Remarks on ergodicity of stationary irreducible transient Markov chains

Summary Consider a stationary process {X _n(), – < n < . If the measure of the process is finite (the measure of the whole sample space finite), it is well known that ergodicity of the process {X _n(), - < n < and of each of the subprocesses {X _n(), 0 n < , {X _n(), – < n 0 are equivalent (see [3]). We shall show that this is generally not true for stationary processes with a sigma-finite measure, specifically for stationary irreducible transient Markov chains. An example of a stationary irreducible transient Markov chain {X _n(), - < n <} with {itX_n(), 0 n < < ergodic but {X _n(), < n 0 nonergodic is given. That this can be the case has already been implicitly indicated in the literature [4]. Another example of a stationary irreducible transient Markov chain with both {X _n(), 0 n < and {itX _n(),-< < n 0} ergodic but {X _n(), - < n < nonergodic is presented. In fact, it is shown that all stationary irreducible transient Markov chains {X _n(), - < n < < are nonergodic.This research was supported in part by the Office of Naval Research.John Simon Guggenheim Memorial Fellow.     

Embeddings of Banach Spaces Into Banach Lattices and the Gordon–Lewis Property

In this paper we first show that if X is a Banach space and is a left invariant crossnorm on lX, then there is a Banach lattice L and an isometric embedding J of X into L, so that I J becomes an isometry of lX onto l_m J(X). Here I denotes the identity operator on l and l_m J(X) the canonical lattice tensor product. This result is originally due to G. Pisier (unpublished), but our proof is different. We then use this to prove the main results which characterize the Gordon–Lewis property GL and related structures in terms of embeddings into Banach lattices.     

Selfdecomposable distributions for maxima of independent random vectors

Summary In the present paper the limit laws for conveniently normalized multivariate sample extremes are characterized by means of the decomposability of probability distributions. Continuous automorphisms ofR ^d =[–,]^d with respect to the operation v defined by x y=(max(x _i, y_i),i=1... d) are treated as norming mappings. An integral representation of the limit distributions is found using their log-concavity and a decomposition ofR ^d in orbits of the norming family. Finally an example is given as an illustration.Research supported in part by the Committee of Science, Bulgarian Concil of Ministers, under contract no. 60/1987     

On a Property of Subexponential Distributions

Let F be a distribution function (d.f.) on [0, ) with finite first moment m >0. We define the integrated tail distribution function F ₁ of F by F ₁(t)=m^-1 ₀ ^t (1- F(u))du, t0. In this paper, we obtain sufficient conditions under which implications FSF ₁S and F ₁S FS hold, where S is the class of subexponential distributions.     

Distribution of the supremum of sums of independent variables with negative drift

Let {_n} be a sequence of identically distributed independent random variables,M₁=<0,M ₁ ² <;S ₀=0,S _n =₁+₂,+...+ _{n, n}1;¯ S=sup {S _n n=0.} The asymptotic behavior ofP(¯ St) as t is studied. If _t ^P (₁x dx=0((t)), thenP(¯ St)– 1/¦¦ _t P (₁x dx=0((t)) (t) is a positive function, having regular behavior at infinity.Translated from Matematicheskie Zametki, Vol. 22, No. 5, pp. 763–770, November, 1977.The author thanks B. A. Rogozin for the formulation of the problem and valuable remarks.     

Future independent times and Markov chains

Summary A random timeT is a future independent time for a Markov chain (X _n ) ₀ ifT is independent of (X _T+n ) _n ^/ =0 and if (X _T+n ) _n ^/ =0 is a Markov chain with initial distribution and the same transition probabilities as (X _n ) ₀ . This concept is used (with the conditional stationary measure) to give a new and short proof of the basic limit theorem of Markov chains, improving somewhat the result in the null-recurrent case.This work was supported by the Swedish Natural Science Research Council and done while the author was visiting the Department of Statistics, Stanford University     

On pathwise behavior of multiserver queues

In this paper, by exploiting recent results on the pathwise behavior of the workload process in single server, work conserving queues of theG/G/1/ type, we show that the workload of multiserver, work conserving queues ofG/G/m/ (m<) (andG/G/) queues satisfies an o(t) growth condition, provided that the time average of the work brought into the system is less thanm form < (and finite form=).     

Über eine Verallgemeinerung der Nicoletti-Aufgabe für Funktional-Differentialgleichungen mit voreilendem Argument

We consider a functional differential equation (1) (t)=F(t,) fort[0,+) together with a generalized Nicoletti condition (2)H()=. The functionF: [0,+)×C ⁰[0,+)B is given (whereB denotes the Banach space) and the value ofF (t, ) may depend on the values of (t) fort[0,+);H: C ⁰[0,+)B is a given linear operator and B. Under suitable assumptions we show that when the solution :[0,+)B satisfies a certain growth condition, then there exists exactly one solution of the problem (1), (2).     

Characterization,structure and analysis on abelian ℒ∞ groups

An abelian topological group is an group if and only if it is a locally -compactk-space and every compact subset in it is contained in a compactly generated locally compact subgroup. Every abelian groupG is topologically isomorphic to G ₀ where ₀ andG ₀ is an abelian group where every compact subset is contained in a compact subgroup. Intrinsic definitions of measures, convolution of measures, measure algebra,L ₁-algebra, Fourier transforms of abelian groups are given and their properties are studied.     

