Optimality

196 V. H. de la Peña, R. Ibragimov and S. Sharakhmetov
to the study of convergence of the measures of intertemporal dependence of the time
series, including the above multivariate Pearson coefficient φ, the relative entropy δ,
the divergence measures D ψ and the mean information for discrimination between
the dependence and independence I(f0, f; Φ). We obtain the following Theorem 7.2
which deals with the convergence in distribution of m-dimensional statistics of time
series.
Let h : R m → R be an arbitrary function of m arguments, Y be some r.v. and
let ψ be a convex function increasing on [1,∞) and decreasing on (−∞, 1) with
ψ(1) = 0. In what follows, D → represents convergence in distribution. In addition,
{ξn i } and{ξt} stand for dependent copies of{X n i } and{Xt}.
Theorem 7.2. For the double array {Xn i }, i = 1, . . . , n, n = 0,1, . . . let func-
tionals φ2 n,n = φ2 Xn 1 ,Xn 2 ,...,Xn n , δn,n = δXn 1 ,Xn 2 ,...,Xn n , Dψ n,n = D ψ
Xn 1 ,Xn 2 ,...,Xn, ρ(q) n,n =
n
ρ (q)
X n 1 ,Xn 2 ,...,Xn n , q∈ (0, 1),Hn,n = (1/2ρ (q)
n,n) 1/2 , n = 0, 1,2, . . . denote the corresponding
distances. Then, as n→∞, if
n�
i=1
ξ n i
D
→ Y
and either φ 2 n,n→ 0, δn,n→ 0, D ψ n,n→ 0, ρ (q)
n,n→ 0 orHn,n→ 0 as n→∞, then
as n→∞,
n�
i=1
X n i
D
→ Y.
For a time series{Xt} ∞ t=0 let the functionals φ 2 t = φ 2 Xt,Xt+1,...,Xt+m−1 , δt =
δXt,Xt+1,...,Xt+m−1, D ψ
t = D ψ
, ρ(q)
Xt,Xt+1,...,Xt+m−1 t = ρ (q)
, q∈ (0, 1),
Xt,Xt+1,...,Xt+m−1
Ht = (1/2ρ (q)
t ) 1/2 , t = 0, 1,2, . . . denote the m-variate Pearson coefficient, the
relative entropy, the multivariate divergence measure associated with the function
ψ, the generalized Tsallis entropy and the Hellinger distance for the time series,
respectively.
Then, if, as t→∞,
h(ξt, ξt+1, . . . , ξt+m−1) D → Y
and either φ 2 t → 0, δt→ 0, D ψ
t → 0, ρ (q)
t → 0 orHt→ 0 as t→∞, then, as
t→∞,
h(Xt, Xt+1, . . . , Xt+m−1) D → Y.
From the discussion in the beginning of the present section it follows that in the
case of Gaussian processes{Xt} ∞ t=0 with (Xt, Xt+1, . . . , Xt+m−1)∼N(µt,m,Σt,m),
the conditions of Theorem 7.2 are satisfied if, for example,|Rt,m(2Im− Rt,m)|→1
or|Σt,m|/ �m−1 i=0 σ2 t+i → 1, as t → ∞, where Rt,m denote correlation matrices
corresponding to Σt,m and (σ2 t , . . . , σ2 t+m−1) = diag(Σt,m). In the case of processes
{Xt} ∞ t=1 with distributions of r.v.’s X1, . . . , Xn, n≥1, having generalized Eyraud–
Farlie–Gumbel–Morgenstern copulas (3.3) (according to [70], this is the case for any
time series of r.v.’s assuming two values), the conditions of the theorem are satisfied
if, for example, φ2 t = �m �
c=2 i1