Lebesgue Density Research Articles

Analogues of the Lebesgue Density Theorem for Fractal Sets of Reals and Integers

Analogues of the Lebesgue Density Theorem for Fractal Sets of Reals and Integers

The space of density continuous functions

We denote by Rd the set of real numbers, R, endowed with the density topology. A function f : Rd ~ Rd is said to be density continuous, if it is continuous with respect to the topology on Rd in both the domain and range. The set of density continuous functions has been studied in several limited ways. Bruckner [1] and Niewiarowski [3] have studied density continuous functions which are homeomorphisms under the standard topology on R. Ostaszewski has investigated the local behavior of density continuous functions [4] and has investigated their behavior as a semigroup [5]. In this paper, we consider the composition of the set of density continuous functions. The structure of this set seems to be quite complicated. Ostaszewski [5] has noted that it is not dosed under uniform convergence. In Example 2 we show that it is not a vector space. Corollary 3 shows that each real-analytic function is density continuous, but Example 1 is a C ~ function which is not density continuous. It is not difficult to construct a density continuous function which is not continuous. On the other hand, every density continuous function must be approximately continuous. In what follows, the right (left) unilateral derivatives of a function f are represented as f+ ( f ) . The Lebesgue measure of a set A is denoted by IAI and the Lebesuge density (right, left Lebesgue density) of A at a point x is written as d(A ,x ) (d+(A ,x ) ,d (A ,x ) ) . The set of functions which are infinitely differentiable on R is written as C ~176 Finally, if A and B are two sets such that sup A __< inf B, then we write A << B. Before stating the main result, we first present the following lemma.

The Hausdorff Dimension of Graphs of Density Continuous Functions

The density topology on the real line consists of all measurable sets whose all points are their points of Lebesgue density one. A real-valued function of a real variable that is continuous with respect to the density topology on both the domain and the range is termed density continuous. By constructing graphs of density continuous functions as invariant sets of systems of affine maps on the unit square we show that the Hausdorff dimensions of graphs of density continuous functions vary continuously between one and two.

The Hausdorff dimension of graphs of density continuous functions

The density topology on the real line consists of all measurable sets whose all points are their points of Lebesgue density one. A real-valued function of a real variable that is continuous with respect to the density topology on both the domain and the range is termed density continuous. By constructing graphs of density continuous functions as invariant sets of systems of affine maps on the unit square we show that the Hausdorff dimensions of graphs of density continuous functions vary continuously between one and two.

An approach to consensus and certainty with increasing evidence

We investigate conditions under which conditional probability distributions approach each other and approach certainty as available data increase. Our purpose is to enhance Savage's (1954) results, in defense of ‘personalism’, about the degree to which consensus and certainty follow from shared evidence. For problems of consensus, we apply a theorem of Blackwell and Dubins (1962), regarding pairs of distributions, to compact sets of distributions and to cases of static coherence without dynamic coherence. We indicate how the topology under which the set of distributions is compact plays an important part in determining the extent to which consensus can be achieved. In our discussion of the approach to certainty, we give an elementary proof of the Lebesgue density theorem using a result of Halmos (1950).

Edgeworth Expansions in Functional Limit Theorems

Expansions for the distribution of differentiable functionals of normalized sums of i.i.d. random vectors taking values in a separable Banach space are derived. Assuming that an $(r + 2)$th absolute moment exist, the CLT holds and the distribution of the $r$th derivative $r \geq 2$ of the functionals under the limiting Gaussian law admits a Lebesgue density which is sufficiently many times differentiable, expansions up to an order $O(n^{-r/2 + \varepsilon})$ hold. Applications to goodness-of-fit statistics, likelihood ratio statistics for discrete distribution families, bootstrapped confidence regions and functionals of the uniform empirical process are investigated.

