On independent statistical decision problems and products of diffusions

Summary LetZ _t be a null recurrent diffusion on ^p with generatorG=(1/r)·(r ) for smooth positiver. This note constructs an independent recurrent diffusionZ _ton ¹ such that (Z _t, Z_t)is transient in ^p+1. This resolves negatively an old question in simultaneous estimation: Is there an admissible but not Bayes estimator(X) of the mean of a multivariate normal distribution for quadratic loss with the property: for every admissible (X), whereX is normal and independent ofX, (, ) remains admissible in the combined problem obtained by summing the component losses?Work supported by NSF at Mathematical Sciences Research Institute, Berkeley     

An infinite dimensional analogue of the Albeverio-Röckner's theorem on Fukushima's conjecture

LetX be the solution of the SDE:dX _t = (X _t)dB _t +b(X _t)dt, with andb C _b (R) such that >0 for some constant , andB a real Brownian motion. Let be the law ofX onE=C([0, 1],R) andk E* – {0}, whereE* is the topological dual space ofE. Consider the classical form: _k (u, v)=u / kv / kd, whereu andv are smooth functions onE. We prove that, if _k is closable for anyk in a dense subset ofE* and if the smooth functions are contained in the domain of the generator of the closure of _k , must be a constant function.     

Note on angle-preserving mappings

Summary Let (V, K, q) be aq-regular metric vector space over a commutative field with quadratic formq and letA(V, K, q) be the corresponding affine-metric space. A metric collineation ofA(V, K, q) is a product of a translation and a semilinear bijection ( ₁, ₂) (where ₂ AutK) such that, for a K\{0}, we haveq ₁ = ₂ q. For linesA + KB, A + KC whereA, B, C V\{X Vq(X) = 0} we define an angle-measure < _q (A +KB, A +KC) f(B, C)² q(B)^–1 q(C)^–1 wheref is the bilinear form corresponding toq. For a point tripleA, B, C we define < _q ABC < _q (K(A – B),K(C – B)) whenever the right-hand side is defined. Now assume |K| > 5. In order to get minimal conditions for metric collineations we prove: If 0, 4 is an occurring angle-measure and if is a permutation of the point set such that exactly the point triples with measure are mapped to point triples with measure 0, 4, then is already a metric collineation.     

Ergodicity of nonlinear first order autoregressive models

Criteria are derived for ergodicity and geometric ergodicity of Markov processes satisfyingX _n+1 =f(X _n )+(X _n ) _n+1 , wheref, are measurable, { _n } are i.i.d. with a (common) positive density,E| _n |>. In the special casef(x)/x has limits, , asx– andx+, respectively, it is shown that <1, <1, <1 is sufficient for geometric ergodicity, and that <-1, 1, 1 is necessary for recurrence.     

Sharp Large Deviation Estimates for the Stochastic Heat Equation

We consider the family {X , 0} of solution to the heat equation on [0,T]×[0,1] perturbed by a small space-time white noise, that is _t X = X +b({X })+({X }) . Then, for a large class of Borelian subsets of the continuous functions on [0,T]×[0,1], we get an asymptotic expansion of P({X }A) as 0. This kind of expansion has been handled for several stochastic systems, ranging from Wiener integrals to diffusion processes.     

Convergence of epigraphs and of sublevel sets

Let LSC(X) be the set of the proper lower semicontinuous extended real-valued functions defined on a metric spaceX. Given a sequence f _n in LSC(X) and a functionf LSC(X), we show that convergence of f _n tof in several variational convergence modes implies that for each , the sublevel set at height off is the limit, in the same variational sense, of an appropriately chosen sequence of sublevel sets of thef _n, at height _n approaching . The converse holds true whenever a form of stability of the sublevel sets of the limit function is verified. The results are obtained by regarding a hyperspace topology as the weakest topology for which each member of an appropriate family of excess functionals is upper semicontinuous, and each member of an appropriate family of gap functionals is lower semicontinuous. General facts about the representation of hyperspace topologies in this manner are given.     

