Criteria for large deviations

Abstract:

We give the general variational form of \[ \limsup (\int _X e^{h(x)/t_{\alpha }}\mu _{\alpha }(dx))^{t_{\alpha }}\] for any bounded above Borel measurable function $h$ on a topological space $X$, where $(\mu _{\alpha })$ is a net of Borel probability measures on $X$, and $(t_{\alpha })$ a net in $]0,\infty [$ converging to $0$. When $X$ is normal, we obtain a criterion in order to have a limit in the above expression for all $h$ continuous bounded, and deduce new criteria of a large deviation principle with not necessarily tight rate function; this allows us to remove the tightness hypothesis in various classical theorems.