Suppose that $\{X_i\}$ is an $\mathrm{AR}(r)$, defined by:
$X_{i}= h(i) + \varepsilon_i $,
$h(i)=\alpha_1 X_{i-1} + \dots + \alpha_{r} X_{i-r}$
where $\{\varepsilon_i\}$ are i.i.d. ${\cal N}(0,1)$.
QUESTION: Does $(X_{i}-h_i)X_{i-k}$, for $k=1..r$, satisfies the Law of Iterated Logarithm -- LIL?
In other words, is the equation (EQ1) below true?
$$ \limsup\limits_{n \rightarrow \infty} \frac{ \sum\limits_{i=1}^n (X_{i}-h_i)X_{i-k} }{\sqrt{n \log\log n}} = c $$
for $c\in(0,\infty)$.
More details
Using Meyn & Tweedie, I could conclude that:
- The AR process is a Linear State Space Model -- LSS (page 24)
- LSS is a Markov Chain (page 9)
- LSS with some conditions satisfies LIL (Theorem 17.0.1) for $V(x)=|x|^2 +1$ (page 456)
But, this $V(x)$, apparently, is not sufficient to provide "(EQ1)".
Basically, the Theorem 17.0.1 provides the LIL for $g^2 \leq V$.
In our case, $g((X_{i-r},\dots,X_i))=(X_{i}-h_i)X_{i-k} $
Thank you.
Paul