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September 2000 - January 2016

## Publications

Publications (97)

In this article, a longitudinal functional model including both fixed and random effects which depend on measurement periods is considered. A new procedure is presented to estimate the parameters included in the fixed and random effect terms of the model. We put light on the performance of our estimation procedure by two simulation studies and a re...

Simple harmonizable processes, introduced by Soltani and Parvardeh (Theory Probab Appl 50(3):448–462, 2006), form a fairly large class of second order processes that includes stationary processes and periodically correlated processes. The spectral density of a simple process is supported by certain curves in \([0,2\pi )^2\). In this article we proc...

In this article, we introduce a weighted periodogram in the class of smoothed periodograms as a consistent estimator for the spectral density matrix of a periodically correlated process. We derive its limiting distribution that appears to be a certain finite linear combination of Wishart distribution. We also provide numerical derivations for our s...

In this article, we consider a real second-order discrete-time stationary process and use the corresponding finite Fourier transforms to construct an embedded Hilbertian stationary process. We provide the moving average representation for the embedded process on a certain Hilbert function space and use it for estimation of the moving average coeffi...

In this paper, we introduce a test statistics to test whether a discrete time periodically correlated model with a given spectral density explains an observed time series. Our testing procedure is based on an application of the asymptotic distribution of the periodogram established in Soltani and Azimmohseni (Stat Plan Inference 137:1236–1242, 2007...

It is well known that no strongly harmonizable SαS process has classical ARMA representation with respect to a SαS noise sequence with independent values. In this article, we introduce certain strongly harmonizable ARMA SαS models, in which the noise sequences have orthogonal values in certain Hilbert spaces. Our study on strongly harmonizable ARMA...

In this article we consider the sequences of sample and popu- lation covariance operators for a sequence of arrays of Hilbertian random elements. Then, under the assumptions that sequences of the covari- ance operators norm are uniformly bounded and the sequences of the principal component scores are uniformly summable, we prove that the convergenc...

An interesting class of continuous distributions, called Cauchy-type mixture, with potential applications in modelling erratic phenomena is introduced by Soltani and Tafakori [A class of continuous kernels and Cauchy type heavy tail distributions. Statist Probab Lett. 2013;83:1018–1027]. In this work, we provide more insights into the Cauchy-type m...

We let be a separable Banach space, and let be a sequence of independent and identically distributed random elements in . Then we prove that for a given strongly periodic sequence of bounded linear operators , the order one autoregressive system equations in set on integers, possesses a unique almost sure strictly periodically correlated solution;...

We prove strong consistency for the k-th order random Stieltjes partial sums (RSPS) whenever the underlying distribution function is supported by a finite interval; and derive explicit formula for its kth moment about zero. Then we provide formulas for the mean and variance of RSPS. We examine the power distribution with density function
, where θ...

In this article we present a testing procedure for spatial scan statistics when the underlying population characteristics are not known. Specifically, the test procedure is designed for the situation when the number of affected cases in the population is random. We further assume that the number of contaminated case in the geographic region of inte...

Let (formula presented) be a random algebraic polynomial, where A0;A1;::: is a sequence of independent random variables belong to the domain of attraction of the normal law. Thus Aj 's for j = 0; 1;::: possess the characteristic functions exp(formula presented), where Hj (t)'s are complex slowly varying functions. Under the assumption that there ex...

Simple harmonizable processes (SHP) introduced by Soltani and Parvardeh (2006) are a large class of nonstationary processes which includes stationary and periodically correlated (PC) processes. Detection and estimation of SHP structure are important problems when dealing with nonstationary data. In this paper, we study the spectral properties of si...

In this article we establish a simulation procedure to generate values for a real discrete time multivariate stationary process, based on a factor of spectral density matrix. We prove the convergence of the simulator, at each time epoch, to the actual process, and provide the corresponding rate of convergence. We merely assume that the spectral den...

In this paper we assume that X1, X2, . . . is a sequence of independent continuous centered random variables with finite variances σ21,σ22,. . .. Then we present a central limit theorem for the randomly weighted averages Sn=R1X1+···+RnXn, where the random weights R1, . . ., Rn are the cuts of (0, 1) by the order statistics of a random sample of siz...

