Harmonic renewal measures and bivariate domains of attraction in fluctuation theory

Summary Each probability measure C on a first orthant is associated with a harmonic renewal measure G. Specifically we consider (N, S _N ) the ladder (time, place) of a random walk S _n. Using bivariate G we show that when S ₁ is in a domain of attraction so is (N, S _N). This unifies and generalizes results of Sinai, Rogosin.     

Spitzer's condition and ladder variables in random walks

Summary Spitzer's condition holds for a random walk if the probabilities _n =P{ _n > 0} converge in Cèsaro mean to , where 0<<1. We answer a question which was posed both by Spitzer [12] and by Emery [5] by showing that whenever this happens, it is actually true that n converges to . This also enables us to give an improved version of a result in Doney and Greenwood [4], and show that the random walk is in a domain of attraction, without centering, if and only if the first ladder epoch and height are in a bivariate domain of attraction.     

Pinching theorems of hypersurfaces in a unit sphere

Let Mn be a complete hypersurface in Sn+1(1) with constant mean curvature. Assume that Mn has n−1 principal curvatures with the same sign everywhere. We prove that if RicM≤C₋(H), either S?S₊(H) or RicM?0 or the fundamental group of Mn is infinite, then S is constant, S=S₊(H) and Mn is isometric to a Clifford torus with . These rigidity theorems are still valid for compact hypersurface without constancy condition on the mean curvature.     

Products of independent,normally attracted random variables

Summary LetR _n denote the correlation coefficient of ann-sample of pairs (X _i ,Y _i ), each distributed as (X, Y). AssumeX andY are independent and in the domain of attraction of the Normal law. It is shown that this entailsXY being in that domain of attraction, and thatd _n R _n N(0, 1) in distribution for constantsd _n satisfying lim sup(n ^1/2/d _n )1. Examples illustrate details of these limit theorems.     

Nonabelian K-theory: The nilpotent class of K1 and general stability

A functorial filtration GL _n =S^–1L _n S⁰L _n S ⁱ L _n E _n of the general linear group GL _{n, n} 3, is defined and it is shown for any algebra A, which is a direct limit of module finite algebras, that S^–1 L _n (A)/S⁰L _n (A) is abelian, that S⁰L _n (A) S¹L _n (A) is a descending central series, and that S ⁱ L _n (A) = E _n(A) whenever i the Bass-Serre dimension of A. In particular, the K-functors k ₁ S ⁱ L _n =S ⁱ L _n /E _n are nilpotent for all i 0 over algebras of finite Bass-Serre dimension. Furthermore, without dimension assumptions, the canonical homomorphism S ⁱ L _n (A)/S ⁱ⁺¹ L _n (A)S ⁱ L _{n+ 1}(A)/S ⁱ⁺¹ L _{n + 1} (A) is injective whenever n i + 3, so that one has stability results without stability conditions, and if A is commutative then S⁰L _n (A) agrees with the special linear group SL _n (A), so that the functor S⁰L _n generalizes the functor SL _n to noncommutative rings. Applying the above to subgroups H of GL _n (A), which are normalized by E _n(A), one obtains that each is contained in a sandwich GL _n (A, ) H E _n(A, ) for a unique two-sided ideal of A and there is a descending S⁰L _n (A)-central series GL _n (A, ) S⁰L _n (A, ) S¹L _n (A, ) S ⁱ L _n (A, ) E _n(A, ) such that S ⁱ L _n (A, )=E _n(A, ) whenever i Bass-Serre dimension of A.Dedicated to Alexander Grothendieck on his sixtieth birthday     

A Donsker-Varadhan type of invariance principle

Summary LetX ₁,X ₂,h. be i.i.d. random variables in the domain of attraction of a stable lawG, and denote S_n = X₁ + X_n, L_n(, A) =n_–1 satisfiesa(n) ^–1 S _n G. Large deviation probability estimates of Donsker-Varadhan type are obtained forL _n (, ·), and these are then used to study the behavior of small values of (S _n /a(n). These latter results are analogues of Strassen's results which described the behavior of large values of (S _n /a(n)) when the limit law was Gaussian. The limiting constants are seen to depend only on the limit lawG and not on the distribution ofX ₁. The techniques used are those developed by Donsker and Varadhan in their theory of large deviations.This work was partially supported by the National Science Foundation.     

