On mixing in infinite measure spaces

Summary Two concepts of mixing for null-preserving transformations are introduced, both coinciding with (strong) mixing if there is a finite invariant measure. The authors believe to offer the correct answer to the old problem of defining mixing in infinite measure spaces. A sequence of sets is called semiremotely trivial if every subsequence contains a further subsequence with trivial remote -algebra (=tail -field). A transformation T is called mixing if (T ^–n A) is semiremotely trivial for every set A of finite measure; completely mixing if this is true for every measurable A. Thus defined mixing is exactly the condition needed to generalize certain theorems holding in finite measure case. For invertible non-singular transformations complete mixing implies the existence of a finite equivalent invariant mixing measure. If no such measure exists, complete mixing implies that for any two probability measures ₁,₂, in total variation norm.Research of this author is supported by the National Science Foundation (U.S.A.) under grant GP 7693.     

On Quantiles and the Central Limit Question for Strongly Mixing Sequences

Three classes of strictly stationary, strongly mixing random sequences are constructed, in order to provide further information on the borderline of the central limit theorem for strictly stationary, strongly mixing random sequences. In these constructions, a key role is played by quantiles, as in a related construction of Doukhan et al.⁽¹¹⁾     

Weak Mixing of Random Walks on Groups

Let G be a locally compact -compact group with right Haar measure m and a regular probability measure on G. We say that is weakly mixing if for all gL (G) and all fL ¹(G) with fdm=0 we have n ^{–1 n} _k=1| ^k *f,g|0. We show that is weakly mixing if and only if is ergodic and strictly aperiodic. To prove this we use and prove some results about unimodular eigenvalues for general Markov operators.     

Note on weakly mixing transformations

Summary LetT be a weakly mixing transformation with respect to a probability measureP on a metric space (X, d). Suppose further that every open ball of (X, d) has positive measure. Then we show that, for anyP-measurable setA withP(A) > 0, lim supD _k (T ⁿ A) =D _k (X) fork = 2, 3,, whereD _k (B) is the geometric diameter of orderk of a subsetB ofX. It is shown further that D _k can be replaced by essD _k , in the case whenTB is measurable wheneverB is measurable. These results complement a previous one due to R. E. Rice for strongly mixing transformations and improve a result of C. Sempi on weakly mixing transformations.     

On the convergence of the kadanoff transformation towards trivial fixed points

Summary We consider the Kadanoff transformation T (depending on a positive parameter p) acting on probability measures on the space {+1, –}^d. A measure is called a non-trivial fixed point of T, if it is extremal in the set of T-invariant measures but is not a product measure. We describe the set of trivial fixed points and show that non-trivial fixed points exist provided that d2 and p large enough. A strong mixing condition on implies convergence of T ⁿ towards a trivial fixed point. In particular this applies to the two-dimensional Ising model except at the critical point. What happens at the critical point still remains unknown.Research supported by the Deutsche Forschungsgemeinschaft (Sonderforschungsbereich 123)     

Mixing and some integro-functional equations

Summary It is proved that the operatorP: L ¹ (0, ) L ¹(0, ), given byPg(z) = _z/c [g(x)/cx]dx, is completely mixing, i.e.,P ⁿ g ₁ 0 forg L ¹(0, ) with g dx = 0. This implies that, forc (0, 1), each continuous and bounded solution of the equationf(x)= ₀ ^cx f(t)dt/(cx) (x (0, 1]) is constant.     

Estimates for the -Neumann problem and nonexistence of <Emphasis Type="Italic">C</Emphasis><Superscript>2</Superscript> Levi-flat hypersurfaces in

The Markov moment problem is to characterize sequences admitting the representation s_n=₀¹x^nf(x)dx, where f(x) is a probability density on [0,1] and 0f(x)c for almost all x. There are well-known characterizations through complex systems of non-linear inequalities on {s_n}. Necessary and sufficient linear conditions are the following: s₀=1, and for all and . Here, is the forward difference operator. This result is due to Hausdorff. We give a new proof with some ancillary results, for example, characterizing monotone densities. Then we make the connection to de Finettis theorem, with characterizations of the mixing measure.in final form: 18 June 2003     

Asymptotic limit law for the close approach of two trajectories in expanding maps of the circle

Summary Given two pointsx, yS ¹ randomly chosen independently by a mixing absolutely continuous invariant measure of a piecewise expanding and smooth mapf of the circle, we consider for each >0 the point process obtained by recording the timesn>0 such that |f ⁿ (x)–f ⁿ (y)|. With the further assumption that the density of is bounded away from zero, we show that when tends to zero the above point process scaled by –1 converges in law to a marked Poisson point process with constant parameter measure. This parameter measure is given explicity by an average on the rate of expansion off.Partially supported by FAPESP grant number 90/3918-5     

A note on threshold theorems for epidemics with bunching

This note begins by reviewing the Kermack-McKendrick and Whittle Threshold Theorems for the general epidemic. It then extends these results to the case of the general epidemic with bunching where thexy homogeneous mixing term is replaced byxy/(x+y), 01.Research supported by Office of Naval Research Contract N00014-84-K-0568.     

