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 $\alpha \in \mathbb{R},$ $ μ\in \mbox{dom}\, A$, one has \begin{equation*} \frac1{T} \ln {\mathbf E} \big( e^{μ(Z(T)-αT)} \big|Z(s), \ s\le 0 \big) = A(μ) +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 ${\epsilon }_T\to 0$ sufficiently slowly as $T\to \infty ,$ one has \begin{equation*} \lim_{T\to\infty}\frac1T \ln {\mathbf P} \big( {Z(T)}/T-α \in (β-\varepsilon_T, β+\varepsilon_T) \big| Z(s), \ s\le 0\big) = - D(β) \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 $\ln \mathbf{P}(z_T\in (f{)}_{{\epsilon }_T})$ as $T\to \infty$, where $f\in \mathbb{D}(0,1),$ $(f{)}_{\epsilon }$ is the $\epsilon$-neighborhood of $f$ in the space $\mathbb{D}(0,1)$ with respect to the uniform metric, and ${\epsilon }_T\to 0$ 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 $T$.

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