Consider a m-vector linear process
where are i.i.d. m-vector random variables with p.d.f. on , are matrices depending on a parameter vector .
Assume the following conditions are satisfied
A1 i) For some
where denotes the sum of the absolute values of the entries of .
ii) Every is continuously two times differentiable with respect to , and the derivatives satisfy
for where .
iii) for and can be expanded as follows:
where , satisfy
iv) Every is continuously two times differentiable with respect to , and the derivatives satisfy
A3 The continuous derivative of exists on .
From A1 the linear process can be expressed as
From A1 it can be seen that
Let and be the probability distributions of and , respectively. Then
For two different values , the likelihood-ration is
Then the following theorem follows
The sequence of experiments
is asymptotically equicontinuous on compact subset of . That is
(i) , the log-likelihood ratio admits, under the hypothesis (i.e. ), as , the asymptotic representation
(note that by Taylor expansion )
(ii) Under ,
(iii) and all , the mapping
is continuous w.r.t. the variational distance
A proof of the theorem will follow in a future post.