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

Operator stable laws: Series representations and domains of normal attraction

LetX,X ₁,X ₂,... be i.i.d. random vectors in ^d. The limit laws that can arise by suitable affine normalizations of the partial sums,S _n=X ₁+...+X _n, are calledoperator-stable laws. These laws are a natural extension to ^d of the stable laws on. Thegeneralized domain of attraction of [GDOA()] is comprised of all random vectorsX whose partial sums can be affinely normalized to converge to . If the linear part of the affine transformation is restricted to take the formn ^–B for some exponent operatorB naturally associated to thenX is in thegeneralized domain of normal attraction of [GDONA()]. This paper extends the theory of operator-stable laws and their domains of attraction and normal attraction.     

Affine normability of partial sums of I.I.D. random vectors: A characterization

Summary Let X, X ₁,X ₂,... be i.i.d. d-dimensional random vectors with partial sums S _n . We identify the collection of random vectors X for which there exist non-singular linear operators T _n and vectors _n ^d such that {(T _n (S _n – _n )),n>=1} is tight and has only full weak subsequential limits. The proof is constructive, providing a specific sequence {T _n }. The random vector X is said to be in the generalized domain of attraction (GDOA) of a necessarily operator-stable law if there exist {T _n } and { _n } such that (T _n (S _n – _n )). We characterize the GDOA of every operator-stable law, thereby extending previous results of Hahn and Klass; Hudson, Mason, and Veeh; and Jurek. The characterization assumes a particularly nice form in the case of a stable limit. When is symmetric stable, all marginals of X must be in the domain of attraction of a stable law. However, if is a nonsymmetric stable law then X may be in the GDOA of even if no marginal is in the domain of attraction of any law.This paper was presented in the special session on Asymptotic Behavior of Sums at the Annual IMS Meeting in Cincinnati, August 18, 1982This research was supported in part by National Science Foundation Grants MCS-81-01895 and MCS-83-01326This research was supported in part by National Science Foundation Grants MCS-80-64022 and MCS-83-01793     

On the unique finite (SI) decomposition of the Jordan operator ⊕ i=1 n S(θ) for certain inner function θ

In this paper, we have proven that for the Jordan blockS() withS() (SI), _i=1 ⁿ S() =S() ⁽ⁿ⁾ (n 1) has unique finite (SI) decomposition up to a similarity. As result, we obtain that ifV is a Volterra operator onH=L ²([0, 1]), thenV ⁽ⁿ⁾ has unique finite (SI) decomposition.This project was supported by National Natural Science Foundation of China.     

On the torsion theories of Morita equivalent rings

We generalize the well-known fact that for a pair of Morita equivalent ringsR andS their maximal rings of quotients are again Morita equivalent: If _n (M) denotes the torsion theory cogenerated by the direct sum of the firstn+1 injective modules forming part of the minimal injective resolution ofM then _n (R)= _n (S) where is the category equivalenceR-ModS-Mod. Consequently the localized ringsR _{n (R)} andS n _(S) are Morita equivalent.     

Non-symmetric association schemes of symmetric matrices

Let X_n be the set of n×n symmetric matrices over a finite field F_q,where q is a power of an odd prime.For S_1,S_2 ∈ X_n,we define (S_1,S_2)∈ R_0 iff S_1=S_2;(S_1,S_2)∈R_(r,ε)iff S_1-S_2 is congruent to where?=1 or z,z being afixed non-square element of F_q.Then X_n=(X_n,{R_0,R_(r,ε)|1≤r≤n,?=1 or z}) is a non-symmetric association scheme of class 2n on X_n.The parameters of X_n have been computed.And we also prove that X_n is commutative.     

The real spectrum of an ideal and KO-theory exact sequences

A construction in abstract real algebra is used to define invariants S _n(A) of commutative rings, with or without identity. If A=C(X) is the ring of continuous real functions on a compact space, then S _n(A) = k0^–n(X), and, for any A, S _n(A) Z[1/2]-W _n(A) Z[1/2], where the W _n(A) are the Witt groups of A. In addition, a short exact sequence of rings yields a long exact sequence of the groups S _n. The functors S _n(A) thus provide a solution of a problem proposed by Karoubi. This paper primarily deals with the exact sequences involving a ring A and an ideal I A. Work supported in part by NSF Grant DMS85-06816.     