On weakly mixing Markov operators and non-singular transformations

Summary Let P be a Markov operator on L (X, , m). Theorem 1: (i) P is weakly mixing (ii) For every fL there is a sequence {n_t} of density 1 such that all w ^*-cluster points of are constants (iii) For every fL there is a {k_j} with w ^*-convergent to a constant. Theorem 2: If P is induced by a non-singular transformation , P is weakly mixing For every A, { ^–n(A)} has a remotely trivial subsequence. The existence of a finite invariant measure is not required in these results.     

A skew-product which is Bernoulli

We show that if is the shift on sequences of {0,1} and is the entropy zero transformation used by Ornstein in constructing a counter-example toPinsker's conjecture, then the skew-product transformationT defined byT(x,y)=(x, ^x0 y) is Bernoulli. ThisT is conditionally mixing with respect to the independent generator for , a partition with full entropy.This research was done while the first author was a visitor at Stanford, supported in part by NSF Grant MP-575-08324.     

Optimal tests for no contamination in symmetric multivariate normal mixtures

SenGupta and Pal (1991,J. Statist. Plann. Inference,29, 145–155) have recently obtained the locally optimal test for zero intraclass correlation coefficient in symmetric multivariate normal mixtures, with known mixing proportion, for the case when the common mean,m, and the common variance, ², are known. Here, we establish that even under the general situation, when some or none ofm and ² are known, simple optimal tests can be derived, which are locally most powerful similar, whose exact cut-off points are already available and which retain all the previous optimality properties, e.g. unbiasedness, monotonicity and consistency. Some power tables are presented to demonstrate the favorable performances of these tests.     

Sur le théorème de Berry-Esseen pour les suites faiblement dépendantes

Résumé Nous étendons la méthode de démonstration du théorème de Berry-Esseen proposée par Bergström aux suites de variables aléatoires faiblement dépendantes. En particulier, nous montrons que, pour les suites stationnaires de variables aléatoires réelles bornées, la vitesse de convergence dans le théorème limite central en distance de Lévy est de l'ordre den ^–1/2 dès que la suite ( _p)p>0 des coefficients de mélange uniforme satisfait la condition _p>0 p _p <

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About the Berry-Esseen Theorem for weakly dependent sequences

We extend the method of Bergström for the rates of convergence in the central limit theorem to weakly dependent sequences. In particular, we prove that, for stationary and uniformly mixing sequences of real-valued and bounded random variables, the rate of convergence in the central limit theorem is of the order ofn ^–1/2 as soon as the sequence ( _p)p>0 of uniform mixing coefficients satisfies _p>0 p _p <.

     

<Emphasis Type="Italic">m</Emphasis>-Dissipativity of Some Gradient Systems with Measurable Potential

We consider the operator in L ²(B, ν) and in L ¹(B, ν) with Neumann boundary condition, where U is an unbounded function belonging to for some q ∈(1, ∞), B is the possibly unbounded convex open set in where U is finite and ν(dx) = C exp (−2U (x))dx is a probability measure, infinitesimally invariant for N ₀. We prove that the closure of N ₀ is a m-dissipative operator both in L ²(B, ν) and in L ¹(B, ν). Moreover we study the properties of ergodicity and strong mixing of the measure ν in the L ² case.      

Ergodic properties of certain surjective cellular automata

We consider one-dimensional cellular automata, i.e. the mapsT:P P (P is a finite set with more than one element) which are given by (Tx) _i =F(x _i+l , ...,x _i+r ),x=(x _i )P for some integerslr and a mappingFP ^r–l+1P. We prove that ifF is right- (left-) permutative (in Hedlund's terminology) and 0l<r (resp.l<r0), then the natural extension of the dynamical system (P , , ,T) is a Bernoulli automorphism ( stands for the (1/p, ..., 1/p)-Bernoulli measure on the full shiftP ). Ifr<0 orl>0 andT is surjective, then the natural extension of the system (P , , ,T) is aK-automorphism. We also prove that the shift ²-action on a two-dimensional subshift of finite type canonically associated with the cellular automatonT is mixing, ifF is both right and left permutative. These results answer some questions raised in [SR].     

