk}→0 and . It is shown that such convergence, with suitably chosen A_n's and necessarily stable limit laws, holds for all such arrays {α_nk} provided it holds for the special case α_nk=1/n, 1≤k≤n. Necessary and sufficient conditions for such convergence are classical. Conditions are given for the convergence of the moments of the sequence {F_n(x)}, as well as for its convergence in mean. The second part of the paper deals with the almost sure convergence of sums , where a_n≠0, b_n>0, andmax _1≤k≤n a_k/b_n→0. The strong law is said to hold if there are constants A_n for which S_n→0 almost surely. Let N(0)=0 and N(x) equal the number of n≥1 for which b_n/|a_n|<x if x>0. The main result is as follows. If the strong law holds,EN (|X₁|)<∞. If for some 0<p≤2, then the strong law holds with if 1≤p≤2 and A_n=0 if 0<p<1. This extends the results of Heyde and of Jamison, Orey, and Pruitt. The strong law is shown to hold under various conditions imposed on F(x), the coefficients a_n and b_n, and the function N(x). Proceedings of the Seminar on Stability Problems for Stochastic Models, Moscow, 1993.     

Moments of randomly stopped sums-revisited

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 .     

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.     

Uniform in bandwidth consistency of kernel estimators of the tail index

We consider a class of kernel estimators [^(t)]_n,h\hat{\tau}_{n,h} of the tail index of a Pareto-type distribution, which generalizes and includes the classical Hill estimator [^(a)]_n,k\hat{a}_{n,k}. It is well-known that [^(a)]_n,k\hat{a}_{n,k} is a consistent estimator of the tail index if and only if k→ ∞ and k/n→0. Under suitable assumptions on the kernel, [^(t)] _n,h\hat{\tau} _{n,h} is consistent whenever the bandwidth is taken to be a sequence of non-random numbers satisfying h _n →0 and nh _n → ∞. We extend this result and prove the consistency uniformly over a certain range of bandwidths. This permits the treatment of estimators of the tail index based upon data-dependent bandwidths, which are often used in practice. In the process, we establish a uniform in bandwidth result for kernel-type regression estimators with a fixed design, which will likely be of separate interest.     

Minimal rates of entropy convergence for completely ergodic systems

If (X,T) is a completely ergodic system, then there exists a positive monotone increasing sequence {a _n } _n ^1/∞ with lim _n →∞a _n =∞ and a positive concave functiong defined on [1, ∞) for whichg(x)/x ² isnot integrable such that for all nontrivial partitions α ofX into two sets.     

Penultimate limiting forms in extreme value theory

Summary Let {X _n}_n≧1 be a sequence of independent, identically distributed random variables. If the distribution function (d.f.) ofM _n=max (X ₁,…,X _n), suitably normalized with attraction coefficients {α_n}_n≧1(α_n>0) and {b _n}_n≧1, converges to a non-degenerate d.f.G(x), asn→∞, it is of interest to study the rate of convergence to that limit law and if the convergence is slow, to find other d.f.'s which better approximate the d.f. of(M _n−b_n)/a_n thanG(x), for moderaten. We thus consider differences of the formF ⁿ(a_nx+b_n)−G(x), whereG(x) is a type I d.f. of largest values, i.e.,G(x)≡Λ(x)=exp (-exp(−x)), and show that for a broad class of d.f.'sF in the domain of attraction of Λ, there is a penultimate form of approximation which is a type II [Ф _α(x)=exp (−x^−α), x>0] or a type III [Ψ _α(x)= exp (−(−x)^α), x<0] d.f. of largest values, much closer toF ⁿ(a_nx+b_n) than the ultimate itself.     

A note on asymptotics of linear combinations of iid random variables

Let X₁,X₂, ... be iid random variables, and let a _n = (a ₁,n, ..., a _n,n ) be an arbitrary sequence of weights. We investigate the asymptotic distribution of the linear combination $ S_{a_n } $ S_{a_n } = a _1,n X ₁ + ... + a _n,n X _n under the natural negligibility condition lim _n→∞ max{|a _k,n |: k = 1, ..., n} = 0. We prove that if $ S_{a_n } $ S_{a_n } is asymptotically normal for a weight sequence a _n , in which the components are of the same magnitude, then the common distribution belongs to $ \mathbb{D} $ \mathbb{D} (2).     

On extending backwards positive definite sequences

The two-sided Hamburger moment problem¹, also called the strong one [4], has been extensively studied in recent years in connection with rational approximation. We propose to consider the question of when a sequence, say {a _n } _n=0 can be extended backwards so that the resulting sequence {a _n } _n=–N has an integral representation of the Hamburger type. This was settled (without any proof) under different circumstances in [6]. Here we wish to discuss this completely, as well as the possibility of extending {a _n } _n=0 to {a _n } _n– .     