An invariance principle for the local time of a recurrent random walk

Summary Let (S _j ) be a lattice random walk, i.e. S _j =X ₁ +...+X _j , where X ₁,X ₂,... are independent random variables with values in the integer lattice and common distribution F, and let , the local time of the random walk at k before time n. Suppose EX ₁=0 and F is in the domain of attraction of a stable law G of index > 1, i.e. there exists a sequence a(n) (necessarily of the form n ¹l(n), where l is slowly varying) such that S _n /a(n) G. Define , where c(n)=a(n/log log n) and [x] = greatest integer x. Then we identify the limit set of {g _n (, ·) n1} almost surely with a nonrandom set in terms of the I-functional of Donsker and Varadhan.The limit set is the one that Donsker and Varadhan obtain for the corresponding problem for a stable process. Several corollaries are then derived from this invariance principle which describe the asymptotic behavior of L _n (, ·) as n.Research partially supported by NSF Grant #MCS 78-01168. These results were announced at the Fifteenth European Meeting of Statisticians, Palermo, Italy (September, 1982)     

On the approximation of positive operators and the behaviour of the spectra of the approximants

LetT be a positive linear operator on the Banach latticeE and let (S _n ) be a sequence of bounded linear operators onE which converge strongly toT. Our main results are concerned with the question under which additional assumptions onS _n andT the peripheral spectra (S _n ) ofS _n converge to the peripheral spectrum (T) ofT. We are able to treat even the more general case of discretely convergent sequences of operators.     

Conditioned limit theorems for random walks with negative drift

Summary In this paper we will solve a problem posed by Iglehart. In (1975) he conjectured that if S _n is a random walk with negative mean and finite variance then there is a constant so that (S _[n.]/n ^1/2¦N>n) converges weakly to a process which he called the Brownian excursion. It will be shown that his conjecture is false or, more precisely, that if ES ₁=–a<0, ES ₁ ² <, and there is a slowly varying function L so that P(S ₁>x)x ^–q L(x) as x then (S _[n.]/n¦S _n >0) and (S _[n.]/n¦N>n) converge weakly to nondegenerate limits. The limit processes have sample paths which have a single jump (with d.f. (1–(x/a)^–q )⁺) and are otherwise linear with slope –a. The jump occurs at a uniformly distributed time in the first case and at t=0 in the second.The research for this paper was started while the author was visiting W. Vervaat at the Katholieke Universiteit in Nijmegen, Holland, and was completed while the author was at UCLA being supported by funds from NSF grant MCS 77-02121     

Solution of the neutron transport equation by means of hermite-S ∞-theory

Summary A stable numerical approximation (H -S ) is obtained through the use of Hermite's method of order (H ) in the spatial integration of the 1D neutron transport equation. The theory for =1 is applied to a one-group shielding problem.Numerical calculations show the new method to converge much faster than earlier versions ofS -theory. Comparison ofH ₁-S with the well-knownS _N -code ANISN indicates a large gain in computing time for the former.

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Zusammenfassung Durch Anwendung der Hermiteschen Integrationsmethode der Ordnung (H ) auf die Ortsintegration der 1D Neutronentransportgleichung entsteht eine numerisch stabile Approximation (H -S ). Diese Methode wurde für =1 auf ein 1 Gruppen-Abschirmungsproblem angewandt.Numerische Rechnungen zeigen die wesentlich raschere Konvergenz der Methode verglichen mit den ursprünglichen Versionen derS -Theorie. Durch Vergleich mit dem bekanntenS _N -Code ANISN wurde gezeigt, dass mit derH ₁-S -Methode ein grosser Rechenzeitgewinn erzielt wird.

     

Second-order Asymptotics on Distributions of Maxima of Bivariate Elliptical Arrays

Let {(ξni, ηni), 1 ≤ i ≤ n, n ≥ 1} be a triangular array of independent bivariate elliptical random vectors with the same distribution function as(S_1, ρ_n S_1 +(1-ρ_n~2S_2)~(1/2)), ρn∈(0, 1), where(S1, S2) is a bivariate spherical random vector. For the distribution function of radius (S_1~2+ S_2~2)~(1/2) belonging to the max-domain of attraction of the Weibull distribution, the limiting distribution of maximum of this triangular array is known as the convergence rate of ρn to 1 is given. In this paper,under the refinement of the rate of convergence of ρn to 1 and the second-order regular variation of the distributional tail of radius, precise second-order distributional expansions of the normalized maxima of bivariate elliptical triangular arrays are established.     

Some monotonicity properties of symmetric pólya densities and their exponential families

Summary In this note we observe that for independent symmetric random variables X and Y, when the pdf of X is PF, the conditional distributions of ¦Y¦ given S = X + Y form a MLR family. We then show that for a function : R ⁿR that is symmetric in each coordinate and increasing on (0, )ⁿ, E((S₁,...,S_n)¦S_n = s) is even and increasing in ¦s¦. Here S₁,...,S_n are partial sums with independent symmetric PF summands. Application is made to sequential tests that minimize the maximum expected sample size when the model is a one-parameter exponential family generated by a symmetric PF density.Work supported by NSF grants MPS 72-05082 AO2 and MCS 75-23344