Mod-phi convergence I: Normality zones and precise deviations

Understand

In this paper, we use the framework of mod-$\phi$ convergence to prove precise large or moderate deviations for quite general sequences of real valued random variables $(X_{n})_{n \in \mathbb{N}}$, which can be lattice or non-lattice distributed.

• We establish precise estimates of the fluctuations $P[X_{n} \in t_{n}B]$, instead of the usual estimates for the rate of exponential decay $\log( P[X_{n}\in t_{n}B])$.
• Our approach provides us with a systematic way to characterise the normality zone, that is the zone in which the Gaussian approximation for the tails is still valid.
• Besides, the residue function measures the extent to which this approximation fails to hold at the edge of the normality zone.