Some improvements on empirical bayes testing in lebesgue-exponential families

We consider the empirical Bayes linear loss two-action problem in exponential families, λxβ(λ)u(x), determined by a measure with Lebesgue density u, where u>0 if and only if x>a, and the parameter λ has a distribution G. Based on a sequence of observations X1,X2.. ,Xn, i.i.d. according to the marginal distribution of x, estimates of the posterior mean are used to define estimates for the Bayes test. Rates of convergence of the excess risk are obtained under certain integrability conditions. These results are proved under weaker hypotheses than those of Johns and Van Ryzin (1972). Our approach is based on a new technique of bias reduction of kernel type density estimates.

Some nonstandard methods in combinatorial number theory

A combinatorial result about internal subsets of *N is proved using the Lebesgue Density Theorem. This result is then used to prove a standard theorem about difference sets of natural numbers which provides a partial answer to a question posed by Erdos and Graham.

On the Minimax Value in the Scale Model with Truncated Data

Let $X$ be a positive random variable with Lebesgue density $f_\theta(x)$, where $\theta$ is the scale parameter, and let $Y$ be a positive random variable independent of $X$. We consider two models of truncation: the LHS model, where the data consist only of those observations of $X$ for which $X > Y$; and the RHS model, where the data consist of those observations of $X$ for which $X \leq Y$. Consider the problem of estimating $\theta^s, s \neq 0$, under a normalized squared error loss function. It is shown that under appropriate assumptions, if $f_1(\cdot)$ varies regularly at 0 (or $+ \infty$), then the minimax value in the RHS (LHS) model is equal to 1 for arbitrarily large sample size. This implies the existence of trivial minimax and admissible estimators, which do not depend on the sample at all, in contrast with the scale model without truncation.

A short zero-one law proof of a result of Abian

In this note a new and very short zero-one law proof of the following theorem of Abian is presented. The subset of the unit interval [0, 1) consisting of those real numbers whose Hamel expansions do not use a given basis element of a prescribed Hamel basis, has outer Lebesgue measure one and inner measure zero. Let {a, b, c, ...} be a Hamel basis for the real numbers. LetA be the subset of the unit interval [0, 1) consisting of those real numbers whose Hamel expansions do not use the basis elementa. Sierpinski [4, p. 108] has shown thatA is nonmeasurable in the sense of Lebesgue. Abian [1] has improved Sierpinski's result by showing thatm* (A), the outer measure ofA, is one and thatm * (A), the inner measure ofA, is zero. In this note a very short proof, using a zero-one law, of Abian's result will be presented. The following zero-one law is an immediate consequence of the Lebesgue Density Theorem [2, p. 290].

Supplements to operator-stable and operator-semistable laws on Euclidean spaces

Operator-stable laws and operator-semistable laws (introduced as limit distributions by M. Sharpe and R. Jajte, respectively) are characterized by decomposability properties. Disintegration of their corresponding Lévy measures requires appropriate cross sections. Furthermore in both situations the Lévy measures constitute a Bauer simplex whose extreme boundary can be explicitly given. Finally the infinitely differentiable Lebesgue density of an operator-semistable law is shown to be even analytic in some cases.

A note on strong unimodality and dispersivity

A distribution R is ‘more dispersed' than a distribution P if any two quantiles of R are more widely separated than the corresponding quantiles of P. Lewis and Thompson (1981) prove that a non-degenerate distribution P is strongly unimodal if and only if it is ‘dispersive', i.e. if P∗ Q1 is more dispersed than P∗ Q2 whenever Q1 is more dispersed than Q2. For P admitting a positive Lebesgue density, we prove that P is strongly unimodal if and only if P ∗ Q is more dispersed than P for every distribution Q. Hence the latter property also characterizes the dispersivity of P.

