A Comparison Theorem on Moment Inequalities Between Negatively Associated and Independent Random Variables

Let {X _i, 1in} be a negatively associated sequence, and let {X* _i , 1in} be a sequence of independent random variables such that X* _i and X _i have the same distribution for each i=1, 2,..., n. It is shown in this paper that Ef( ⁿ _i=1 X _i)Ef( ⁿ _i=1 X* _i ) for any convex function f on R ¹ and that Ef(max_1kn ⁿ _i=k X _i)Ef(max_1kn ^k _i=1 X* _i ) for any increasing convex function. Hence, most of the well-known inequalities, such as the Rosenthal maximal inequality and the Kolmogorov exponential inequality, remain true for negatively associated random variables. In particular, the comparison theorem on moment inequalities between negatively associated and independent random variables extends the Hoeffding inequality on the probability bounds for the sum of a random sample without replacement from a finite population.     

A New Law of the Iterated Logarithm in Rd with Application to Matrix-Normalized Sums of Random Vectors

Let (X _n , n1) be a sequence of independent centered random vectors in R ^d . We study the law of the iterated logarithm lim sup _n(2 log log B _n )^–1/2 B ^–1/2 _n S _n =1 a.s., where B _n is the covariance matrix of S _n = ⁿ _i=1 X _i , n1. Application to matrix-normalized sums of independent random vectors is given.     

Convergence of Appell Polynomials of Long Range Dependent Moving Averages in Martingale Differences

We study limit distribution of partial sums S_N,k(t) = _{s = 1} ^{[N t]} A_k(X_s) of Appell polynomials of the long-range dependent moving average process X_t> = _{i t} b_{t - i i}, where {_i} is a strictly stationary and weakly dependent martingale difference sequence, and b_i i^{d - 1} (0 < d < 1/2). We show that if k(1-2 d)<1, then suitably normalized partial sums S_N,k(t) converge in distribution to the kth order Hermite process. This result generalizes the corresponding results of Surgailis, and Avram and Taqqu obtained in the case of the i.i.d. sequence { _i}.     

Strong approximation of additive functionals

LetX ₁,X ₂,...be a sequence of i.i.d. random variables and putS ₀=0,S _n =X ₁+...+X _n . A strong approximation type result is given forA _N = _i=1 ^N f(S _i ) whereF(x),xR is a real valued function. A similar result is given for ₀ ^t g(B(s))ds. Some weak convergence type implications are also discussed.1991 Mathematics Subject Classification: Primary: 60F15, Secondary: 60J15.Supported by the Hungarian National Foundation for Scientific Research, Grant No. 1905.Supported by an NSERC Canada Grant at Carleton University.Supported by a PSC CUNY Grant No.662349.     

Stein-type improvements of confidence intervals for the generalized variance

Based on independent random matices X: p×m and S: p×p distributed, respectively, as N _pm (, I _m ) and W _p (n, ) with unknown and np, the problem of obtaining confidence interval for || is considered. Stein's idea of improving the best affine equivariant point estimator of || has been adapted to the interval estimation problem. It is shown that an interval estimator of the form |S|(b ^–1, a ^–1) can be improved by min{|S|, c|S +XX'|}(b ^–1, a ^–1) for a certain constant c depending on (a, b).     

Exceptional Sequences Determined by their Cartan Matrix

We investigate complete exceptional sequences E=(E ₁,¨,E _n ) in the derived category D ^b of finite-dimensional modules over a canonical algebra, equivalently in the derived category D ^b X of coherent sheaves on a weighted projective line, and the associated Cartan matrices C(E)=( [E _i ],[E _j ]). As a consequence of the transitivity of the braid group action on such sequences we show that a given Cartan matrix has at most finitely many realizations by an exceptional sequence E, up to an automorphism and a multi-translation (E ₁,¨,E _n )(E ₁[i ₁],¨,E _n [i _n ]) of D ^b . Moreover, we determine a bound on the number of such realizations. Our results imply that a derived canonical algebra A is determined by its Cartan matrix up to isomorphism if and only if the Hochschild cohomology of A vanishes in nonzero degree, a condition satisfied if A is representation-finite.     