A Class of Iterative Algorithms and Solvability of Nonlinear Variational Inequalities Involving Multivalued Mappings

The solvability of the following class of nonlinear variational inequality (NVI) problems based on a class of iterative procedures, which possess an equivalence to a class of projection formulas, is presented.Determine an element x ^* K and u ^* T(x ^*) such that u ^*, x – x ^* 0 for all x K where T: K P(H) is a multivalued mapping from a real Hilbert space H into P(H), the power set of H, and K is a nonempty closed convex subset of H. The iterative procedure adopted here is represented by a nonlinear variational inequality: for arbitrarily chosen initial points x ⁰, y ⁰ K, u ⁰ T(y ⁰) and v ⁰ T(x ⁰), we have u ^k + x ^k+1 – y ^k , x – x ^k+1 0, x K, for u ^k T(y ^k ) and for k 0where v ^k + y ^k – x ^k , x – y ^k 0, x K and for v ^k T(x ^k ).     

Mixing examples in the class of piecewise monotone and continuous maps of the unit interval

A process (T, P) is said to have the “ ” property if there is a uniform, positive lowerbound δ on the separation between theT-P names of (almost) every pair of pointsx≠y. A finite group rotation with partition into distinct points provides a trivial example. Given any process having the property we show that there exists a Bernoulli shiftB so thatT×B is measurably isomorphic to the natural extension of a piecewise monotone, continuous, and expanding map of the unit interval. This construction is applied to produce interval maps which are ergodic but not weak-mixing, weak-mixing but not mixing, and mixing but not exact with respect to their unique absolutely continuous invariant measures, in contrast with the results known for piecewiseC ^1+∈ expansive interval maps. In obtaining these examples we identify a number of nontrivial classes of automorphismsT which admit processes having the property. Supported by NSERC grant OGP0046586 90.     

Limit Theorems for the Maximum Terms of a Sequence of Random Variables with Marginal Geometric Distributions

Let X _n1 ^* , ... X _nn ^* be a sequence of n independent random variables which have a geometric distribution with the parameter p _n = 1/n, and M _n ^* = \max\{X _n1 ^* , ... X _nn ^* }. Let Z ₁, Z₂, Z₃, ... be a sequence of independent random variables with the uniform distribution over the set N _n = {1, 2, ... n}. For each j N _n let us denote X _nj = min{k : Z_k = j}, M _n = max{X_n1, ... X_nn}, and let S _n be the 2nd largest among X _n1, X_n2, ... X_nn. Using the methodology of verifying D(u_n) and D'(u_n) mixing conditions we prove herein that the maximum M _n has the same type I limiting distribution as the maximum M _n ^* and estimate the rate of convergence. The limiting bivariate distribution of (S_n, M_n) is also obtained. Let _n, _n N_n, , and T _n = min{M(A_n), M(B_n)}. We determine herein the limiting distribution of random variable T _n in the case _n , _n/_n > 0, as n .     

Smoothness of solutions of a nonlinear ode

Smoothness of aC -functionf is measured by (Carleman) sequence {M _k} ₀ ; we sayfC _M [0, 1] if|f ^(k) (t)|CR ^k M _k,k=0, 1, ... withC, R>0. A typical statement proven in this paper isTHEOREM: Let u, b be two C -functions on [0, 1]such that (a) u=u ²+b, (b) |b ^(k) (t)|CR ^k (k!) , >1,k_–.Then |u^(k)(t)|C₁R^k((k–1)!),k_–.The first author acknowledges the hospitality of Mathematical Research Institute of the Ohio State University during his one month visit there in the spring of 1999     

On Differences of Semicubical Powers

For X,Y,>0, let and define I ₈(X,Y,) to be the cardinality of the set. In this paper it is shown that, for >0, Y ²/X ³=O(), =O(Y ³/X ³) and X=O (Y ²), one has I ₈(X,Y,)=O(X ² Y ²+X min (X ^{3/2} Y ³, X ^{11/2} Y ^{–1})+X min (^{1/3} X ² Y ³, X ^{14/3} Y ^{1/3})), with the implicit constant depending only on . There is a brief report on an application of this that leads, by way of the Bombieri-Iwaniec method for exponential sums, to some improvement of results on the mean squared modulus of a Dirichlet L-function along a short interval of its critical line.