Convergence of a Test Statistic

115 Views Asked by At

I'm reading a paper of Shao and Zhang: Testing for Change Points in Time series. In this paper they claim the following: The are testing whether there is a change in the mean of a time series. So

$H_{0}:\mathbb{E}\left[X_{1}\right]=\mathbb{E}\left[X_{2}\right]=\ldots=\mathbb{E}\left[X_{n}\right]=\mu$

$H_{1}:\mathbb{E}\left[X_{1}\right]=\ldots =\mathbb{E}\left[X_{k^{*}}\right]\neq \mathbb{E}\left[X_{k^{*}+1}\right]=\ldots\mathbb{E}\left[X_{n}\right]$

They define the following statistic:

$T_{n}\left(\left[nr\right]\right)=\sum_{\ell=1}^{\left[nr\right]}\left(X_{\ell}-\overline{X_{n}}\right)$ for $r\in\left[0,1\right]$.

They claim that the Sequence $T_{n}\left(\left[nr\right]\right)$ converges to a Brownian Bridge under $H_{0}$ as $n\rightarrow\infty$. But what happens under $H_{1}$ and why?