Relative stability of trimmed sums

Summary This paper gives extensions of Mori's strong law for ^(r) S _n =S _n –X _n ^(1)} ...–X _n ^(r) , where S _n =X₁+X₂+...+X _n ,X _i are iidrv's and (X _i ⁿ ()) is (X _i ) arranged in decreasing order of absolute magnitude. The methods differ from Mori's. Continuity of the distribution of the X _i is assumed throughout. Necessary and sufficient conditions for relative stability (^(r) S _n /B _n ±1 a.s. for some B _n ), including a generalised condition of Spitzer's and a dominated ergodic theorem, are proved. A one-sided version of the relative stability results is also given. A theorem of Kesten's is generalised to show that if (^(r) S _n –A _n )/B _n is bounded almost surely for constants A _n ,B _n + then for some _n . A corollary to this is that if ¦ ^(r) S _n ¦/B _n is bounded away from 0 and + a.s. then ^(r) S _n is relatively stable. This generalises a result of Chow and Robbins, apart from the continuity assumption.     

Décomposition d"une théorie du potentiel en des sous-espaces elliptiques et paraboliques

Let V be a proper kernel on a measurable space (X, B), and E _V the cone of excessive functions generated by V. We give a necessary and sufficient condition to decompose the space (X, E _V ) in an ordered and countable family of subspaces (X _n, E _{V n} ), which are elliptic or parabolic. The X _i 's are finely open and measurable and form a partition of X. The kernel W = V _i on X _i is subordinate to V, and has a triangular matrix. Résumé. Soit V un noyau propre sur un espace mesurable (X, B), et E _V le cône des fonctions excessives engendré par V. Nous donnons une condition nécéssaire et suffisante pour décomposer l"espace (X, E _V ) en une famille dénombrable et ordonnée (X _n , E _{V n} ) de sous-espaces elliptiques ou paraboliques. Les X _i sont des ouverts fins mesurables et forment une partition de X. Le noyau W = V _i sur X _i est subordonné à V, et sa matrice est triangulaire.     

Interacting Brownian particles and the Wigner law

Summary In this paper, we study interacting diffusing particles governed by the stochastic differential equationsdX _j (t)= _n dB _j (t) –D _jØ_n(X ₁,...,X _n)dt,j=1, 2,...,n. Here theB _jare independent Brownian motions in ^d , and Ø _n (X ₁,...,X _n)= _{n ij} V(X _iX _j) + _ni U(X ₁). The potentialV has a singularity at 0 strong enough to keep the particles apart, and the potentialU serves to keep the particles from escaping to infinity. Our interest is in the behaviour as the number of particles increases without limit, which we study through the empirical measure process. We prove tightness of these processes in the case ofd=1,V(x)=–log|x|,U(x)=x ²/2 where it is possible to prove uniqueness of the limiting evolution and deduce that a limiting measure-valued process exists. This process is deterministic, and converges to the Wigner law ast. Some information on the rates of convergence is derived, and the case of a Cauchy initial distribution is analysed completely.Supported by SERC grant number GR/H 00444     

Iterative Averaging of Entropic Projections for Solving Stochastic Convex Feasibility Problems

The problem considered in this paper is that of finding a point which iscommon to almost all the members of a measurable family of closed convexsubsets of R₊₊ ⁿ , provided that such a point exists.The main results show that this problem can be solved by an iterative methodessentially based on averaging at each step the Bregman projections withrespect to f(x)=_i=1 ⁿx_i· ln x_i ofthe current iterate onto the given sets.     

An Invariance Principle for Triangular Arrays

Let A _{n, i} be a triangular array of sign-symmetric exchangeable random variables satisfying nE(A ² _{n, i} )1, nE(A ⁴ _{n, i} )0, n ² E(A ² _{n, 1} A ² _{n, 2})1. We show that ^[nt] _i=1 A _ni, 0t1, converges to Brownian motion. This is applied to show that if A is chosen from the uniform distribution on the orthogonal group O _n and X _n(t)=^[nt] _i=1 A _ii, then X _n converges to Brownian motion. Similar results hold for the unitary group.     

Comparison of exact constants in Kolmogorov-type inequalities for periodic and nonperiodic functions of many variables

We investigate the correlation between the constants K(ℝⁿ) and , where

Limit theorems for trimmed sums

This paper studies the heavily trimmed sums (*) _{[ns] + 1} ^[nt] X _j ⁽ⁿ⁾ , where {X _j ⁽ⁿ⁾ } _{j = 1} ⁿ are the order statistics from independent random variables {X ₁,...,X _n } having a common distributionF. The main theorem gives the limiting process of (*) as a process oft. More smoothly trimmed sums like _{j = 1} ^[nt] J(j/n)X _j ⁽ⁿ⁾ are also discussed.     

