Page 37 - Textos de Matemática Vol. 47

THE TAYLOR PROPERTY IN AR AND BILINEAR PROCESSES 27
In what follows, we denote by KX the kurtosis coe cient of Xt, that is, KX = μ4,X/(μ2,X)2, where μk,X is the centered moment of order k of Xt.
2. The Taylor property in first-order non-negative autoregressive models
Let X = (Xt, t 2 Z) be a real stochastic process such that
Xt = Xt 1 + "t (2.1)
where 0 < < 1 and " = ("t , t 2 Z) is a sequence of non-negative identically distributed random variables with moments up to the fourth order and such that E "it|"t 1 = mi, i = 1, 2, 3, t 2 Z, with "t 1 the -algebra of the past of "t, that is, "t 1 = ("t 1, "t 2, ...). So, E "it = E E "it|"t 1 = mi, i = 1, 2, 3. Naturally, let us denote E "4t by m4, t 2 Z.
The following lemma summarizes, in these conditions, the expressions of the autocorrelation functions of X and X2 for h 2 N.
Lemma 2.1. (Gon¸calves, Martins and Mendes-Lopes, 2014) We have
a) ⇢X (h)= h, h2N.
2 h h 2 h C o v X t , X t2
+ 2 m 1 1 V ( X t2 ) , h 2 N .

b ) ⇢ X 2 ( h ) =
We note that ⇢X2 is positive, since Cov Xt, Xt2 0 as Xt is non-negative.
The following result follows immediately.
Theorem 2.2. (Gon¸calves, Martins and Mendes-Lopes, 2014) The process X defined in (2.1) satisfies the Taylor property if and only if
Cov Xt,Xt2 < 1 . V ( X t2 ) 2 m 1
The studies on the Taylor property previously referred have shown a strong relation between its occurrence and strong values of the kurtosis of the process X. In order to explore this relation, we rewrite the previous condition in terms of this coe cient.
Corollary 2.3. The process X defined in (2.1) satisfies the Taylor property if and only if
KX >1 2E(Xt)μ3,X. V ( X t2 )
Let us note that, as KX 1, we may conclude that if μ3,X 0 then the Taylor property is present in the process X, or equivalently, if the genera- tor process is symmetric or right-skewed distributed, taking into account that μ3,X = μ3,"/(1 3).