Showing papers in "Stochastic Analysis and Applications in 1987"
TL;DR: In this article, sequences of random variables are considered satisfying certain mixing conditions and some alternative characterizations are discussed, illustrative examples are given for each case considered and various moment in equalities are extensively discussed in a systematic manner.
Abstract: In this work, sequences of random variables are considered satisfying certain mixing conditions. After the relevant definitions are presented, some alternative characterizations are discussed. Also, illustrative examples are given for each case considered. Finally , various moment in equalities are extensively discussed in a systematic manner. These equalities are interesting on their own right and also useful in statistical applications. Certain such applications will be presented in a separate report to avoid overloading the present one
122 citations
TL;DR: For weighted sums, general strong laws of large numbers of the form almost certainly are established where the random variables are stochastically dominated by a random variable or are all 0 as discussed by the authors.
Abstract: For weighted sums, general strong laws of large numbers of the form almost certainly are established where the random variables are stochastically dominated by a random variable are suitable conditional expectations or are all 0. The hypotheses involve both the behavior of the tail of the distribution of ∣Y∣ and the growth behavior of the constants bn/an. In general, no assumptions are made regarding the joint distributions of . As special cases, both old and new results are obtained
79 citations
TL;DR: In this paper, it was shown that for the identically distributed random variables the conditions a.i.d. and EX = 0 imply that a.s. converges to zero almost surely.
Abstract: Let be random variables and a triangular array of constants. In this paper we find the various conditions on {ani } and {Xn } under which converges to zero almost surely. It is shown that for the identically distributed random variables the conditions a.s. A generalization of Marcinkiewicz's law of large numbers is given. It is shown that for the i.i.d. case the conditions and EX = 0 imply that a.s
61 citations
TL;DR: In this article, the spmg-convergence of set valued supermartingales with values in a separable Banach space has been studied in the context of information systems with goal uncertainty.
Abstract: The starting point of this paper is a recent work. of de Korvin – Kleyle (Stoch Anal. Appl. 3 (1985)) on the convergence of set valued supermartingales with values in a separable Banach space. Since their motivation is to model information systems with goal uncertainty, they introduce the spmg–convergence. Here we present another version of this theorem with a different type of convergence. In fact in finite dimensions this mode is convergence in the Eausdroffmetric. Finally i we generalize to set valued quasimartingales
19 citations
TL;DR: In this paper, a necessary and sufficient condition is given for the strong law of large numbers to hold for sequences of exchangeable random variables and a characterization of symmetric exchangeability is given.
Abstract: A necessary and sufficient condition is given for the strong law of large numbers to hold for sequences of exchangeable random variables. This result is contrasted with the traditional results for independent random variables where moment conditions are generally sufficient. In addition, laws of large numbers are given for symmetrically, exchangeable random variables, and a characterization of symmetric exchangeability is given
17 citations
TL;DR: In this paper, a sequence of independent, identically distributed random variables and sequences of constants is considered and necessary and sufficient conditions are provided for the normed weighted sum to converge almost certainly to 0.
Abstract: Consider a sequence of independent, identically distributed random variables and sequences of constants . Sets of necessary and/or sufficient conditions are provided for to obey the general strong law of large numbers with norming constants that is, for the normed weighted sum to converge almost certainly to 0. An example is also given showing that constant 0, n≥l, and Y1 is bounded from below but not from above
15 citations
TL;DR: In this paper, the authors provided an estimate for the probable number of zeros of the random trigonometric polynomial where are independent normal random variables with mean m and variance.
Abstract: This paper provides an estimate for the probable number of zeros of the random trigonometric polynomial where are independent normal random variables with mean m and variance . It is shown that the result is valid even for m infinite
13 citations
TL;DR: This paper is concerned with optimal policy approximations based on asymptotic renewal theory, and their accuracy conditions, for a class of periodic-review inventory systems.