Applied statistical decision theory has wide applications in decision-making fields of studies, such as economic, business management and industrial managements. In this work, following Pratt et al.’s [Introduction to statistical decision theory. 3rd ed. Cambridge, MA: The MIT Press; 2001] approach, we provide theoretical and practical formulations...

We introduce semi-Markov fields and provide formulations for the basic terms in the semi-Markov theory. In particular we define and consider a class of associated reward fields. Then we present a formula for the expected reward at any multidimensional time epoch. The formula is indeed new even for the classical semi-Markov processes. It gives the e...

We establish a hypotheses testing procedure equivalent to the Kulldorff (19977.
Kulldorff, M. (1997). A spatial scan statistic. Commun. Statist. Theor. Meth. 26(6): 1481–1496.View all references) spatial scan hypotheses test for cluster detection, then provide transparent test statistics for cluster detection in a spatial setting. We also specify t...

In this article, we consider Hilbertian spatial periodically correlated autoregressive models. Such a spatial model assumes periodicity in its autocorrelation function. Plausibly, it explains spatial functional data resulted from phenomena with periodic structures, as geological, atmospheric, meteorological and oceanographic data. Our studies on th...

In this work we introduce and study discrete time periodically correlated stable processes and multivariate stationary stable processes related to periodic and cyclic flows. Our study involves producing a spectral representation and a spectral identification for such processes. We show that the third component of a periodically correlated stable pr...

A consistency criteria is given for a certain class of finite positive measures on the surfaces of the finite dimensional unit balls in a real separable Hilbert space. It is proved, through a Kolmogorov type existence theorem, that the class induces a unique positive measure on the surface of the unit ball in the Hilbert space. As an application, t...

Soltani and Mohammadpour (200615.
Soltani , A. R. ,
Mohammadpour , M. (2006). Moving average representations for multivariate stationary processes. J. Time Ser. Anal. 27(6):831–841. [CrossRef], [Web of Science ®]View all references) observed that in general the backward and forward moving average coefficients, correspondingly, for the multivariat...

We give an affirmative answer to the conjecture raised in Soltani and Roozegar [On distribution of randomly ordered uniform incremental weighted averages: divided difference approach. Statist Probab Lett. 2012;82(5):1012-1020] that a certain class of power semicircle distributions, parameterized by n, gives the distributions of the average of n ind...

We shed light on an interesting class of one sided continuous kernels on [0,∞)[0,∞). Then we present its properties, provide new integral formulas including an extension for the Kotz and Ostrovskii (1996) mixture representation, introduce a class of Cauchy type distributions, and finally enlarge the class of one sided stable densities. This study p...

In this article, we prove that with probability one the k-th order upper random Stieltjes sum defined on a random sample from a distribution supported by a finite interval converges to the corresponding k-th moment distribution. When the underlying distribution is uniform U(0,θ), we prove that the adjusted method of moments (AMM) estimator, introdu...

We give a new numerical testing hypothesis procedure by using the adjusted method of moments estimators, introduced by Soltani and Homei (2009). By using simulated data, we put light on the effectiveness of this procedure by making a comparison between its performance with the three well known numerical testing procedures, namely, Pal et al. (2007)...

We prove that the limit of a sequence of Pettis integrable bounded scalarly measurable weak random elements, of finite weak norm, with values in the dual of a non-separable Banach space is Pettis integrable. Then we provide basic properties for the Pettis conditional expectation, and prove that it is continuous. Calculus of Pettis conditional expec...

Soltani and Shirvani (Comput Stat 25:155–161, 2010) provided a characterization and a simulation method for truncated stable random variables when the characteristic exponent
$\alpha \ne 1 $
, and left the case
$\alpha =1$
open. The case of
$\alpha =1$
is treated in this article.

In this article we employ certain techniques in divided differences to relate the generalized Stieltjes transform of the distribution of a randomly weighted average of independent random variables X1,â€¦,Xm to the generalized Stieltjes transforms of the distribution functions F1,â€¦,Fm; Xiâˆ¼Fi,i=1,â€¦,m. The random weights are assumed to be cuts o...

In this article, the problem of testing hypothesis for unknown parameters is considered. A new method is proposed based on a combination of the computational approach test, introduced by Pal, Lim and Ling (2007), and adjusted method of moments (AMM), introduced by Soltani and Homei (2009). In this method, there is no need to any formulation for the...

