THEORETICAL DEPENDENCE CONCEPTS FOR STOCHASTIC PROCESSES

Abstract

We in this paper obtained the theoretical results for multivariate stochastic processes which help us to extended negatively orthant dependent(ENOD) structures among hitting times of the processes. In addition, some applications are given to illustrate these concepts.

1. Introduction

We first present some definitions in this section.

Definition 1.1 (Lehmann(1966)). Random variables X and Y are said to be negatively dependent(ND) if

for all x, y ∈ R. A collection of random variables is said to be pairwise negatively dependent(PND) if every pair of random variables in the collection satisfies (1.1). It is important to note that (1.1) implies that

for all x, y ∈ R. Moreover, it follows that (1.2) implies (1.1), and, hence, (1.1) and (1.2) are equivalent. However, (1.1) and (1.2) are not equivalent for a collection of 3 or more random variables. Consequently, the following definition is needed to define sequences of negatively dependent random variables.

Definition 1.2 (Joag-Dev and Proschan(1983)). A sequence {Xi, 1 ≤ i ≤ n} of random variables is said to be negatively upper orthant dependent(NUOD) if for all real numbers x1, ··· xn,

and it is said to be negatively lower orthant dependent(NLOD) if for all real numbers x1, ··· xn,

A sequence {Xi, 1 ≤ i ≤ n} of random variables is said to be negatively orthant dependent(NOD) if it is both (1.3) and (1.4).

Definition 1.3 (Liu(2009)). A sequence {Xi, 1 ≤ i ≤ n} of random variables is said to be extended negatively upper orthant dependent(ENUOD) if for all real numbers x1, ··· xn, there exists a constant M > 0 such that

and it is said to be extended negatively lower orthant dependent(ENLOD) if for all real numbers x1, ··· xn, there exists a constant M > 0 such that

A sequence {Xi, 1 ≤ i ≤ n} of random variables is said to be extended negatively orthant dependent(ENOD) if it is both (1.5) and (1.6).

Lehmann(1966) introduced various concepts of dependence for two random variables. Esary and Proschan(1972) were later developed the stronger notio ns of bivariate dependence. Ahmed et al.(1978), Ebrahimi and Ghosh(1981), and Joag-Dev and Proschan(1983) obtained multivariate versions of various bivariate positive and negative dependence as described by Lehmann, and Esary and Proschan. In addition, for other related dependence concepts, many authors had been generalized and extended to several directions and their concept has been very useful in reliability theory and applications; (see Baek(1995), Barlow and Proschan(1975), Block et al.(1981, 1983, 1988), Brindly and Thompson(1972), Choi and Baek(2013), Glaz and Johnson(1982)). Recently, Liu(2009) introduced the concepts of extended negatively dependence in the multivariate case. Concepts of dependence have subsequently been extended to stochastic processes in different directions by many authors;(see Berman (1977), (1978) and Cox and Isham(1980), Ebrahimi(1987, 1994), Ebrahimi and Ramallingam(1988, 1989), Friday(1981), Marshall and Shaked(1983)).

Multivariate stochastic processes arise when instead of observing a single process we observe several processes, say X1(t); ···, Xn(t), simultaneously. For example, we may want to study the simultaneous variation of current and voltage, or temperature, pressure and volume over time and we also may be interested in studying inflation rates and money supply, unemployment and interest rates. We could, of course, study each quantity on its own and treat each as a separate univariate process. Although this would give us some information about each quantity it could never give information about the interrelationship between various quantities. This leads us to introduce some concepts of negative dependence for multivariate stochastic processes. To introduce the new ideas involved in the study of multivariate processes we consider the multivariate stochastic processes.

The main purpose of this paper introduce various concepts of extended negative dependence for multivariate stochastic processes, namely extended negatively orthant dependent(ENOD), negatively associated(NA), right corner set decreasing(RCSD), and right tail decreasing in sequence(RTDS). The theoretical results of these concepts are studied in Section 3. Some applicat ions of these concepts are developed in Section 4.

 

2. Preliminaries

In this section we present definitions and notations which will be used through out this paper. Suppose that {X1(t)|t≥0} ···, {Xn(t)|t≥0} are stochastic processes. The state space of Xi(t) will be taken to be a subset Ei of real line R+ = [0, ∞]; i = 1, 2, ···, n. For any state ai ∈ Ei, i = 1,2, ···, n, we now define the random times as follows,

that is, Si(ai) is the first hitting times that the process Xi(t) reaches ai.

