  This work concerns random dynamics of hyperbolic entire and meromorphic
functions of finite order and whose derivative satisfies some growth condition
at infinity. This class contains most of the classical families of
transcendental functions and goes much beyond. Based on uniform versions of
Nevanlinna's value distribution theory we first build a thermodynamical
formalism which, in particular, produces unique geometric and fiberwise
invariant Gibbs states. Moreover, spectral gap property for the associated
transfer operator along with exponential decay of correlations and a central
limit theorem are shown. This part relies on our construction of new positive
invariant cones that are adapted to the setting of unbounded phase spaces. This
setting rules out the use of Hilbert's metric along with the usual contraction
principle. However these cones allow us to apply a contraction argument
stemming from Bowen's initial approach.
