If π denotes the number of digits in the regular continued fraction expansion that can be determined from π digits in the decimal expansion, then Lochsβ Theorem states that the fraction π π...Show moreIf π denotes the number of digits in the regular continued fraction expansion that can be determined from π digits in the decimal expansion, then Lochsβ Theorem states that the fraction π π converges Lebesgue almost surely to a fraction of two entropies as π β β. These are the entropies of the interval maps that generate these expansions. Lochsβ Theorem has been generalized to pairs of interval maps that both belong to a class of piecewise monotonic transformations that generate expansions and that admit an invariant density with suitable ergodic properties. The first aim of this thesis is to review sufficient conditions on interval maps to belong to this class. For this, we first of all recover the famous existence result for invariant densities by Lasota and Yorke for expanding piecewise monotonic interval maps. As an example of a nonexpanding piecewise monotonic interval map, we also consider the Liverani-Saussol-Vaienti (LSV) map and provide a new proof of the already known result that such a map admits an invariant probability density if and only the corresponding parameter lies in (0, 1). Motivated by the practical use of beta encoders, one of the main goals in this thesis is to extend Lochsβ Theorem to expansions generated by a class of random piecewise monotonic interval maps. We review sufficient conditions on random interval maps to belong to this class. For two random interval maps π and π in this class, we show that, if π denotes the number of digits in the π-expansion that can be determined from π digits in the π-expansion, then, roughly speaking, the fraction π π converges Lebesgue almost surely to a fraction of two fiber entropies as π β β. As a second important goal, we prove that the skew product of an LSV map with parameter in (0, 1) and another LSV map with parameter in [1, β) and with underlying Bernoulli shift admits an invariant probability density.Show less