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Let $x$ be a random vector over $\mathbb{R}^n$, for some $T \subseteq \mathbb R$ we have a function $g : \mathbb{R} ^n \times T \to \mathbb R$. Let $a_n$ be a positive sequence with zero limit and $x_1, \ldots, x_n$ a random sample with the same distributión of $x$.

A uniform law of large numbers could be something of the flavour
$$\sup_{t \in T} |E (g(x, t)) - \frac{1}{n}\sum_{i=1}^n g(x_i, t)| = \mathcal{O}_p( a_n) $$ I wish to extend this notion a bit, where $a_n$ is actually dependent of $t$. The mathematical definition would be similar to the big-O convergence.

Def: Given $\varepsilon > 0$ there exists a constant $D < \infty$ such that for all $n \in \mathbb{N}$ and for all $t \in T$

$$P\left( |E(g(x,t)) - \frac{1}{n}\sum_{i=1}^n g(x_i, t)| \leq D a_n(t) \right) \geq 1-\varepsilon$$

This definition implies the previous one if $a_n(t) = a_n$. Note that i have to get rid of the $\sup$ because I found no way to use it in the new definition.

I wish to know if there is a name and notation for this type of convergence also any reference would be appreciated.

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