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Suppose we have a family of random variables $\{ X_{m,n} \}_{m=1}^\infty$ that are not necessarily mutually independent or identically distributed, but that all do satisfy $X_{m,n} = O_p(1)$ as $n \to \infty$, i.e., for each $m \in \mathbb{N}_+$ we have

$$\forall_{\varepsilon>0} \exists_{N_\varepsilon, \delta_\varepsilon} : \mathbb{P}( | X_{m,n} | > \delta_\varepsilon ) \leq \varepsilon \, \forall_{n > N_\varepsilon}$$

Additionally, we have a sequence of numbers $\{ a_n \}_{n=1}^\infty$ where each $a_n \in \mathbb{N}$. Define

$$S_n = \sum_{m=1}^{a_n} X_{m,n}$$

Can we conclude that $S_n = O_p(a_n)$?

Example. If the $a_n = a$ are independent of $n$, this holds since $O_p(1) + O_p(1) = O_p(c)$ for any constant $c$. This case pertains to Sum of bounded in probability random variables. The question here is primarily about what happens when the $a_n$ depend on $n$.

Note 1. When $a_n = n$ and the random variables are identically distributed, this question is similar to Is average of little o_p(1) still little o_p(1)?. Unfortunately, the question wasn't answered.

Note 2. When $a_n = n$, an answer is given in Convergence to infinity of a sum of independent random variables under the stronger assumptions that the $X_{m,n}$ are almost surely bounded and independent.

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