Let $(X_n)$ be a sequence of nonnegative random variables, and consider the following null and alternative hypotheses:
$$H_0: X_n=c+o_p(1) \quad \text{as}\quad n\to\infty$$
$$H_1: X_n=O_p(1/n) \quad \text{as}\quad n\to\infty$$
where $c>0$ is an unknown constant. How can I construct a test statistic $\xi_n=f(X_n)$ to test this null hypothesis? The statistic $\xi_n$ should converge to a null distribution under $H_0$ and be consistent under $H_1$.
If we consider the sequence $Y_n=n X_n$, then the previous two hypotheses become
$$H_0: Y_n\overset{p}\to\infty \quad \text{as}\quad n\to\infty$$
$$H_1: Y_n=O_p(1) \quad \text{as}\quad n\to\infty$$
Any ideas how to proceed here? Thanks a lot for your help.
Note that $1/\sqrt{n}$ goes to 0 slower than $1/n$. Now consider $\xi_n= 1\{X_n>1/\sqrt{n}\}$.
Under $H_0$, let $Z_n:=X_n-c=o_p(1)$. Then if $1/\sqrt{n}<c/2$:
$$P(\xi_n=0)=P(X_n\leq 1/\sqrt{n})=P(Z_n\leq 1/\sqrt{n}-c)\leq P(Z_n \leq -c/2)\to 0.$$ So $\xi_n \to 1$ (in distribution).
Under $H_1$, let $Y_n:=nX_n= O_p(1)$. Then $$P(\xi_n=0)=P(X_n\leq 1/\sqrt{n})=P(Y_n\leq\sqrt{n})\to1.$$ So $\xi_n \to 0$ (in distribution).
I wouldn't use this in practice, but in theory it works.