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.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Browse other questions tagged or ask your own question.