Parabolic subgroups of twisted Chevalley groups over a semilocal ring

It is shown that, under minor additional assumptions, the standard parabolic subgroups of a Chevalley group G (, R) of twisted type =A_l,l odd, D_l, E₆ over a commutative semilocal ring R with involution are in one-to-one correspondence with the -invariant parabolic nets of ideals of R of type , i.e., with the sets, of ideals of R such that: (l) whenever; (2) = for all ; (3) =R for > 0. For Chevalley groups of normal types, analogous results were obtained in Ref. Zh. Mat. 1976, 10A151; 1977, 10A 301; 1978, 6A476.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 94, pp. 21–36, 1979.     

Mouvement moyen et système dynamique gaussien

Summary In the situation of the classical mean motion, we haven planets moving in the plane, planetk+1 being a satellite of planetk. A classcal result then states that planetn has a mean motion,i.e. its mean angular speed between time 0 and timet has a limit whent. We show in this article that any real gaussian dynamical system can be interpreted as the limit of this situation, whenn. From a given nonatomic probability measure on [0,], we construct a transformationT of the complex brownian path (B _u)_0u1 which preserves Wiener measure.T is defined as the limit of a sequenceT _n, whereT _n acts as the motion of 2ⁿ planets. In this way we get a real gaussian dynamical system, whose spectral measure is the symetric probability on [-,] obtained from . The transformationT can be inserted in a flow (T ^t) _t, and the orbitstZ _t=B ₁T ^t still have almost surely a mean motion, which is the mean of .     

Parabolic subgroups of Chevalley groups over a semilocal ring

Let G be the Chevalley group over a commutative semilocal ring R which is associated with a root system . The parabolic subgroups of G are described in the work. A system =() of ideals in R ( runs through all roots of the system ) is called a net of ideals in the commutative ring R if ₊ for all those roots and for which + is also a root. A net is called parabolic if =R for >0. The main theorem: under minor additional assumptions all parabolic subgroups of G are in bijective correspondence with all parabolic nets . The paper is related to two works of K. Suzuki in which the parabolic subgroups of G are described under more stringent conditions.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 75, pp. 43–58, 1978.     

On asymptotically efficient estimation for a semiparametric regression model

1.IntroductionConsiderthemodelY=X"0 g(T) E,(1'1)whereX"~(xl,',xo)areexplanatoryvariablesthatenterlinearly,Pisakx1vectorofunknownparameters,Tisanotherexplanatoryvariablesthatentersinanonlinearfashion,g')isanunknownsmoothfunctionofTinR',(X,T)andeareindependent,andeistheerrorwithmean0andvariancea2.Trangesoveranondegeneratecompact1-dimensionalilltervalC*;withoutlossofgenerality,C*=[0,1].Chenl2]discussedasymptoticnormalityofestimatorsP.of0byusingpiecewisepolynthacaltoapproximateg.Speckmanls…     

Sum of Divisors in a Ring of Gaussian Integers

We construct an asymptotic formula for a sum function for _a (), where _a () is the sum of the ath powers of the norms of divisors of the Gaussian integer on an arithmetic progression ₀ (mod ) and in a narrow sector ₁ arg < ₂. For this purpose, we use a representation of _a (n) in the form of a series in the Ramanujan sums.     

Inequalities in Products of Minors of Totally Nonnegative Matrices

Let _I,I be the minor of a matrix which corresponds to row set I and column set I. We give a characterization of the inequalities of the form _{I,I K,K J,J L,L} which hold for all totally nonnegative matrices. This generalizes a recent result of Fallat, Gekhtman, and Johnson.     

Non uniqueness of invariant means for amenable group actions

It is shown, that for the action of a -compact group, being amenable as an abstract discrete group, on a locally compact measure space (X, , ), is not the unique invariant mean. Furthermore, this paper gives a characterisation of probability spaces, having a unique invariant mean for the action of an amenable group.     

Darstellung einer Ordnung von Maßen durch Stoppzeiten

Summary Given a stochastic matrixP on the state spaceI an ordering for measures inI can be defined in the following way: iff(f)(f) for allf in a sufficiently rich subcone of the cone of positiveP-subharmonic functions. It is shown that, if, are probability measures with , then in theP-process (X _n)_n0 having as initial distribution there exists a stopping time such thatX is distributed according to. In addition, can be chosen in such a way, that for every positive subharmonicf with(f)< the submartingale (f(X _n))_n0 is uniformly integrable.     

