On large deviation principles for general random processes

Abstract

Let $Z=\{Z(t): t\in \mathbb R\}$ be a stochastic process with trajectories in space $\mathbb D (\mathbb R)$. It is assumed that there exists an essentially smooth function $A:\mathbb R\to (-\infty, \infty] $ such that, for all $\alpha \in \mathbb R, $ $ \mu\in \mbox{dom}\, A$, one has \begin{equation*} \frac1{T} \ln {\mathbf E} \big( e^{\mu (Z(T)-\alpha T)} \big|Z(s), \ s\le 0 \big) = A(\mu) +o(1) \end{equation*} uniformly on the event $C(T):=\{|Z(0)/T - \alpha |< \eta_T \} $, where $ \eta_T \to 0$ as $T\to\infty.$ Under this condition, a uniform conditional local large deviation principle (l.l.d.p.) is established: for any fixed $\alpha, \beta\in \mathbb R$ and a positive function $\eta_T=o(1)$, for $\varepsilon_T \to 0$ sufficiently slowly as $T\to\infty,$ one has \begin{equation*} \lim_{T\to\infty}\frac1T \ln {\mathbf P} \big( {Z(T)}/T-\alpha \in (\beta-\varepsilon_T, \beta +\varepsilon_T) \big| Z(s), \ s\le 0\big) = - D(\beta ) \end{equation*} uniformly on $C(T)$, where $D$ is the Legendre transform of the function $A$. This result is used to establish a conditional l.l.d.p. for the finite-dimen\-sional distributions of the process $ \{ z_T(s) = Z(sT)/T: s\in [0,1]\}$. Under additional conditions on the magnitude of oscillations of the trajectories $z_T$, a functional l.l.d.p. is obtained for the asymptotics of $\ln {\mathbf P} (z_T\in (f)_{\varepsilon_T})$ as $T\to\infty$, where $f\in \mathbb D(0,1),$ $(f)_\varepsilon$ is the $\varepsilon$-neighborhood of $f$ in the space $ \mathbb D(0,1)$ with respect to the uniform metric, and $\varepsilon_T \to 0$ sufficiently slowly. The obtained results can be extended to a more general triangular array scheme where the process itself $Z=Z^{(T)}$ also depends on the parameter $T$.