Testing epidemic change in nearly nonstationary process with statistics based on residuals

We study an epidemic type change in innovations of a first order autoregressive process $ y_{n,k} = \varphi_n y_{n,k-1} + \epsilon_{k} + a_{n,k}$, where $\phi_n$ is either a constant in $(-1,1)$ or a sequence in $(0,1)$, converging to 1. For $k$ inside some unknown interval $\mathbb{I}_n^\ast=(k^\ast,k^\ast+\ell^\ast]$, $a_{n,k}=a_n$ while $a_{n,k}=0$ for $k$ outside $\mathbb{I}_n^\ast$. When $a_n\neq 0$, we have an epidemic deviation from the usual (zero) mean of innovations. Since innovations are not observed, we build uniform increments statistics on residuals $(\widehat{\epsilon}_k)$ of the process $y_{n,k}$. We assume that innovations $(\epsilon_k)$ are regularly varying with index $p \ge 2$ or satisfies integrability condition $\lim_{t \to \infty} t^p P(|\epsilon_1| > t) = 0$ for $p > 2$ and $E\epsilon_k^2 < \infty$ for $p=2$. We find the limit distributions of the tests under no change and prove consistency under short epidemics that is $\ell^\ast=O(n^\beta)$ for some $0<\beta\le 1/2$.