LENGTH-BIASING, CHARACTERIZATIONS OF LAWS AND THE MOMENT PROBLEM

A positive random variable X with a finite mean has an induced length-biased law represented by Y, and Y is stochastically larger than X. An independent uniform random contraction of Y, UY, has the same law as X if and only if the latter is exponential. This property is extended to non-uniform contractions and a more general notion of length-biasing. The distributional equality of X and W leads to a functional equation for the moment function of X, which has either Infinitely many solutions or none. When U is constant, X can have a log-normal law, but it can also have laws with the same moment sequence as this log-nod law. The case where U has a certain beta, or generalized beta, law give t3 characterizations of generalized gamma laws, or to products of independent copies of them. This occurs even when these laws are not determined by their moment sequences.     

Characterizations,length-biasing,and infinite divisibility

SupposeL(X) is the law of a positive random variableX, andZ is positive and independent ofX. Admissible solution pairs (L(X),L(Z)) are sought for the in-law equation $\hat X \cong X o Z$ °Z, where $L\left( {\hat X} \right)$ is a weighted law constructed fromL(X), and ° is a binary operation which in some sense is increasing. The class of weights includes length biasing of arbitrary order. When ° is addition and the weighting is ordinary length biasing, the class of admissibleL(X) comprises the positive infinitely divisible laws. Examples are given subsuming all known specific cases. Some extensions for general order of length-biasing are discussed.     

Criteria for the Unique Determination of Probability Distributions by Moments

A positive probability law has a density function of the general form Q ( x )exp(− x ^1/λ L ( x )), where Q is subject to growth restrictions, and L is slowly varying at infinity. This law is determined by its moment sequence when λ< 2, and not determined when λ> 2. It is still determined when λ= 2 and L ( x ) does not tend to zero too quickly. This paper explores the consequences for the induced power and doubled laws, and for mixtures. The proofs couple the Carleman and Krein criteria with elementary comparison arguments.     

DISTRIBUTIONAL CHARACTERIZATIONS THROUGH SCALING RELATIONS

Investigated here are aspects of the relation between the laws of X and Y where X is represented as a randomly scaled version of Y. In the case that the scaling has a beta law, the law of Y is expressed in terms of the law of X. Common continuous distributions are characterized using this beta scaling law, and choosing the distribution function of Y as a weighted version of the distribution function of X, where the weight is a power function. It is shown, without any restriction on the law of the scaling, but using a one‐parameter family of weights which includes the power weights, that characterizations can be expressed in terms of known results for the power weights. Characterizations in the case where the distribution function of Y is a positive power of the distribution function of X are examined in two special cases. Finally, conditions are given for existence of inverses of the length‐bias and stationary‐excess operators.     

NUMBERS OF OBSERVATIONS NEAR ORDER STATISTICS

Limit theorems are obtained for the numbers of observations in a random sample that fall within a left‐hand or right‐hand neighbourhood of the kth order statistic. The index k can be fixed, or tend to infinity as the sample size increases unboundedly. In essence, the proofs are applications of the classical Poisson and De Moivre–Laplace theorems.     

Explicit Distributional Results In Pattern Formation II

The paper derives the joint generating function of a collection of pattern statistics associated with binary sequences. The models discussed cover independent and some dependent Bernoulli trials, including Markov dependent ones. The results cover, in particular, the moment generating function of the random search time for certain general binary patterns in the popular Knuth-Morris-Pratt algorithm and hence shed more light into its performance.     

Some properties of the exponential distribution class with applications to risk theory

This paper derives some equivalent conditions for tail equivalence of a distribution $G$ and the convolution $G1H$, where $G$ belongs to the exponential distribution class and $H$ is another distribution. This generalizes some existing sufficient conditions and gives further insight into closure properties of the exponential distribution class. If $G$ also is $O$-subexponential, then the new conditions are satisfied. The obtained results are applied to investigating asymptotic behavior for the finite-time ruin probability in a discrete-time risk model with both insurance and financial risks, where the distributions of the insurance risk or the product of the two risks may not belong to the convolution equivalence distribution class.     

Stochastic Algorithms,Symmetric Markov Perfect Equilibrium,and the ‘curse’ of Dimensionality

This paper introduces a stochastic algorithm for computing symmetric Markov perfect equilibria. The algorithm computes equilibrium policy and value functions, and generates a transition kernel for the (stochastic) evolution of the state of the system. It has two features that together imply that it need not be subject to the curse of dimensionality. First, the integral that determines continuation values is never calculated; rather it is approximated by a simple average of returns from past outcomes of the algorithm, an approximation whose computational burden is not tied to the dimension of the state space. Second, iterations of the algorithm update value and policy functions at a single (rather than at all possible) points in the state space. Random draws from a distribution set by the updated policies determine the location of the next iteration's updates. This selection only repeatedly hits the recurrent class of points, a subset whose cardinality is not directly tied to that of the state space. Numerical results for industrial organization problems show that our algorithm can increase speed and decrease memory requirements by several orders of magnitude.     

Characterization of laws by balancing weighting against a binary operation

LetL(X) be the law of a positive random variableX, andZ be positive and independent ofX. Solution pairs (L(X), L(Z)) are sought for the in-law equation $\hat X \cong X \circ Z$ where $L(\hat X)$ is a weighted law constructed fromL(X), and ° is a binary operation which in some sense is increasing. The class of weights includes length biasing of arbitrary order. When ° is the maximum operation a complete solution in terms of a product integral is found for arbitrary weighting. Examples are given. An identity for the length biasing operator is used when ° is multiplication to establish a general solution in terms of an already solved inverse equation. Some examples are given.