A note on strong unimodality and dispersivity

A distribution R is ‘more dispersed' than a distribution P if any two quantiles of R are more widely separated than the corresponding quantiles of P. Lewis and Thompson (1981) prove that a non-degenerate distribution P is strongly unimodal if and only if it is ‘dispersive', i.e. if P∗ Q 1 is more dispersed than P∗ Q 2 whenever Q1 is more dispersed than Q 2 . For P admitting a positive Lebesgue density, we prove that P is strongly unimodal if and only if P ∗ Q is more dispersed than P for every distribution Q. Hence the latter property also characterizes the dispersivity of P.

Stochastic scheduling problems I — General strategies

The paper contains an introduction to recent developments in the theory of non-preemptive stochastic scheduling problems. The topics covered are: arbitrary joint distributions of activity durations, arbitrary regular measures of performance and arbitrary precedence and resource constraints. The possible instability of the problem is demonstrated and hints are given on stable classes of strategies available, including the combinatorial vs. analytical characterization of such classes. Given this background, the main emphasis of the paper is on the monotonicity behaviour of the model and on the existence of optimal strategies. Existing results are presented and generalized, in particular w.r.t. the cases of lower semicontinuous performance measures or joint duration distributions having a Lebesgue density.

Occupation Densities

This is a survey article about occupation densities for both random and nonrandom vector fields $X: T \rightarrow \mathbb{R}^d$ where $T \subset \mathbb{R}^N$. For $N = d = 1$ this has previously been called the "local time" of $X$, and, in general, it is the Lebesgue density $\alpha(x)$ of the occupation measure $\mu(\Gamma) =$ Lebesgue measure $\{t\in T: X(t)\in \Gamma\}$. If we restrict $X$ to a subset $A$ of $T$ we get a corresponding density $\alpha(x, A)$ and we will be interested in its behavior both in the space variable $x$ and the set variable $A$. The first part of the paper deals entirely with nonrandom, nondifferentiable vector fields, focusing on the connection between the smoothness of the occupation density and the level sets and local growth of $X$. The other two parts are concerned, respectively, with Markov processes $(N = 1)$ and Gaussian random fields. Here the emphasis is on the interplay between the probabilistic and real-variable aspects of the subject. Special attention is given to Markov local times (in the sense of Blumenthal and Getoor) as occupation densities, and to the role of local nondeterminism in the Gaussian case.

Stochastic metastability and Hamiltonian dynamics

Stochastic metastability is stability weakened by chance. In a definition of stability certain conditions must be satisfied for sufficiently small displacements. Corresponding to a definition of stability we introduce a definition of (stochastic) metastability, in which the same conditions must be satisfied with probability approaching unity as the maximum displacement approaches zero. Measurable sets are essential to metastability as open sets are to stability. Because of Arnol’d diffusion, the invariant tori of conservative Hamiltonian systems of n ≽ 3 degrees of freedom are not usually dynamically stable with respect to changes in initial conditions. However, we prove using the Lebesgue density theorem that if the proper invariant tori of phase space fill a bounded domain of positive measure, then they are almost all dynamically metastable. Metastability is difficult to distinguish from stability in physical systems or numerical experiments, so the result is consistent with the difficulty of observing Arnol’d diffusion in the neighbourhood of an invariant torus in practice. Probability and measure are essential to the general theory of stability of nonlinear systems.

Lebesgue density influences Hausdorff measure; large sets surface‐like from many directions

In a typical counter-example construction in geometric measure theory, starting from some initial set one obtains by successive reductions a decreasing sequence of sets Fn, whose intersection has some required property; it is desired that ∩ Fn shall have large Hausdorf F dimension. It has long been known that this can often be accomplished by making each Fn+1 sufficiently “dense” in Fn. Our first theorem expresses this intuitive idea in a precise form that we believe to be both new and potentially useful, if only for simplifying the exposition in such cases. Our second theorem uses just such a construction to solve the problem that originally stimulated this work: can a Borel set in ℝk have Hausdorff dimension k and yet for continuum-many directions in every angle have at most one point on each line in that direction? The set of such directions must have measure zero, since in fact in almost all directions there are lines that meet the Borel set (of dimension k) in a set of dimension 1: this can easily be deduced from Theorem 6.6 of Mattila [5], which generalized Marstrand's result [4] for the case k = 2.