Definition 2.1. Stochastic processes {X1(t)|t≥0}, ···, {Xn(t)|t≥0} are said to be extended negatively upper orthant dependent(ENUOD) if for all ti and ai, i = 1, 2, ···, n, there exists a constant M > 0 such that

and they are said to be extended negatively lower orthant dependent (ENLOD) if for all ti and ai, i = 1, 2, ···, n, there exists a constant M > 0 such that

Stochastic processes {X1(t)|t≥0}, ···, {Xn(t)|t≥0} are said to be extended negatively orthant dependent(ENOD) if it is both ENUOD and ENLOD.

It is clear that stochastic processes {X1(t)|t≥0}, ···, {Xn(t)|t≥0} are called NOD if (2.1) and (2.2) hold when M = 1. Obviously, an NOD stochastic processes must be an ENOD stochastic processes. Therefore, the ENOD structure is substantially more comprehensive than the NOD structure and ENOD structure can reflect not only a negative dependence structure but also positive one, to some extend.

Definition 2.2. Stochastic processes {X1(t)|t≥0}, ···, {Xn(t)|t≥0} are said to be negatively associated if for all ai, i = 1, 2, ···, n, and increasing functions f and g for which the covariance exists,

Definition 2.3. Stochastic processes {X1(t)|t≥0}, ···, {Xn(t)|t≥0} are said to be right corner set decreasing(RCSD) if for all ai and ti, i = 1, ···, n,

decreasing in t′1, ···, t′n for every choice of t1, ···, tn.

Definition 2.4. Stochastic processes {X1(t)|t≥0}, ···, {Xn(t)|t≥0} are said to be right tail decreasing in sequence(RTDS) if for all ai and ti, i = 1, 2, ···, n,

decreasing in t1, ···, ti−1.

 

3. Theoretical Results for Multivariate Stochastic Processes

Definition 3.1. Suppose that stochastic processes {X1(t)|t≥0}, ···, {Xn(t)|t≥0} are RCSD. Then {X1(t)|t≥0}, ···, {Xn(t)|t≥0} are RTDS.

Proof. Let stochastic processes {X1(t)|t≥0}, ···, {Xn(t)|t≥0} are RCSD.

Then is decreasing in t′1, ···, t′n for all choices of t1, ···, tn. Therefore, for fixed j,

is decreasing in t′1, ···, t′n for all choices of tj. Now putting t′i → 0 for all i = 1, ···, j-1, we can obtain that

is decreasing in t′1, ···, t′j-1 for all choices of tj. Since j is arbitrary, {X1(t) t≥0}, ···, {Xn(t)|t≥0} are RTDS.

Next, we now show that RTDS implies ENOD.

Theorem 3.2. Suppose that the stochastic processes {X1(t)|t≥0}, ···, {Xn(t)|t≥0} are RTDS, then they are ENOD.

Proof. We prove only this result for RTDS implies ENUOD.

The proof of the ENLOD is similar to that of ENUOD.

Theorem 3.3. Suppose that stochastic processes {X1(t)|t≥0}, ···, {Xn(t)|t≥0} are ENOD and let f1, ···, fn are nonnegative increasing functions. Let Y1, Y2, ···, Yn be independent of {X1(t)|t≥0}, ···, {Xn(t)|t≥0} and P(S1(a1) > t1), ···, P(Sn(an) > tn) be increasing or decreasing in both a1, a2, ···, an and t1, t2, ···, tn, then {f1(X1(t))+Y1|t≥0}, ···, {fn(Xn(t))+Yn|t≥0} are ENOD.

Proof. We prove only this result for ENUOD. Let Si(ai) = in f{s|fi(Xi(s))+Yi ≥ ai} and Ti(bi)=in f{t|Xi(t) ≥ bi}, i=1, ···, n. Then

The proof of ENLOD is similar to that proof of ENUOD.

Remark 3.1. Theorem 3.3 can be proved for NA(RCSD(RTDS)) concepts of dependence.

Theorem 3.4. Suppose that stochastic processes (a) {X1(t)|t≥0}, ···, {Xn(t)|t≥0} are ENOD, (b) {X1(t)|t≥0}, ···, {Xn(t)|t≥0} are multivariate stochastic processes with state spaces {a1, b1}, {a2, b2}, ···, {an, bn}, respectively, and (c) max (a1, b1), max (a2, b2), ···, max(an, bn) are absorbing states. Then for all a1, a2, ···, an and t1, t2, ···, tn,

Proof. We prove only this result for ENUOD. Without loss of generality assume that a1 < b1, a2 < b2, ···, an < bn. Now,

Next, we prove that the next limit theorem demonstrates preservation of the ENOD among the hitting times.

