Let $(\Omega, \mathcal F, \mathbb P)$ be a probability space and consider a sequence $X = (X_n)_{n\in \mathbb N}$ of integrable random variables, let $X_\infty\in L_1$ and $\alpha > 0$.
Say that $X$ is convergent of order $\alpha$ to $X_\infty$ if there exists a $C\geq 0$ s.t.
$$\|X_n- X_\infty\|_1 \leq C\frac{1}{n^\alpha}, \forall n \geq 1.$$
In this case I will write $X \to_\alpha X_\infty$. The motivation for this comes from the Euler-Maruyama method for numerically solving SDE's.
One can prove some nice properties, e.g.
- $(X)_{n\in \mathbb N} \to_\alpha X$
- $X \to_\alpha X_\infty$, $n : \mathbb N \to \mathbb N$ monotone $\Rightarrow$ $(X_{n(k)})_k \to X_\infty$
- $X \to_\alpha X_\infty$ and $X'\to_\alpha X_\infty$ $\Rightarrow X\cap X'\to_\alpha X_\infty$
where $(X\cap X')_{2n} = X_n$ and $(X\cap X')_{2n-1} = X'_n$.
What about the following one:
Is is true that $X\to_\alpha X_\infty$ and ($X \in A $ eventually $\Leftrightarrow X'\in A$ eventually) $\Rightarrow X'\to_\alpha X_\infty$?
Here $X\in A$ eventually if $X_n\in A$ for large $n$.
It would be nice to have (*), but I don't see how this could be proven.
I was thinking of $A_k := \{ X : \|X - X_\infty\|_1 \leq C \frac{1}{k^\alpha}\}$, then $X\in A_k$ eventually, hence $X'\in A_k$ eventually. In other words
$$\forall k\in \mathbb N : \exists i \geq 0 : \forall j \geq i : \|X_j' - X_\infty\|_1 \leq C \frac{1}{k^\alpha}$$
But it doesn't seem like that's enough to conclude that $X'\to_\alpha X_\infty$.
(*) In this case this subsequential structure could be induced by a convergence structure.
I think the following map be a counterexample. For any sequence $X$, define $X'$ by $X'_n = X_{\sqrt{n}}$, where the index is rounded down. Certainly $X'$ has the same eventuality set as $X$. However, $$ \|X'_n-X_\infty\|_1=\|X_{\sqrt{n}}-X_\infty\|_1\le C/(\sqrt{n})^\alpha=C/n^{\alpha/2} $$ so you can only say $X'\to_{\alpha/2}X_\infty.$