I am trying to understand the asymptotic behaviour of stochastic processes of the form $\{X_{t,T}\}_{t=1, \dotsc,T}$ where the time is rescaled to $[0, 1]$, i.e. with $T \rightarrow \infty$ we get more and more observations of the series on the time interval $[0, 1]$.
Locally stationary processes
I am trying to understand the concept of "locally stationary processes" as introduced in this paper, page 4:
A stochastic process $X_{t,T}$ is locally stationary if there exist $p>0, C<\infty$ such that:
For each $u \in [0, 1]$ there exists a positive random variable $U_{t,T}(u)$ with $E(U^p) < \infty$ and a stationary process $\{\tilde X_t(u)\}$ such that:
$|X_{t,T} - \tilde X_t(u)| \leq \left( |\frac{t}{T}-u| + \frac{1}{T}\right)U_{t,T}(u)$ almost sure
I.e. for $t/T \approx u$, the process $\{X_{t,T}\}$ can be approximated by the process $\{\tilde X_t(u)\}$ in a stochastic sense.
What to prove
I am trying to prove that a (nonstationary) time-varying AR(1) process, $X_{t, T} =\varepsilon_t + \alpha(t/T) X_{t-1,T}$ is locally stationary w.r.t. the corresponding (stationary) AR(1) process $\tilde X_t(u) = \varepsilon_t + \alpha(u) \tilde X_{t-1}(u)$. The function $\alpha$ is smooth.
My ideas
Using the triangle inequality, we get $|X_{t,T} - \tilde X_{t}(u)| \leq \underbrace{|X_{t,T} -\tilde X_{t}(t/T)|}_{(*)} + \underbrace{|\tilde X_{t}(t/T) - \tilde X_{t}(u)|}_{(**)}$.
It is well known that an AR(1) process can be written as a particular MA($\infty$) process, so we can replace the terms in $(**)$ by the following infinite sums:
$\left| \sum_{i=0}^{\infty} \alpha(t/T)^i \varepsilon_{t-i} - \sum_{j=0}^{\infty} \alpha(u)^j \varepsilon_{t-j}\right|$
... and from there we can proceed. (Probably all the inequalities work similarly for $(*)$.)
Any ideas on how to proceed?