From WIkipedia
Given a Polish space $\mathcal{X}$, let $\{ \mathbb{P}_N\}$ be a sequence of Borel probability measures on $\mathcal{X}$, let $\{a_N\}$ be a sequence of positive real numbers such that $\lim_N a_N=+\infty$, and finally let $I:\mathcal{X}\to [0,+\infty]$ be a lower semicontinuous functional on $\mathcal{X}$. The sequence $\{ \mathbb{P}_N\}$ is said to satisfy a large deviation principle with speed $\{a_n\}$ and rate $I$ if, and only if, for each Borel measurable set $E \subset \mathcal{X}$, $ -\inf_{x \in E^\circ} I(x) \le \varliminf_N a_N^{-1} \log\big(\mathbb{P}_N(E)\big) \le \varlimsup_N a_N^{-1} \log\big(\mathbb{P}_N(E)\big) \le -\inf_{x \in \bar{E}} I(x) , $ where $\bar{E}$ and $E^\circ$ denote respectively the closure and interior of $E$.
I was wondering how the above formal definition corresponds to the following informal interpretation:
the theory of large deviations concerns the asymptotic behaviour of remote tails of sequences of probability distributions. ... Roughly speaking, large deviations theory concerns itself with the exponential decay of the probability measures of certain kinds of extreme or tail events, as the number of observations grows arbitrarily large.
In particular,
- what are the certain kinds of extreme or tail events in the formal definition? $E \subset \mathcal{X}$ is any Borel measurable set, not necessarily tail or extreme events.
- How is the "exponential decay" represented in the formal definition? Is it represented as $a_n$ being a exponential function of $n$?
- What are the interpretations for $a_n$ and $I$ in the formal definition?
Thanks!