Abstract: This paper is concerned with optimal policy approximations based on asymptotic renewal theory, and their accuracy conditions, for a class of periodic-review inventory systems We compare the performances of two such approximations with those of the optimal policies using a wide range of demand distributions and parameter settings Accuracy conditions are derived from new bounds on the optimal policy and from empirical considerations related to the rate of approach of the renewal function to its linear asymptot. Algorithms used in computing asymptotic approximations and optimal policies are also based on new theoretical findings, including a sufficient optimality condition
12 citations
TL;DR: In this paper, the authors deal with discrete-time approximation of nonlinear filtering problems in which both the state and observation processes are described by stochastic delay equations, and the approach consists in approximating the original problem by Euler-type discretetime model for which an optimal filter can be obtained by an explicit recursive procedure.
Abstract: This paper deals with discrete–time approximation of nonlinear filtering problems in which both the state and observation processes are described by stochastic delay equations. The approach consists in approximating the original problem by Euler–type discrete–time model for which an optimal filter can be obtained by an explicit recursive procedure. The validity of the approach is substantiated by verifying the weak convergence of the approximating model to our original problem
9 citations
TL;DR: The law of the iterated logarithm for solutions of Stochastic Differential Equations (SDEs) driven by continuous semiraartingales was shown in this paper.
Abstract: We prove the law of the iterated logarithm for solutions of Stochastic Differential Equations (SDEs) driven by continuous semiraartingales, under suitable conditions. This extends a result of Kulinich for classical diffusions to solutions of SDEs which are not necessarily Markov
9 citations
TL;DR: In this article, a 4-dimensional nonlinear system of ordinary differential equations was constructed by Antonelli and Seymour to model the European rabbit viral disease known as myxomatosis.
Abstract: A 4–dimensional nonlinear system of ordinary differential equations was recently constructed by Antonelli and Seymour to model the European rabbit viral disease known as myxomatosis. This model is based on the documented Rothschild's effect, that is, on the hormonal control of flea reproduction by the rabbit. Our model incorporates density dependent rabbit reproductivity and total caloric intake over time. Three nonlinear filtering problems are considered here in which the signal process is always the entire 4–dimensional system but where the observations processes are, least squares estimation of the number or density of rabbits, estimation of flea numbers, and simultaneous estimation of rabbit densities and total caloric intake, respectively. Extensive use is made of H. Kunita's backward Stratonovich calculus and stochastic partial differential equations theory to obtain exact solution measures of the Zakai equation for these three problems
TL;DR: In this article, the existence of optional and predictable projections of stochastic processes X taking values in a Banach space E is proved, and if the range of X is contained in a compact set and if X is cadlag (respectively caglad), then the optional projection possesses the same property.
Abstract: In this note we shall prove the existence of optional and predictable projections of stochastic processes X taking values in a Banach space E. Furthermore, if the range of X is contained in a compact set and if X is cadlag (respectively caglad), then the optional (respectively predictable) projection possesses the same property. Finally, we shall prove that every E-valued martingale has a cadlag modification
TL;DR: The existence of an invariant distribution was then used to justify the approximation of the stochastic system by a switched Markov linear model if the piecewise linear regions are large “contracting” ones or small “expanding’ ones relative to the input noise variance.
Abstract: This paper is concerned with the properties of piecewise linear discrete-time dynamic systems driven by white Gaussian noise. The properties of the deterministic system are explored, and condition for the existence of invariant distributions are derived. The existence of an invariant distribution was then used to justify the approximation of the stochastic system by a switched Markov linear model if the piecewise linear regions are large “contracting” ones or small “expanding” ones relative to the input noise variance. The approach is expected to be useful for constructing approximate nonlinear filtering schemes for such systems
TL;DR: In this article, a nonparametric inference method for an irreversible k-compartmental system is proposed, where the time of transitions between the transient states is assumed to be unobservable.
Abstract: Methods of nonparametric inference are proposed for an irreversible k-compartmental system. It is assumed that the time of transitions between the transient states is unobservable. Explicit expression for the estimator of the transition rate is provided. Asymptotic result is given. A comparsion is made with the maximum likelihood estimator which relies on observing the process continuously.