γ-Subdifferential and γ-convexity of functions on a normed space

In this paper, the -subdifferential is introduced for investigating the global behavior of real-valued functions on a normed spaceX. Iff: DX attains its global minimum onD atx ^*, then 0 f(x ^*). This necessary condition always holds, even iff is not continuous orx ^* is at the boundary of its domain. Nevertheless, it is useful because, by choosing a suitable ₊, many local minima cannot satisfy this necessary condition. For the sufficient conditions, the so-called -convex functions are defined. The class of these functions is rather large. For example, every periodic function on the real line is a -convex function. There are -convex functions which are not continuous everywhere. Every function of bounded variation can be represented as the difference of two -convex functions. For all that, -convex functions still have properties similar to those of convex functions. For instance, each -local minimizer off is at the same time a global one. Iff attains its global minimum onD, then it does so at least at one point of its -boundary.This research was supported by the Alexander von Humboldt Foundation. The author thanks Professors R. Bulirsch, K. H. Hoffmann, and H. G. Bock for inviting him to Munich and Augsburg where this research was done.     

φ-Factors for Function Seminorms

Let (, A, ) be a measure space, a function seminorm on M, the space of measurable functions on , and M the space {f M : (f) < }. Every Borel measurable function : [0, ) [0, ) induces a function : M M by (f)(x) = (|f(x)|). We introduce the concepts of -factor and -invariant space. If is a -subadditive seminorm function, we give, under suitable conditions over , necessary and sufficient conditions in order that M be invariant and prove the existence of -factors for . We also give a characterization of the best -factor for a -subadditive function seminorm when is -finite. All these results generalize those about multiplicativity factors for function seminorms proved earlier.     

Solution of a problem on the identification of parameters by the distribution of the maximum random variable: a multivariate normal case

In this paper we solve the problem of unique factorization of products ofn-variate nonsingular normal distributions with covariance matrices of the form , _ij =p _{i j} forij, = _i ² ,j=j,p0.     

Solvability of partial differential equations of infinite order in certain classes of entire functions

In this paper it is shown that under conditions of applicability of the operator to the class [,] =(I,s), ₂ 1, ₂), ₁, ₂< the equation y=f has a particular solution of this class vf[, ]. The general form of a solution of the homogeneous equation y=0 is established. The growth of a solution is investigated by means of a system of conjugate orders and a system of conjugate types. A solvability result is also obtained in the class , where T is a certain set in R ₊ ² depending on the operator .Translated from Matematicheskie Zametki, Vol. 19, No. 2, pp. 225–236, February, 1976.In conclusion, the author would like to express his thanks to his adviser, Yu. F. Korobeinik.     

Cohomology of Compact Locally Symmetric Spaces

We obtain a necessary condition for a cohomology class on a compact locally symmetric space S()=X (a quotient of a symmetric space X of the non-compact type by a cocompact arithmetic subgroup of isometries of X) to restrict non-trivially to a compact locally symmetric subspace S _H()=Y of X. The restriction is in a 'virtual' sense, i.e. it is the restriction of possibly a translate of the cohomology class under a Hecke correspondence. As a consequence we deduce that when X and Y are the unit balls in ⁿ and ^m , then low degree cohomology classes on the variety S() restrict non-trivially to the subvariety S _H (); this proves a conjecture of M. Harris and J-S. Li. We also deduce the non-vanishing of cup-products of cohomology classes for the variety S().     

On the speed of convergence in the central limit theorem of log-likelihood ratio processes

Let , the parameter space, be an open subset ofR ^k ,k1. For each , let the r.v.'sX _n ,n=1, 2,... be defined on the probability space (X, P ) and take values in (S,S,L) whereS is a Borel subset of a Euclidean space andL is the -field of Borel subsets ofS. ForhR _k and a sequence of p.d. normalizing matrices _n = _n ^{k × k} (₀ set _n ^* = ^* = ₀ + _n h, where ₀ is the true value of , such that ^*, . Let _n (^*, _*)( be the log-likelihood ratio of the probability measure with respect to the probability measure , whereP _n is the restriction ofP over _n = (X ₁,X ₂,...,X _n . In this paper we, under a very general dependence setup obtain a rate of convergence of the normalized log-likelihood ratio statistic to Standard Normal Variable. Two examples are taken into account.