On Probability and Moment Inequalities for Supermartingales and Martingales

The probability inequality for sum S _n = _j=1 ⁿ X _j is proved under the assumption that the sequence S _k , k= , forms a supermartingale. This inequality is stated in terms of the tail probabilities P(X _j >y) and conditional variances of the random variables X _j , j= . The well-known Burkholder moment inequality is deduced as a simple consequence.     

Probabilistic characterizations of the Generalized Domain of Attraction of the multivariate normal law

For independent identically distributed random vectors,X _i , we give necessary and sufficient probabilistic conditions for their common distribution to belong to the Generalized Domain of Attraction of the multivariate normal law. The first condition says that after projecting onto any direction, , the sum of squares, _i ¹⁼¹ X _i , ², properly normalized, converges to one in probability uniformly over the unit sphere. The second condition says that max X _i , ²/ _n ⁱ⁼¹ X _i , ² converges to zero in probability uniformly over the unit sphere.     

More on P-Stable Convex Sets in Banach Spaces

We study the asymptotic behavior and limit distributions for sums S _n =b_n ^-1 _i=1 ⁿ _i,where _i, i 1, are i.i.d. random convex compact (cc) sets in a given separable Banach space B and summation is defined in a sense of Minkowski. The following results are obtained: (i) Series (LePage type) and Poisson integral representations of random stable cc sets in B are established; (ii) The invariance principle for processes S _n(t) =b_n ^-1 _i=1 ^[nt] _i, t[0, 1], and the existence of p-stable cc Levy motion are proved; (iii) In the case, where _i are segments, the limit of S _n is proved to be countable zonotope. Furthermore, if B = R ^d , the singularity of distributions of two countable zonotopes Y_{p 1}, ₁,Y_{p 2}, ₂, corresponding to values of exponents p ₁, p ₂ and spectral measures ₁, ₂, is proved if either p ₁ p ₂ or _{1 2}; (iv) Some new simple estimates of parameters of stable laws in R ^d , based on these results are suggested.     

The absolute convergence of weighted sums of dependent sequences of random variables

Summary The sum a _n X _n of a weighted series of a sequence {X _n } of identically distributed (not necessarily independent) random variables (r.v.s.) is a.s. absolutely convergent if for some in 0<1, ¦a _n ¦ < and E¦X _n ¦ < ; if a _n =z ⁿ for some ¦z¦<1 then it suffices that E(log¦X _n ¦)₊<. Examples show that these sufficient conditions are not necessary. For mutually independent {X _n } necessary conditions can be given: the a.s. absolute convergence of X _n z ⁿ (all ¦z¦<1) then implies E(log¦X _n ¦)₊ < , while if the X _n are non-negative stable r.v.s. of index , ¦a _n X _n ¦< if and only if ¦a _n ¦ < .     

∑II∑ threshold formulas

A II formula has the form, where eachL is either a variable or a negated variable. In this paper we study the computation of threshold functions by II formulas. By combining the proof of the Fredman-Komlós bound [5, 10] and a counting argument, we show that fork andn large andkn/2, every II formula computing the threshold functionT _k ⁿ has size at least exp . Fork andn large andkn ^2/3, we show that there exist II formulas for computingT _k ⁿ with size at most exp .     

Limit theorems for sums of dependent random variables

Summary In Lai and Stout [7] the upper half of the law of the iterated logarithm (LIL) is established for sums of strongly dependent stationary Gaussian random variables. Herein, the upper half of the LIL is established for strongly dependent random variables {X _i} which are however not necessarily Gaussian. Applications are made to multiplicative random variables and to f(Z _i ) where the Z _i are strongly dependent Gaussian. A maximal inequality and a Marcinkiewicz-Zygmund type strong law are established for sums of strongly dependent random variables X _i satisfying a moment condition of the form E¦S _a,n ¦^pg(n), where , generalizing the work of Serfling [13, 14].Research supported by the National Science Foundation under grant NSF-MCS-78-09179Research supported by the National Science Foundation under grant NSF-MCS-78-04014     

On the convergence of orthogonal series to +∞

For any sequence of numbers _n0, _n=1 a _n ² =, a uniformly bounded orthonormal system of continuous functions _n(x) which is complete in L₂ (0, 1), and a sequence of numbers b_n(0< b_na_n) are constructed such that _n=1 ^Emphasis> b_nn(x)= everywhere on (0, 1).Translated from Matematicheskie Zametki, Vol. 11, No. 5, pp. 499–508, May, 1972.