Strongly correlated random fields as observed by a random walker

Summary We are given a random walk S ₁, S ₂, ... on ℤ^ν, ν≧1, and a strongly correlated stationary random field ξ(x), xεℤ^ν, which is independent of the random walk. We consider the field as observed by a random walker and study partial sums of the form . It is assumed that the law corresponding to the random walk belongs to the domain of attraction of a non-degenerate stable law of index β, 0<β≦2. We further suppose that the field ξ satisfies the non-central limit theorem of Dobrushin and Major with a scaling factor . Under the assumption αk<β it is shown that converges weakly as n→∞ to a self-similar process {Δ _t , t≧0} with stationary increments, and Δ _t can be represented as a multiple Wiener-It? integral of a random function. This extends the noncentral limit theorem of Dobrushin and Major and yields a new example of a self-similar process with stationary increments.     

Continued radicals

If a ₁,a ₂,…,a _n are nonnegative real numbers and , then f ₁○f ₂○⋅⋅⋅○f _n (0) is a nested radical with terms a ₁,…,a _n . If it exists, the limit as n→∞ of such an expression is a continued radical. We consider the set of real numbers S(M) representable as a continued radical whose terms a ₁,a ₂,… are all from a finite set M. We give conditions on the set M for S(M) to be (a) an interval, and (b) homeomorphic to the Cantor set.      

Singular solutions of some nonlinear parabolic equations with spatially inhomogeneous absorption

We study the limit behaviour of solutions of with initial data k δ ₀ when k → ∞, where h is a positive nondecreasing function and p > 1. If h(r) = r ^β , β > N(p − 1) − 2, we prove that the limit function u _∞ is an explicit very singular solution, while such a solution does not exist if β ≤  N(p − 1) − 2. If lim inf _{r→ 0} r ² ln (1/h(r))  >  0, u _∞ has a persistent singularity at (0, t) (t ≥  0). If , u _∞ has a pointwise singularity localized at (0, 0).     

On Some Properties of Randomly Indexed Sequences of Random Elements

Let be a probability space with a nonatomic measure P and let (S,ρ) be a separable complete metric space. Let {N _n ,n≥1} be an arbitrary sequence of positive-integer valued random variables. Let {F _k ,k≥1} be a family of probability laws and let X be some random element defined on and taking values in (S,ρ). In this paper we present necessary and sufficient conditions under which one can construct an array of random elements {X _n,k ,n,k≥1} defined on the same probability space and taking values in (S,ρ), and such that , and moreover as  n→∞. Furthermore, we consider the speed of convergence to X as n→∞.      

Limiting distribution of sums of nonnegative stationary random variables

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=X_n,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=f_n(X_j). 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.     

Two renewal theorems for general random walks tending to infinity

Summary. Necessary and sufficient conditions for the existence of moments of the first passage time of a random walk S _n into [x, ∞) for fixed x≧ 0, and the last exit time of the walk from (−∞, x], are given under the condition that S _n →∞ a.s. The methods, which are quite different from those applied in the previously studied case of a positive mean for the increments of S _n , are further developed to obtain the “order of magnitude” as x→∞ of the moments of the first passage and last exit times, when these are finite. A number of other conditions of interest in renewal theory are also discussed, and some results for the first time for which the random walk remains above the level x on K consecutive occasions, which has applications in option pricing, are given. Received: 18 September 1995/In revised form: 28 February 1996     

A boundedness theorem for a certain nth order differential equation

Summary Consider the equation(1.2) in which a₁, a₂, ..., a_n−1 are constants and the functions f_n(x) and p(t) (both continuous) together with are all bodnded. A recent investigation by Reissig[1] shows that if f_n(x)sgn x>0 (|x|≥h>0) then subject to certain conditions, which are stated explicitly in[1], the solutions of such an equation(1.2) are all ultimately bounded The object of the psesent paper is to generalize that result to the equation(1.1) in which φ_n is a bounded function depending on all the variables shown, and each coefficient ϕ_i (i=2,3,..., n−1) satisfies as |ζ|→∞ for some constant a_i. Entrata in Redazione il 20 agosto 1970.     

非Lipschitz条件下严格伪压缩映射不动点的逼近

Abstract. Without the Lipschitz assumption and boundedness of K in arbitrary Banach spaces,the Ishikawa iteration     

Some finitely additive versions of the strong law of large numbers

LetX be a non-empty set,H= X{su\t8, \gs = \lj{in1}x\lj{in2}x,σ=γ ₁×γ ₂×… be an independent strategy onH, and {Y _n} be a sequence of coordinate mappings onH. The following strong law in a finitely additive setting is proved: For some constantr≧1, if \GS _n=1 ^\t8 {\GS(\vbY _n \vb^2r )n ¹⁺ⁿ < \t8 andσ(Y _n)=0 for alln=1, 2, …, then \1n\gS{inj-1}/{sun} Y{inj}Y _jconverges to 0 withσ-measure 1 asn → ∞. The theorem is a generalization of Chung’s strong law in a coordinate representation process. Finally, Kolmogorov’s strong law in a finitely additive setting is proved by an application of the theorem. This research was based in part on the author’s doctoral dissertation submitted to the University of Minnesota, and was written with the partial support of the United States Army Grant DA-ARO-D-31-124-70-G-102.