Probability Interpretation of the Integral of Fractional Order

We establish a relation between stable distributions in probability theory and the fractional integral. Moreover, it turns out that the parameter of the stable distribution coincides with the exponent of the fractional integral. It follows from an analysis of the obtained results that equations with the fractional time derivative describe the evolution of some physical system whose time degree of freedom becomes stochastic, i.e., presents a sum of random time intervals subject to a stable probability distribution. We discuss relations between the fractal Cantor set (Cantor strips) and the fractional integral. We show that the possibility to use this relation as an approximation of the fractional integral is rather limited.     

稀疏过程在破产问题中的应用

本讨论一类人寿保险的风险过程，其中保单到达服从齐次Poisson过程。而描述退保及索赔发生的计数过程分别为这一过程的q-稀疏与p-稀疏．对此模型给出其破产概率的具体上界，并与其它一类风险模型进行比较．     

Lacunary Measures and Self-similar Probability Measures in Function Spaces

The paper deals with multifractal quantities for some types of Radon measures,especiallyself-similar probability measures,and their relations to Besov spaces.     

The evolution of a truncated gaussian probability density through time—modelling animal liveweights after selection

The form of the probability density derived from the evolution in time of a previously truncated frequency distribution of animal Liveweights is of interest in animal husbandry. Truncated frequency distributions arise when the heavier animals are sold for slaughter and the lighter animals retained. The demands of modern quality assurance schemes require that, given information on animal growth, the farmer is able to estimate the number of animals that would meet the specifications at some time in the future after truncation. Assuming that animal growth can be described by a linear stochastic differential equation, we derive an explicit expression for the probability density of animal Liveweights at any time after the truncation of an initial Gaussian density. It is shown that this probability density converges rapidly to a Gaussian density, so that after about 20 days of typical growth rates for lambs, the resulting density is practically indistinguishable from Gaussian.     

一个强极限定理

设是可列非齐次马氏链，本文通过利用［１］中提出的在Ｗｉｅｎｅｒ概率空间的一种实现，而给出了一个对任意可列非齐次马氏链普遍成立的强极限定理。     

Some new classes of stochastic order relations among arithmetic random variables, with applications in actuarial sciences

Recently, Fishbum and Lavalle (1995) and Lefèvre and Utev (1996) have considered some stochastic order relations specific for arithmetic random variables. The present work is concerned with these orderings, together with two other classes of stochastic order relations closely related. First, attention is paid to characterizations and various properties of all these orderings. Then, sufficient conditions of crossing-type for the two new classes of orderings are derived and extrema among discrete random variables are deduced. This is applied in actuarial sciences to obtain new bounds for the classical single life premiums as well as for the probability of ruin in the compound binomial risk model.     

Assessing the Error Probability of the Model Selection Test

The asymptotic error probability of Linhart's model selection test isevaluated, and compared with the nominal significance level. We examine thecase where the expected discrepancies of the candidate models from the truemodel are asymptotically equal. The local alternatives method is employed inthe limiting operation of the asymptotic evaluation. Although the errorprobability under the null hypothesis is actually shown to be equal to orless than the level for most situations, intolerable violations of the errorcontrol are observed for nested models: It is often erroneously concludedthat the smaller model is significantly better than the larger model. Toprevent this violation, a modification of Linhart's test statistic isproposed. The effectiveness of the proposed test is confirmed throughtheoretical analysis and numerical simulations.     

Monitoring the packet gap of real-time packet traffic

Real-time packet traffic is characterized by a strict deadline on the end-to-end time delay and an upper bound on the information loss. Due to the high correlation among consecutive packets, the individual packet loss does not well characterize the performance of real-time packet sessions. An additional measure of packet loss is necessary to adequately assess the quality of each real-time connection. The additional measure considered here is the average number of consecutively lost packets, also called the average packet gap. We derive a closed form for the average packet gap for the multiclassG/G/m/B queueing system in equilibrium and show that it only depends on the loss behavior of two consecutive packets. This result considerably simplifies the monitoring process of real-time packet traffic sessions. If the packet loss process is markovian, the consecutive packet loss has a geometric distribution.     

Estimating Invariant Probability Densities for Dynamical Systems

Knowing a probability density (ideally, an invariant density) for the trajectories of a dynamical system allows many significant estimates to be made, from the well-known dynamical invariants such as Lyapunov exponents and mutual information to conditional probabilities which are potentially more suitable for prediction than the single number produced by most predictors. Densities on typical attractors have properties, such as singularity with respect to Lebesgue measure, which make standard density estimators less useful than one would hope. In this paper we present a new method of estimating densities which can smooth in a way that tends to preserve fractal structure down to some level, and that also maintains invariance. We demonstrate with applications to real and artificial data.