This is simplified from the assumptions in many papers regarding the existence of limit distribution of a least square estimator.
Assume $\{x_n\}$ is a sequence of random variables such that the mean converge in probability to some real number $c$: $$\frac {x_1+\cdots+x_n}{n}\to^p c$$
Is it true that $$\frac {\sum_{i=n/2}^{n/2+\sqrt n}x_i}{\sqrt n}\to^pc?$$
or can we say as $n\to\infty$: $$\frac {\sum_{i=n/2}^{n/2+\sqrt n}x_i}{\sqrt n}=O_p(1)?$$