Vtyjkyuk

Let $(Omega, mathcal F, mathbb P)$ be a probability space and consider a sequence $X = (X_n)_nin mathbb N$ of integrable random variables, let $X_inftyin L_1$ and $alpha > 0$.

Say that $X$ is convergent of order $alpha$ to $X_infty$ if there exists a $Cgeq 0$ s.t.

$$|X_n- X_infty|_1 leq Cfrac1n^alpha, forall n geq 1.$$

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.

• $(X)_nin mathbb N to_alpha X$


• $X to_alpha X_infty$, $n : mathbb N to mathbb N$ monotone $Rightarrow$ $(X_n(k))_k to X_infty$


• $X to_alpha X_infty$ and $X'to_alpha X_infty$ $Rightarrow Xcap X'to_alpha X_infty$

where $(Xcap X')_2n = X_n$ and $(Xcap X')_2n-1 = X'_n$.

What about the following one:

Is is true that $Xto_alpha X_infty$ and ($X in A $ eventually $Leftrightarrow X'in A$ eventually) $Rightarrow X'to_alpha X_infty$?

Here $Xin A$ eventually if $X_nin A$ for large $n$.

It would be nice to have (*), but I don't see how this could be proven.

I was thinking of $A_k := X - X_infty$, then $Xin A_k$ eventually, hence $X'in A_k$ eventually.
In other words

$$forall kin mathbb N : exists i geq 0 : forall j geq i : |X_j' - X_infty|_1 leq C frac1k^alpha$$

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 $X'_n = X_sqrtn$, where the index is rounded down. Certainly $X'$ has the same eventuality set as $X$. However,
$$
|X'_n-X_infty|_1=|X_sqrtn-X_infty|_1le C/(sqrtn)^alpha=C/n^alpha/2
$$
so you can only say $X'to_alpha/2X_infty.$