In this article we shall consider a class of strongly T-periodically correlated processes with values in a separable complex Hilbert space . The periodograms of these processes and their statistical properties were previously studied by the authors. In this paper we derive the asymptotic distribution of the periodogram, that appears to be a certain...

We consider periodically correlated autoregressive processes of order p in Hilbert spaces. Our studies on these processes
involve existence, strong law of large numbers, central limit theorem and parameter estimation.

We consider periodically correlated autoregressive processes in Hilbert spaces. Our studies on these processes involve existence, covariance structure, estimation of the covariance operators, strong law of large numbers and central limit theorem.

Double arrays of n rows and p columns can be regarded as n drawings from some p-dimensional population. A sequence of such arrays is considered. Principal component analysis for each array forms sequences of sample principal components and eigenvalues. The continuity of these sequences, in the sense of convergence with probability one and convergen...

We consider D(t) = [J(t), X (t), Z(rho)(t)], t = 0, where J (t) is a semi-Markov process, X (t) is the age process for J (t), and Z(rho) (t) is a reward process on J (t). In this article we produce a prediction formula to predict Z(rho) (t + s) based on the history D(t), the sigma-field generated by {D(u), u <= t}. In addition, we derive the condit...

We deal with a Sazonov space (X: real separable) valued symmetric α stable random measure Φ with independent increments on the measurable space (Rk, B(Rk)). A pair (k, μ), called here a control pair, for which k: X × Rk → R+, μ a positive measure on (Rk, B(Rk)), is introduced. It is proved that the law of Φ is governed by a control pair; and every...

Kanter (Ann Probab 3(4):697–707, 1975) and Chambers et al. (J Am Stat Assoc 71(354):340–344, 1976) developed a method for
characterizing and simulating stable random variables, X, using nonlinear transformations involving two independent uniform random variables. Their method is scrutinized to provide
a characterization and then develop a method fo...

Forward-moving average coefficients are in general different from their corresponding backward-moving average coefficients in multivariate stationary time series. There is lack of practical methods to derive forward-moving average coefficients from the backward ones. In this article, we establish a new practical approach for obtaining the forward-m...

In this work we shall consider two classes of weakly second-order periodically correlated and strongly second-order periodically correlated processes with values in separable Hilbert spaces. The periodogram for these processes is introduced and its statistical properties are studied. In particular, it is proved that the periodogram is asymptoticall...

A new rich class of generalized two-sided power (TSP) distributions, where their density functions are expressed in terms of the Gauss hypergeometric functions, is introduced and studied. In this class, the symmetric distributions are supported by finite intervals and have normal shape densities. Our study on TSP distributions also leads us to a ne...

A Markov chain is associated to a finite order one-sided moving average of a discrete time stationary Gaussian process. A
method is developed to specify thresholds 0=L0 < L1 < ¼ < Lm < Lm+1=¥{0=L_0 < L_1 < \cdots < L_m < L_{m+1}=\infty} for given on target significant levels p0, ¼, pm, åi=0m pi = 1;{\pi_0, \ldots, \pi_{m},\quad \sum_{i=0}^{m} \pi_{...

A new class of discrete probability distributions with infinite second moments is introduced and studied. The class is induced by stable laws, and contains the discrete stable distributions. Members of this class appear to be suitable substitutes for the Poisson distributions, as underlying statistical distributions for certain erratic phenomena. A...

A weighted Average of n independent continuous random variables X1,...,Xn with random proportions obtained by cutting a simplex by hyper-planes is introduced. A formula between the Stieltjes transforms of the distribution functions of the weighted average and X1,...,Xn is established. The Johnson and Kotz [Johnson, N.L., Kotz, S., 1990. Randomly we...

Time domain calculus of Wiener and Masani together with the von Neumann's alternating projection formula are employed to obtain a time domain algorithm for the best linear interpolator of unrecorded innovations in discrete time multivariate second order stationary processes. From the interpolated innovations of a multivariate discrete-time ARMA pro...

A Hilbert space technique to treat continuous time complex-valued strongly harmonizable symmetric α stable processes was developed in earlier papers. In this work we apply the technique to prove that such a process {X(t),t∈ℝ} will satisfy ∑ n=0 d c n ∂ n X(t)=Z ˙(t), if its spectral density is given by |∑ n=0 d c n (iλ) n -2 . A germ field is intro...

