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I have a LR statistic that I know behaves like:

$$\Lambda_n = \gamma_n + O_p(1),$$

where $\gamma_n$ has asymptotic distribution $\chi^2_k$, $k\geq 1$.

What can I say about the rate of $\Lambda_n$? Is it $O_p(1)$?

https://en.wikipedia.org/wiki/Big_O_notation

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Yes. You have convergence in distribution, this implies boundedness in probability, see lemma 5.3 here. (Sorry for the short answer, that's all there is to it).

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