  The standard Large Deviation Theory (LDT) is mathematically illustrated by
the Boltzmann-Gibbs factor which describes the thermal equilibrium of
short-range-interacting many-body Hamiltonian systems, the velocity
distribution of which is Maxwellian. It is generically applicable to systems
satisfying the Central Limit Theorem (CLT). When we focus instead on stationary
states of typical complex systems (e.g., classical long-range-interacting
many-body Hamiltonian systems, such as self-gravitating ones), the CLT, and
possibly also the LDT, need to be generalised. Specifically, when the
$N\to\infty$ attractor ($N$ being the number of degrees of freedom) in the
space of distributions is a $Q$-Gaussian (a nonadditive $q$-entropy-based
generalisation of the standard Gaussian case, which is recovered for $Q=1$)
related to a $Q$-generalised CLT, we expect the LDT probability distribution to
asymptotically approach a power law. Consistently with available strong
numerical indications for probabilistic models, this behaviour possibly is that
associated to a $q$-exponential (defined as
$e_q^x\equiv\left[1+(1-q)x\right]^{1/(1-q)}$, which is the generalisation of
the standard exponential form, straightforwardly recovered for $q=1$); $q$ and
$Q$ are expected to be simply connected, including the particular case $q=Q=1$.
The argument of such $q$-exponential would be expected to be proportional to
$N$, analogously to the thermodynamical entropy of many-body Hamiltonian
systems. We provide here numerical evidence supporting the asymptotic power-law
by analysing the standard map, the coherent noise model for biological
extinctions and earthquakes, the Ehrenfest dog-flea model, and the random-walk
avalanches.