In this work we shall consider two classes of periodically correlated processes with values in separable Hilbert spaces: weakly
second order and strongly second order. It is proved that the sample Fourier transforms are asymptotically uncorrelated and
the periodograms are asymptotically unbiased for corresponding spectral densities.

The content of this article primarily falls into three sections. Section 1 deals with a basic structural spectral representation theorem for periodically correlated sequences. Section 2 provides a certain class of square integrable functions that is isomorphic to the time domain of the sequence. This complete class is called the spectral domain of...

Let X1, X2,… be a sequence of independent and identically distributed random variables, and let Yn, n = K, K + 1, K + 2,… be the corresponding backward moving average of order K. At epoch n ≥ K, the process Yn will be off target by the input Xn if it exceeds a threshold. By introducing a two-state Markov chain, we define a level of significance (1...

Periodically correlated processes with values in Hilbert spaces are introduced and studied. The harmonizability of such a process is discussed. The covariance operator is characterized. Time-dependent spectra on Hilbert spaces are introduced and a time-dependent spectral density for a periodically correlated process is given.

Discrete time periodically correlated (PC) processes are viewed as the processes with time-dependent spectra. This, together with an auxiliary operator which is defined here is employed to apply classical results on the asymptotic distribution of the periodogram of the univariate white noise (innovations) to derive the asymptotic distributions of t...

In this article, we provide a spectral characterization for a real-valued discrete-time periodically correlated process, and then proceed on to establish a simulation procedure to simulate such a Gaussian process for a given spectral density. We also prove that the simulated process, at each time index, converges to the actual process in the mean s...

Backward and forward moving average (MA) representations are established for multivariate stationary processes. It is observed that in the multivariate case, in contrast to the univariate case, the backward and forward MA coefficients correspondingly, in general, are different. A method is presented to adopt the known techniques in deriving the bac...

Let Q(n)(x) = Sigma(i=0)(n)A(i)x(i) be a random algebraic polynomial where the coefficients A(0), A(1), . . . form a sequence of centered Gaussian random variables. Moreover, assume that the increments Delta(j) = A(j) - A(j- 1), j = 0, 1,2, . . . are independent, A(-1) = 0. The coefficients A(0),A(1), . . . A(n) can be considered as n consecutive o...

Periodically correlated autoregressive nonstationary processes of finite order are considered. The corresponding Yule-Walker equations are applied to derive the generating functions of the
covariance functions, what are called here the periodic
covariance generating functions. We also provide closed formulas
for the spectral densities by using the...

A useful inequality involving correlations between three random variables with mean zero and finite second moments is presented. The inequality is applied to show that the entries on the main diagonal of the spectral density of a periodically correlated Markov process, as derived in Nematollahi and Soltani [2000. Discrete time periodically correlat...

Convergence with probability one (in probability) of sequences of the sample quantiles and the Pearson statistic that are
formed by columns of N� n arrays of random variables and bivariate random vectors respectively is established, n→ ∞. Two applications for the continuity of the Pearson statistics, when sampling is only possible along a sequence...

Autoregressive Gaussian random vectors of order one are introduced, characterized and studied. The characterization in- volves the existence and structural identification of the covariance matrix. Prediction for future values together with necessary and sucient conditions for the stationarity are established. Some ba- sic statistical properties are...

A simple random measure is a finite sum of random measures with disjoint supports. A type of simple random measure which is induced by a multivariate random measure (Φ 1 ,...,Φ m ) and measurable mappings T 1 ,...,T m is introduced and studied. Interestingly it gives rise to introducing a class of processes, called simple, that include stationary p...

The spectral structure of discrete time periodically correlated (as well as multivariate stationary) symmetric [alpha]-stable processes is identified by decomposing such a process uniquely in distribution into one sum of three mutually independent periodically correlated (multivariate stationary) stable processes that are classified as mixed moving...

Certain characterizations for an exchangeable sub-Gaussian random vector are given and a method together with an S-PLUS function for simulating such a vector are introduced. Simulated values are used to plot contours of their empirical density.

In connection with a symmetric α stable random measure Φ on a measurable space (F, ℱ) with values in R , a complete metric space of symmetric finite measures on Sd−1 is constructed, and is employed to characterize the law of Φ by a unique positive measure on ℱ and a unique function on F × R . The stochastic integral ∫Ff dΦ is also defined for certa...

