Let $\\{x_t^m\\}$ and $\\{y_t^{m}\\}$ be collections of sequences (for $m\in [1,2,\dots,M]$) converging to $x$ as $t\to\infty$. Furthermore, suppose that $||x_t^m - x|| \leq ||y_t^{m} - x||$ for all $t,m$.
Are there any assumptions that can be made so that statements about the distance of the average over $m$ of the respective collections of sequences from $x$ can be made? That is, I am curious about sufficient conditions for the following statement:
\begin{equation} ||\frac{1}{M}\sum_{m=1}^M x_t^m - x|| \leq ||\frac{1}{M}\sum_{m=1}^{M} y_t^{m} - x|| \end{equation}
For instance, one way to make this true is to assume a condition such that
\begin{equation} ||\frac{1}{M}\sum_{m=1}^M x_t^m - x|| = \frac{1}{M}\sum_{m=1}^M ||x_t^m - x|| \leq \frac{1}{M}\sum_{m=1}^M||y_t^{m} - x|| = ||\frac{1}{M}\sum_{m=1}^M y_t^{m} - x|| \end{equation}
For instance, if $x_t^m=y_t^{m}=x$, then this is (trivially) satisfied.
Are there other less trivial sufficient conditions where $||\frac{1}{M}\sum_{m=1}^M x_t^m - x|| \leq ||\frac{1}{M}\sum_{m=1}^{M} y_t^{m} - x||$ holds?
For $n\in \Bbb{N}^*$, let $x_n = \frac{1}{n}-\frac{1}{n\sqrt{n}}$ and $y_n = \frac{(-1)^{n+1}}{n}$. One has \begin{equation} \forall n\in \mathbb{N}^*,\quad |x_n| < |y_n| \end{equation} but when $N\to\infty$, \begin{equation} \frac{1}{N}\sum_{n=1}^{N} x_n \sim \frac{\log N}{N}\quad\text{but}\quad \frac{1}{N}\sum_{n=1}^{N} y_n \sim \frac{\log 2}{N} \end{equation} So the average of $x_n$ converges slower to $0$ than the average of $y_n$.