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.     

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.     

The influence of variables in product spaces

LetX be a probability space and letf: X ⁿ → {0, 1} be a measurable map. Define the influence of thek-th variable onf, denoted byI _f (k), as follows: Foru=(u ₁,u ₂,…,u _n−1) ∈X ⁿ⁻¹ consider the setl _k (u)={(u ₁,u ₂,...,u _k−1,t,u _k ,…,u _n−1):t ∈X}. More generally, forS a subset of [n]={1,...,n} let the influence ofS onf, denoted byI _f (S), be the probability that assigning values to the variables not inS at random, the value off is undetermined. Theorem 1:There is an absolute constant c ₁ so that for every function f: X ⁿ → {0, 1},with Pr(f ⁻¹(1))=p≤1/2,there is a variable k so that Theorem 2:For every f: X ⁿ → {0, 1},with Prob(f=1)=1/2, and every ε>0,there is S ⊂ [n], |S|=c ₂(ε)n/logn so that I _f (S)≥1−ε. These extend previous results by Kahn, Kalai and Linial for Boolean functions, i.e., the caseX={0, 1}. Work supported in part by grants from the Binational Israel-US Science Foundation and the Israeli Academy of Science.     

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].     

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.     

Addendum to “An extension of an inequality of Hoeffding to unbounded random variables”: The non-i.i.d. case

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.     

The relation of Banach-Alaoglu theorem and Banach-Bourbaki-Kakutani-Šmulian theorem in complete random normed modules to stratification structure

Let (Ω,A,μ) be a probability space, K the scalar field R of real numbers or C of complex numbers,and (S,X) a random normed space over K with base (ω,A,μ). Denote the support of (S,X) by E, namely E is the essential supremum of the set {A ∈ A: there exists an element p in S such that X _p (ω) > 0 for almost all ω in A}. In this paper, Banach-Alaoglu theorem in a random normed space is first established as follows: The random closed unit ball S ^*(1) = {f ∈ S ^*: X ^* _f ⩽ 1} of the random conjugate space (S ^*,X ^*) of (S,X) is compact under the random weak star topology on (S ^*,X ^*) iff E∩A=: {E∩A | A ∈ A} is essentially purely μ-atomic (namely, there exists a disjoint family {A _n : n ∈ N} of at most countably many μ-atoms from E ∩ A such that E = ∪ _n=1^∞ A _n and for each element F in E ∩ A, there is an H in the σ-algebra generated by {A _n : n ∈ N} satisfying μ(FΔH) = 0), whose proof forces us to provide a key topological skill, and thus is much more involved than the corresponding classical case. Further, Banach-Bourbaki-Kakutani-Šmulian (briefly, BBKS) theorem in a complete random normed module is established as follows: If (S,X) is a complete random normed module, then the random closed unit ball S(1) = {p ∈ S: X _p ⩽ 1} of (S,X) is compact under the random weak topology on (S,X) iff both (S,X) is random reflexive and E ∩ A is essentially purely μ-atomic. Our recent work shows that the famous classical James theorem still holds for an arbitrary complete random normed module, namely a complete random normed module is random reflexive iff the random norm of an arbitrary almost surely bounded random linear functional on it is attainable on its random closed unit ball, but this paper shows that the classical Banach-Alaoglu theorem and BBKS theorem do not hold universally for complete random normed modules unless they possess extremely simple stratification structure, namely their supports are essentially purely μ-atomic. Combining the James theorem and BBKS theorem in complete random normed modules leads directly to an interesting phenomenum: there exist many famous classical propositions that are mutually equivalent in the case of Banach spaces, some of which remain to be mutually equivalent in the context of arbitrary complete random normed modules, whereas the other of which are no longer equivalent to another in the context of arbitrary complete random normed modules unless the random normed modules in question possess extremely simple stratification structure. Such a phenomenum is, for the first time, discovered in the course of the development of random metric theory.     

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 large deviations and density functions

Suppose thatX ₁,X ₂, ... is a sequence of absolutely continuous or integer valued random variables with corresponding probability density functionsf _n (x). Let {φ _n } _n=1 ^∞ be a sequence of real numbers, then necessary and sufficient conditions are given forn ⁻¹ logf _n (φ _n )-n ⁻¹ log P (X _n >φ _n )=0(1) asn→∞.     

Estimation of the variance for strongly mixing sequences

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

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 .     