A discrete time evolutionary second-order process X t is considered. Modulating functions that are the Fourier transforms of certain lattice distributions together with segments of a stationary sequence are employed to form a process X ˜ t N , with N prime. It is proved that X ˜ t N →X t in mean square, under sufficient conditions. A formula in ter...

In this work we study a multivariate reward process Z(t) = (Z1(t),...,Zp(t)), t 0, defined on a semi-Markov pro- cess {J(t), t 0} with a Markov renewal process {(Jn, Tn), n = 0,1,2,...} and non-linear reward functions 1,..., p respec- tively. We follow the definitoin of Soltani 1996, for nonlinear reward processes.Usually in practice, reward functi...

Fatigue is considered as a primary model of failure for metallic structures or mechanical devices subjected to oscillatory stress processes. In this paper we consider certain random polynomials as the underlying stress processes, and recommend the evaluations of fatigue indices. Let Qn(x) = ∑i=0n Aixi, where Aj := ∑k=1m Rk cos jωk, j = 1, 2,..., n,...

A strongly harmonizable continuous time symmetric -stable process is considered. By using covariations, a Hilbert space is formed from the process elements and used for a purpose of moving average representation and prediction.

This works concerns the simulation of an exchangeable stable random vector. A characterization for exchangeability of a stable random vector, in terms of its spectral measure, is given. The Modarres and Nolan's simulating method on stable random vectors is modified to the exchangeable case. FORTRAN subroutines to simulate a desirable exchangeable s...

Assuming (A0;A1;:::;An) is a jointly symmetric ã-stable ran- dom vector, 0 < ã î 2, a general formula for the expected number of real zeros of the random algebraic polynomial n P i=0 A ixi was obtained by Reza- khah in 1997. Rezakhah's formula is applied to the case where (A0;:::;An) is formed by consecutive observations from the Levy stable noise...

Let (S, ∥ ∥) be a Banach space of jointly symmetric α-stable random variables and let ρ be the ∞-ring of Bore1 sets of finite ν measure, where ν is a regular measure in the real line. In this paper we identify every stable random measure by a vector measure . This leads to a method for identifying spectral domain of a certain class of stable proces...

Let X and Y be two Hilbert spaces, and the space of bounded linear transformations from X into Y . Let { A n } ⊂ be a weakly periodic sequence of period T . Spectral theory of weakly periodic sequences in a Hilbert space is studied by H. L. Hurd and V. Mandrekar (Spectral theory of periodically and quasi-periodically stationary S α S sequences, Uni...

Let X and Y be two Hilbert spaces, and \(\mathcal{L}(X,Y)\) the space of bounded linear transformations from X into Y. Let {A
η} ⊂ \(\mathcal{L}(X,Y)\) be a weakly periodic sequence of period T. Spectral theory of weakly periodic sequences in a Hilbert space is studied by H. L. Hurd and V. Mandrekar (1991). In this work we proceed further to charac...

The asymptotic behaviour of the cumulative mean of a reward process 𝒵 ρ , where the reward function ρ belongs to a rather large class of functions, is obtained. It is proved that E 𝒵 ρ ( t ) = C 0 + C 1 t + o (1), t → ∞, where C 0 and C 1 are fully specified. A section is devoted to the dual process of a semi-Markov process, and a formula is given...

The asymptotic behaviour of the cumulative mean of a reward process 풵<sub>ρ</sub>, where the reward function ρ belongs to a rather large class of functions, is obtained. It is proved that E풵<sub>ρ</sub>(t) = C<sub>0</sub> + C<sub>1</sub>t + o(1), t → ∞, where C<sub>0</sub> and C<sub>1</sub> are fully specified. A section is devoted to the dual proc...

A spectral representation for certain sequences of bounded linear transformations between two Hilbert spaces, namely; weakly periodic, is given. In this representation the elements of a weakly periodic sequence, {An, n Z} L(X, Y), are viewed, in the weak sense, as the Fourier coefficients of a certain set function on the Borel sets of [0, 2) with v...

The Fourier analytic approach due to S.M. Berman is considered for a certain class of [alpha]-stable moving average processes, 1 < [alpha] <= 2. It is proved that the local times of such processes satisfy a uniform Hölder condition of order Q1 - 1/[alpha] logQ1/[alpha] for small intervals Q. A decomposition of a stable moving average process into a...

