A Study on some Stochastic models in Time series

A first order autoregressive model is introduced which is an extension of the EAR(1) process of Gaver and Lewis (1980) if the marginals are exponentially distributed. Discrete version of the model is introduced and studied. Some applications of the models are also mentioned.


Introduction
Class -L distributions and geometrically infinitely divisible distributions have important role in time series modelling. A distribution with characteristic function φ(λ) is in class −L if for each ρ, 0 < ρ < 1, φ(λ)/φ(ρnλ) is a characteristic function. A random variable Y is said to be geometrically infinitely divisible if for every p ∈ (0, 1), Y = Np j=1 X (p) j , where N p is a geometric random variable such thatP {N p = k} = p(1 − p) k−1 , k = 1, 2 . . . and X (p) j , j = 1, 2, . . . are independent and identically distributed random variables and Y, N p , X (p) j are independent.
Jayakumar and Pillai(1992) characterized semi α -Laplace distribution which was introduced by Pillai(1985). A distribution with characteristic function f (λ) is called semi α-Laplace if f (λ) = 1 1+ψ(λ) , where ψ(λ) satisfies ψ(λ) = aψ(bλ), 0 < b < 1 and a is the unique solution of ab α = 1 for some 0 < α ≤ 2. For distributions with positive support, if 0 < α < 1, f (λ) is the characteristic funtion of semi MittagLeffler distribution. Gaver and Lewis(1980) proved that only class L distributions can be marginals of stationary first order autoregressive equation X n = ρX n−1 + n , 0 < ρ < 1 and { n } is a sequence of independent and identically distributed random variables. Here we introduce a model which is a generalization of this and is presented in section 2. The model is also solved by assuming exponential marginals under stationarity. The discrete version of this model is given in section 3. In section 4 some applications of the models are given.

The Autoregressive model
Let {X n , n ≥ 1} be a discrete time stochastic process on (0, ∞) defined by where 0 < ρ ≤ 1, 0 ≤ p < 1 and { n } is a sequence of independent and identically distributed random variables such that n is independent of X n−1 . Clearly if a solution exists, then the process is Markovian. Denoting the Laplace transform of the random variable X by φ X (λ), from (1) we arrive at the following.
If we assume that the process is stationary, we get Now we have the following theorem. proof Assume that the model (1) exists under stationarity. Then (3) with ρ = 1 has solution for all p ∈ (0, 1). From (3) we get, Hence X is geometrically infinitely divisible.
Conversely assume that X is geometrically infinitely divisible. Then for each q ∈ (0, 1), there exists a sequence of independent and identically distributed random variables Y Hence the proof of the theorem is complete.
where {X j , j ≥ 1} is a sequence of independent and identically distributed random variables which are identically distributed as X and N c is a geometric random variable with P (N c = n) = c(1 − c) n−1 , n ≥ 1 Thus X is the geometric sum of its own type and by Jayakumar and Pillai(1993), we have the following theorem.
Theorem 2.2 Assume that the process in (1) is stationary. Then X has semi-Mittag -Leffler distribution with exponent α ∈ (0, 1) if and only if X n d = n .

Remark 2.3
If the distribution of X n has real support then Theorem 2.2 hold good, with X n distributed as semi α -Laplace.
The bivariate Laplace transform of (X n , X n−1 ) is given by If X is exponential with mean s, we have φ X (λ) = 1 1+sλ . Therefore, Thus the innovations are mixtures of exponentials. Now we have the following.
Clearly the correlation lies between 0 and 1.

The Integer Valued First-Order Autoregressive model
Here wendefine and study the discrete version of the model in section 2. Let {X n , n ≥ 1} be a non-negative integer valued stochastic process defined by where 0 < ρ ≤ 1, 0 ≤ p < 1 and { n } is a sequence of independent and identically distributed innovations taking non-negative integer values such that n is independent of X n−1 and ρ * X n−1 = X n−1 j=1 N j where N j 's are independent and identically distributed Bernoulli random variables with P (N j = 1) = ρ = 1 − P (N j = 0) , 0 < ρ < 1 Now we will show that in the stationary case the model (7) is properly defined by assuming geometric marginals. The probability generating function of (7) is given by the following.

Conclusion
In this paper we discussed the autoregressive model, the integer valued first order autoregressive model and proved some theorems.