Extension of Subgradient Techniques for Nonsmooth Optimization in Banach Spaces

In this paper we continue the study of the subgradient method for nonsmooth convex constrained minimization problems in a uniformly convex and uniformly smooth Banach space. We consider the case when the stepsizes satisfy _{k=1 k} =, lim _{k k} =0.     

Asymptotics for Sums of Random Variables with Local Subexponential Behaviour

We study distributions F on [0,) such that for some T , F ^*2(x, x+T] 2F(x, x+T]. The case T = corresponds to F being subexponential, and our analysis shows that the properties for T < are, in fact, very similar to this classical case. A parallel theory is developed in the presence of densities. Applications are given to random walks, the key renewal theorem, compound Poisson process and Bellman–Harris branching processes.     

Nonlocal symmetries and the theory of coverings: An addendum to A. M. vinogradov's ‘local symmetries and conservation laws”

For a systemY of partial differential equations, the notion of a covering Y is introduced whereY is infinite prolongation ofY. Then nonlocal symmetries ofY are defined as transformations of which conserve the underlying contact structure. It turns out that generating functions of nonlocal symmetries are integro-differential-type operators.     

Behavior on the real line of entire functions represented by Dirichlet series

Conditions are found under which for an entire function f represented by a Dirichlet series with finite Ritt order on some sequence (x_k), 0 < x_k , as k one has ¦f(x_k)¦=M_t((1 + 0(1) x_k), M_f(x)=sup {¦ f (z) ¦:Re z x}.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 43, No. 2, pp. 265–269, February, 1991.     

Moser-type inequalities in Lorentz spaces

Moser-type estimates for functions whose gradient is in the Lorentz space L(n, q), 1q, are given. Similar results are obtained for solutions uH _inf0 ^sup1 of Au=(f _i ) _{x i} , where A is a linear elliptic second order differential operator and |f|L(n, q), 2q.Work partially supported by MURST (40%).     

On convergence subsystems of orthonormal systems

It is proved that for any sequence {R _k} _k=1 of real numbers satisfyingR _kk (k1) andR _k=o(k log₂ k),k, there exists an orthonormal system {n _k(x)} _n=1 ,x (0;1), such that none of its subsystems {n _k(x)} _k=1 withn _kR_k (k1) is a convergence subsystem.     

Spitzer's Strong Law of Large Numbers in Nonseparable Banach Spaces

It is well known, that for the sums of i.i.d. random variables we have S _n/n 0 a.s. iff _n=1 1/n P(|S _n| > n) < holds for all > 0 (Spitzer's SLLN). The result is also known in separable Banach spaces. It will be shown, that this also holds in nonseparable (= not necessarily separable) Banach spaces without any measurability assumption. In the theory of empirical processes this gives a characterization of Glivenko-Cantelli classes.     

The associativity equation revisited

Summary Consideration of the Associativity Equation,x (y z) = (x y) z, in the case where:I × I I (I a real interval) is continuous and satisfies a cancellation property on both sides, provides a complete characterization of real continuous cancellation semigroups, namely that they are topologically order-isomorphic to addition on some real interval: ( – ,b), ( – ,b], –, +), (a, + ), or [a, + ) — whereb = 0 or –1 anda = 0 or 1. The original proof, however, involves some awkward handling of cases and has defied streamlining for some time. A new proof is given following a simpler approach, devised by Páles and fine-tuned by Craigen.     

Lower functions for asymmetric Lévy processes

Summary Let {X _t } be a ¹ process with stationary independent increments and its Lévy measurev be given byv{yy>x}=x ^–L ₁ (x), v{yy<–x}=x ^–L ₂ (x) whereL ₁,L ₂ are slowly varying at 0 and and 0<1. We construct two types of a nondecreasing functionh(t) depending on 0<<1 or =1 such that lim inf a.s. ast 0 andt for some positive finite constantC.This research is partialy supported by a grant from Korea University     

On a Type of Mixed Norm Spaces

In this paper the mixed norm spaces L(B,p,q) and their duals are investigated. In the case p,q < it is proved that the dual of L(B,p,q) is L(B,p,q), where p-1 + p-1 = 1 and q-1 + q-1 = 1. For p = 2 and q = an isometric isomorphism is discussed between the mixed norm space L(B,2,) and L(B,2), the L-space of 2-valued functions. Here a measurability theorem is proved for 2-valued functions. The dual of an important subspace of L(B,2,) is characterized as a space of vector measures. Finally, as an application we show that if B is finitely generated then the dual of L(B,2,) is L(B,2,1).     

Order of the best spline approximations of some classes of functions

The rate of decrease of the upper bounds of the best spline approximations E_m,n(f)p with undetermined n nodes in the metric of the space L_p(0, 1) (1p) is studied in a class of functionsf(x) for which f ^m+1 (x)L_q(0, 1)1(1qt8) or var {f(m) (x); 0, 1}1 (m=1, 2, ..., the preceding derivative is assumed absolutely continuous). An exact order of decrease of the mentioned bounds is found as n , and asymptotic formulas are obtained for p= and 1q in the case of an approximation by broken lines (m=1). The simultaneous approximation of the function and its derivatives by spline functions and their appropriate derivatives is also studied.Translated from Matematicheskie Zametki, Vol. 7, No. 1, pp. 31–42, January, 1970.