We obtain a certain non-anticipating moving average representation for every regular harmonizable symmetric α stable, SαS, process, 1<α≤2. The noise process in our representation has dependent increments for α<2, but for α=2 it becomes the classical white noise. So in particular, for the case α=2 we recover the well known result that among the stat...

Based on a semi-Markov process J(t), t greater than or equal to 0, a reward process Z(t), t greater than or equal to 0, is introduced where it is assumed that the reward function, rho(k, x) is nonlinear; if the reward function is linear, i.e. rho(k, x) = kx, the reward process Z(t), t greater than or equal to 0, becomes the classical one, which has...

In this paper we provide a characterization for symmetric α-stable harmonizable processes for 1<α≤2. We also deal with the problem of obtaining a moving average representation for stable harmonizable processes discussed by Cambanis and Soltani [3], Makegan and Mandrekar [9], and Cambanis and Houdre
[2]. More precisely, we prove that if Z is an ind...

Our aim in this article is to derive an expression for the best linear predictor of a multivariate symmetric α stable process based on many past values. For this purpose we introduce a definition of dispersion for symmetric a stable random vectors and choose the linear predictor which minimizes the dispersion of the error vector.

Our aim in this article is to derive an expression for the best linear predictor of a multivariate symmetric α stable process based on many past values. For this purpose we introduce a definition of dispersion for symmetric α stable random vectors and choose the linear predictor which minimizes the dispersion of the error vector.

Let $X(t),t \in {\bf R}$, be a symmetric $\alpha$-stable process with independent increments, taking values in ${\bf R}^n $. Let $\mathcal{A} = \overline{\operatorname{sp}} \{ X(t) - X(s),t,s \in {\bf R}\} $. Each $Y \in \mathcal{A}$ is a stable vector, and \[ {\bf E}\exp (i\gamma \cdot Y) = \exp \left( { - \int_S {|\langle {\gamma ,s} \rangle |^\a...

For a function X:T→ℝ d a (γ 1 ,⋯,γ d )-variation is introduced, where T is a rectangle in ℝ N . By using the Fourier analysis techniques and the notion of local time, a sufficient condition for X to have infinite (γ 1 ,⋯,γ d )-variation is given. The result is applied to the study of (γ 1 ,⋯,γ d )-variation of Gaussian fields and (N,d,α)-stable fie...

Let X(t) = [is proportional to]t-[infinity]f(t-s) dZ(s) be a symmetric stable moving average process of index [alpha], 1 < [alpha] [less-than-or-equals, slant] 2. It is proved that when f has a jump discontinuity at a point or when f(x) --> 0 slowly as x [downwards arrow] 0, then almost every sample function of X(t), , is a Janik (J1) function with...

Strong regularity for stationary discrete random fields is discussed. An extension of the classical Beurling's Theorem to functions of several variables is given. Necessary and sufficient conditions for the moving average representation of stationary random fields are obtained. A recipe formula for the best linear extrapolator is also given.

For stable processes which are Fourier transforms of processes with independent increments, we obtain a Wold decomposition, we characterize their regularity and singularity, and, in the discrete-parameter case, we derive their linear predictors. In sharp contrast with the Gaussian case, regular stable processes which are Fourier transforms of proce...

Let X1,X2... be a sequence of i.i.d random variables representing suc- cessive inputs to the moving average process, Yn = 1 K K 1 X i=0 Xn i.

We consider a discrete time periodically correlated process {X.} which is also Markov in the wide sense. We provide closed formulas for the covariance function R (n, m) = EX, X, and for the spectral density f = (fj,) of such a process. Interestingly, we ob- serve that the covariance function, and also the spectral density, is fully specified only b...

By considering certain random vectors on the curve x α +y α =1, we derive new families of distribution functions indexed by 1≤α≤2. The triangle of uniform distributions, the β(1/2,1/2), and the arcsin distribution on (0,1) is furnished by the presented families. Some issues concerning α-symmetric distributions are discussed.

Multivariate reward processes with reward functions of constant rates, defined on a semi-Markov process, first were studied by Masuda and Sumita, 1991. Reward processes with nonlinear reward functions were introduced in Soltani, 1996. In this work we study a multivariate process ()) (,), () (1 t Z t Z t Z p L = , , where are reward processes with n...