Theorem 3.5. Let {X1n(t)|t≥0}, ···, {Xpn(t)|t≥0} be RTDS stochastic processes with distribution functions Hn such that Hn weakly converges to H as n → ∞ where H is the distribution functions of stochastic processes {X1(t)|t≥0}, ···, {Xp(t)|t≥0}. Then {X1(t)|t≥0}, ···, {Xp(t)|t≥0} are ENOD.

Proof. We prove only this result for ENUOD.

The proof of the ENLOD is similar to that of ENUOD.

Remark 3.2. If we change RTDS to ENOD, we can get results of ENOD.

Theorem 3.6. Let be processes and let { (X1n, ···, Xkn|n ≥ 1 } be a k-variate stochastic processes such that (a) (X11, ···, Xk1, (X12, ···, Xk2), ··· are independent and (b) X1i, ···, Xki are ENOD, i = 1,2, ···, (c) N(t) is a Poisson process which is independent of are ENOD.

Proof. We prove only this result for ENLOD.

The proof of the ENUOD is similar to that of ENLOD.

Remark 3.3. Let (a) and (c) hold, and if X1i, ···, Xki are NA(RCSD(RTDS)), i = 1, 2, ···, then we can get that are NA(RCSD(RTDS)).

Theorem 3.7. Let { (X1n, X2n, X3n, ···, Xkn)|n ≥ 1 } be stochastic processes such that (X11, X21, X31, ···, Xk1), (X12, X22, X32, ···, Xk2), ··· are independent and X1i, X2i, X3i, ···, Xki, i = 1, 2, ··· are ENOD. Then {(X1n, X2n, X3n, ···, Xkn|n ≥ 1} is ENOD.

Proof. Let P(T1(x1) > n1, T2(x2) > n2, ···, Tn(xn) > nk)

Now consider the following cases.

Case 1. When n1 = n2 = n3 = ··· = nk,

Case 2. When n1 < n2 < n3 < ··· < nk,

Case 3. The proof of n1 > n2 > n3 > ··· > nk is similar to Case 2.

 

4. Applications

Application 4.1. We consider the uniformly modulated model(see Priestly (1988)) such that non-stationary processes X(t) = {X1(t), X2(t), ···, Xn(t)|t ≥ 0} and Y(t) = {Y1(t), Y2(t), ···, Yn(t)|t ≥ 0} given by

where a(t) is a deterministic continuous function such that a(t) ≥ 0 and Y(t) is non-negative stationary process. If Y(t) is ENOD, we can show that X(t) is ENOD. Using the Theorem 3.3, if fi, i = 1, ···, n are increasing functions, we can obtain that f1(S1(a1)), ···, fn(Sn(an)) are ENOD, where Si(ai) = in f {n|Xi(n) ≥ ai}, i = 1, ···, n.

Application 4.2. Suppose that we are given a system with n components which is subjected to shocks and assume that N(t) be the number of shocks received by time t and let be total damages to components 1, 2, ···, n by time t, respectively, where X1i, X2i, ···, Xni and Z1i, Z2i, ···, Zni are damages to components 1, 2, ···, and n by shocks, respectively. Let (X1i, ···, Xni) and (Z1i, ···, Zni) be ENOD respectively, i = 1, 2, ···, and let (X11, ···, Xn1), (X12, ···, Xn2) ··· and (Z11, ···, Zn1), (Z12, ···, Zn2) ··· are independent and have increasing paths. Then by Theorem 3.6, we can get that {Y1(t)|t≥0}, ···, {Yn(t)|t≥0} are ENOD.

Application 4.3. Let Zi(t), i = 1, 2, ···, n be the strength of system i at time t and let Di, i = 1, 2, ···, n be i.i.d. positive random variables denoting the damage to either system due to the ith shock and N(t) be the number of shocks occurring by time t. Then the stress experienced by either system at time t is given by the process Suppose that Z1(t), ···, Zn(t) are independent processes with decreasing sample paths and that the processes X(t), Z1(t), ···, Zn(t) are independent, then we can obtain using Theorem 3.6 and Application 4.2 that X(t) - Z1(t), X(t) - Z2(t), ···, X(t) - Zn(t), are ENOD processes. Thus the lifetimes of the multivariate systems, namely, Si(ai) = in f{t|X(t)-Zi(t)≥ai}, i = 1, 2, ···, n are ENOD random variables. Therefore, we can get that

and