Universal PI-algebras and algebras of generic matrices

Let Ω[ξ] denote the polynomial algebra (with 1) in commutative indeterminates {ie65-1}, 1 ≦i, j ≦n, 1 ≦k < ∞, over a commutative ring Ω. Thealgebra of generic matrices Ω [Y] is defined to be the Ω-subalgebra ofM _n (Ω[ξ]) generated by the matricesY _k=({ie65-2}), 1 ≦i, j ≦n, 1 ≦k < ∞. This algebra has been studied extensively by Amitsur and by Procesi in particular Amitsur has used it to construct a finite dimensional, central division algebra Ω (Y) which is not a crossed product. In this paper we shall prove, for Ω a domain, that Ω(Y) has exponentn in the Brauer group (Amitsur may already know this fact); consequently, for Ω an infinite field andn a multiple of 4, iff(X ₁, …,X _m) is a polynomial linear in all theX _i but one (similar to Formanek’s central polynomials for matrix rings) andf ² is central forM _n (Ω), thenf is central forM _n (Ω). (The existence of a polynomial not central forM _n (Ω), but whose square is central forM _n(Ω) is equivalent to every central division algebra of degreen containing a quadratic extension of its center; well-known theory immediately shows this is the case of 4‖n and 8χn.) Also, information is obtained about Ω(Y) for arbitary Ω, most notably that the Jacobson radical is the set of nilpotent elements. Partial support for this work was provided by National Science Foundation grant NSF-GP 33591.     

The structure on the real field generated by the standard part map on an o-minimal expansion of a real closed field

Let R be a sufficiently saturated o-minimal expansion of a real closed field, let be the convex hull of ℚ in R, and let st: → ℝ ⁿ be the standard part map. For X ⊆ R ⁿ define st X:= st (X ∩ ). We let ℝ_ind be the structure with underlying set ℝ and expanded by all sets st X, where X ⊆ R ⁿ is definable in R and n = 1, 2,.... We show that the subsets of ℝ ⁿ that are definable in ℝ_ind are exactly the finite unions of sets st X st Y, where X, Y ⊆ R ⁿ are definable in R. A consequence of the proof is a partial answer to a question by Hrushovski, Peterzil and Pillay about the existence of measures with certain invariance properties on the lattice of bounded definable sets in R 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

Uniformly Accurate Quantile Bounds for Sums of Arbitrary Independent Random Variables

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=sup_{1 ￡ k ￡ n}S_kS^{*}_{n}=\sup_{1\le k\le n}S_{k} . We construct upper and lower bounds for s _y and s_y^*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

Stability preservation theorems

The basic result of the paper is the main theorem worded as follows. Let {ie155-01} be a valued field such that {ie155-02} has characteristic p > 0 and let {ie155-03} be an extension of valued fields satisfying the following conditions: (i) there exists a set {ie155-04} for which {ie155-05} is a separating transcendence basis for a field {ie155-06} over F_R; (ii) Γ _R is p-pure in {ie155-07}, i.e., {ie155-08} does not contain elements of order p; (iii) there exists a set B₁ ⊂ F₀^× such that the family {ie155-09} is linearly independent in the elementary p-group {ie155-10}; (iv) F₀ is algebraic over F(B₀ ⋃ B₁). Then the property of being stable for {ie155-11} implies being stable for {ie155-12}. Supported by the Council for Grants (under RF President) and State Aid of Leading Scientific Schools (grant NSh-344.2008.1) and by RFBR (grant No. 08-01-00442-a). __________ Translated from Algebra i Logika, Vol. 47, No. 3, pp. 269–287, May–June, 2008.     

Functions of ann-dimensional Brownian motion that are Markovian

Letf be a continuous function fromR ⁿ toR and letX(t)=(X ₁ (t), …, X _n (t)) be a Brownian motion onR ⁿ . The explicit form off necessary in order to makef(X(t)) a Markov process is determined.     

Precise Rates in the Law of Iterated Logarithm for the Moment of I.I.D. Random Variables

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〈∞.     

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…     

Strongly consistent estimators of k-th order regression curves and rates of convergence

Summary Let X and Y be two jointly distributed real valued random variables, and let the conditional distribution of X given Y be either in a Lebesgue exponential family or in a discrete exponential family. Let r_k be the k-th order regression curve of Y on X. Let X _n=(X ₁,..., X_n) be a random sample of size n on X. For a subset S of the real line R, statistics based on X_n are exhibited and sufficient conditions are given under which is close to O(n ^–1/2) with probability one. To obtain this result, with uf (u known and f unknown) denoting the unconditional (on y) density of X, the problem of estimating r _k (·) is reduced to the one of estimating f ^(k) (·)/f(·) if the density is wrt the Lebesgue measure on R and f ^(k) is the k-th order derivative of f; and to the one of estimating f(·+k)/f(·) if the density is wrt the counting measure on a countable subset of R.