Level-2 large deviation principle for countable Markov shifts without Gibbs states

We consider level-2 large deviations for the one-sided countable full shift without assuming the existence of Bowen's Gibbs state. To deal with non-compact closed sets, we provide a sufficient condition in terms of inducing which ensures the exponential tightness of a sequence of Borel probability measures constructed from periodic configurations. Under this condition we establish the level-2 Large Deviation Principle. We apply our results to the continued fraction expansion of real numbers in $[0,1)$ generated by the R\'enyi map, and obtain the level-2 Large Deviation Principle, as well as a weighted equidistribution of a set of quadratic irrationals to equilibrium states of the R\'enyi map.