TL;DR: In this article, the prophet inequalities of Krengel and Sucheston are applied to problems of order selection, non-measurable stop rules, look-ahead stop rules and iterated maps of random variables.
Abstract: Applications of the original prophet inequalities of Krengel and Sucheston are made to problems of order selection, non–measurable stop rules, look–ahead stop rules, and iterated maps of random variables. Also, proofs are given of two results of Hill and Hordijk concerning optimal orderings of uniform and exponential distributions
TL;DR: In this paper, a random nonlinear operator equation and its approximations on finite dimensional subspaces are considered and weak compactness of distributions of the approximate solutions under some "inverse stability" type condition on the approximating operators is shown.
Abstract: In this paper we consider a random nonlinear operator equation and its approximations on finite dimensional subspaces. We prove the weak compactness of distributions of the approximate solutions under some ‘inverse stability’ type condition on the approximating operators In the last section we give applications of the main result to random integral and differential equations
TL;DR: In this article, the relationship between the vector Ornstein-Uhlenbeck operator L and Ito- and McShane-calculus is discussed, and the variational calculus is used to prove an important property of L.
Abstract: In this paper, we discuss the relationship between the vector Ornstein–Uhlenbeck operator L and Ito- and McShane–Calculus, and use the variational calculus to prove an important property of the operator L. The components Li , 1≤i≥, of L defined on q-dim standard Wiener space are discussed to replace the sum operator σLi in earlier work
TL;DR: In this paper, the authors considered a series system with n repairable components maintained by a single repairman and showed that the policy which always assigns the repairman to the failed component with the smallest failure rate among the failed ones maximizes the expected discounted system operation time irrespective of the values of the repair rates and the discount rate.
Abstract: Consider a series system with n repairable components maintained by a single repairman. The following assumptions are made. Component failure and repair times are independent, exponentially distributed, random variables. Component failures can occur even while the system is not functioning and it is possible to reassign the repairman among failed components instantaneously. It is shown that the policy which always assigns the repairman to the failed component with the smallest failure rate among the failed ones maximizes the expected discounted system operation time irrespective of the values of the repair rates and the discount rate
TL;DR: In this paper, the sensitivity behavior of the Mantel χ2extension test based on ridit scores was examined in biological cases, especially epidemio-1ogical studies, to small incidence rates.
Abstract: The aim of the present study is to examine the sensitivity behavior of the Mantel χ2extension test based on ridit scores, which is widely used in biological cases, especially epidemio1ogical studies, to small incidence rates. Two additional statistical test, the conventional χ2 contingency table analysis and the Solomon–Kullback information χ2 test are also being studied
TL;DR: In this paper, the existence of random solutions for a class of random Volterra integral inclusions in which the orientor field has a stochastic domain was established for nonconvex orientor fields.
Abstract: The existence of random solutions is established for a class of random Volterra integral inclusions in which the orientor field has a stochastic domain. The proof is based on a stochastic analog of the Tietze extension theorem and on a deterministic existence result which are established as well in the paper. In particular the deterministic existence theorem is stated and proved for nonconvex orientor fields
TL;DR: Three techniques are described for solving completely observable stochastic control problems and known optimality conditions are used to develop gradient–type algorithms.
Abstract: Three techniques are described for solving completely observable stochastic control problems. Known optimality conditions are used to develop gradient–type algorithms. Numerical results are presented for illustration and comparison purposes
TL;DR: In this article, the authors give representations of the solution of 1-dimensional stochastic differential equation (SDE) with reflecting barrieres, which can be expressed by the operator "sup inf" and depend on a reflecting Brownian motion determined by solving the deterministic Skorohod eqyation.
Abstract: In this paper we give representations of the solution of 1–dimensional stochastic differential equation (SDE for short) with reflecting barrieres. To this we construct the solution of deterministic Skorohod equation with two reflecting boundaries and show which can be expressed by the operator “sup inf”. Then the solution of given SDE can be represented by a form that depend on a reflecting Brownian motion determined by solving the deterministic Skorohod eqyation