Asymptotic Behavior of Free Boundaries at Their Singular Points

&(u)-=f, u > 0 Q, aQ = D1Q U D2Q; U I 2Q = . VUI a2Q = 0 where 6 is a second order elliptic differential operator and D2Q is the overdetermined free boundary. Systems of this nature appear connection with multiple-phased physical phenomena, such as obstacle problems [L-S], filtration problems [B] and problems the theory of plasticity [T]. The theory of variational inequalities (minimization of a convex functional a Hilbert space) provides a solution to the physical problem the form of a pair (u, Q), where u belongs to C1l(Q), and DQ is a predetermined part of Q (see [L-S], ]B-K]). D2Q, the free boundary, has been studied extensively. In recent work one of the authors, [C], showed that the points of the free boundary that are of positive Lebesgue density with respect to C(Q) are regular points, the sense that a neighbourhood of each such point the boundary is a smooth surface. On the other hand, at those points of the boundary where the Lebesgue density with respect to C(Q) is zero, the behaviour of the boundary has, so far, been discussed only in measure. The lack of a more precise description at such points is due to the fact that the estimates on the growth of the second derivatives of the solution near the boundary are too weak to permit the use of integral arguments for a more precise analysis. In the first part of the paper we obtain, two dimensions, a stronger estimate than [C] for the second derivatives of the solution near the free boundary. In the second part we show that n-dimensional integral arguments apply such a case to prove that, asymptotically, the coincidence set at a point of zero density is arranged along a straight line.

Empirical Bayes Estimation with Convergence Rates in Noncontinuous Lebesgue Exponential Families

Empirical Bayes estimators, asymptotically optimal with rates, are proposed. In the component problem there is a pair $(X, \omega)$ of real valued random variables. The Lebesgue density of $X$, conditional on $\omega$, is of the form $u(x)C(\omega)e^{\omega x}$. Based on a realization of $X$, the problem is squared error loss estimation of $\omega$. Let $G$ be a prior distribution on $\omega$, and $R(G)$ be the Bayes optimal risk wrt $G$. Using $(X_1, \cdots, X_n)$, the observations in the past $n$ such problems, mean square consistent estimators of the derivative of $\log (\int C(\omega)e^{\omega x} dG(\omega))$ are proposed. Then these statistics and the present observation $X$ are used to exhibit estimators $\psi_n$ for the present problem whose risks $R_n$ converge to the Bayes optimal risk $R(G)$ as $n \rightarrow \infty$. In particular, with no assumption on the smoothness or on the form of $u$, a $\psi_n$ for each $\gamma$ in [0, 2) is exhibited. Sufficient conditions are given under which $c_1n^{-4/(4+3\gamma)} \leqq R_n - R(G) \leqq c_2n^{-2\gamma/(4+3\gamma)}$, where $c_1$ and $c_2$ are positive constants. The rhs inequality holds uniformly in $G$ with support in a bounded interval of the real line, while the other holds for a $G$ degenerate at a point and for all $n$ sufficiently large. (Thus with $\gamma$ close to $2, \psi_n$ achieves almost the exact rate.) Examples of families, including one whose $u$ function has infinitely many discontinuities, are given where conditions for the above inequalities are satisfied for $\gamma$ arbitrarily close to 2.

Consistency of a certain class of empirical density functions

Empirical density functionsf n of the class (1.1) are studied. Under certain regularity conditions the rate of probability one convergence (analoguous to the law of the iterated logarithm) is found for $$\mathop {\sup \left| {f_n (t) - f(t)} \right|}\limits_{t \in R} $$ wheref is the Lebesgue density function of the underlying probability measure. The results enable us to choose an “optimal” sequence of bandwiths α n ,n∈N, forf n . Consistency properties of the probability measures determined byf n are discussed.