Multifractal Formalism from Large Deviations

Abstract

It has often been observed that the Multifractal Formalism and the Large Deviation Principles are intimately related. In fact, Multifractal Formalism was heuristically derived using the Large Deviations ideas. In numerous examples in which the multifractal results have been rigorously established, the corresponding Large Deviation results are valid as well. Moreover, the proofs of multifractal and large deviations are remarkably similar. The natural question then is whether under which conditions multifractal formalism can be deduced from the corresponding large deviations results. More specifically, given a sequence of random variables $\{ {X_n} \}_{n\in\N}$, satisfying a Large Deviation Principle, what can be said about the multifractal nature of the level sets $K_\alpha=\{\omega: \lim_{n} \frac{X_n(\omega)}{n}=\alpha\}$. Under some technical assumptions, we establish the upper and lower bounds for multifractal spectra in terms of the large deviation rate functions, and show that many known results of multifractal formalism are covered by our setup.