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 $α\in \mathbb R,$ $ 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 - α|&lt; η<em>T } $, where $η_T \to 0$ as $T\to\infty.$ Under this condition, a uniform conditional local large deviation principle (l.l.d.p.) is established: for any fixed $α, β\in \mathbb R$ and a positive function $η_T=o(1)$, 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-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 $\ln {\mathbf P} (z_T\in (f)<em>{\varepsilon_T})$ as $T\to\infty$, where $f\in \mathbb D(0,1),$ $(f)</em>\varepsilon$ 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 $Z=Z<sup>{(T)}$ also depends on the parameter $T$.

• The paper derives large deviation principles (LDP) for a broad class of random processes without assuming specific structures, relying on conditional moment assumptions and oscillation control.
• Key results establish uniform conditional local, finite-dimensional, and functional LDP using properties like exponential tightness and oscillation bounds, with applications to perturbed processes and noise resilience.
• The authors demonstrate that conditional LDP develop naturally from conditional LLLDP (large local deviation) under specific assumptions, providing a comprehensive framework that incorporates noise robustness, we find powerful methods covering diverse processes, in particular noise robustness conclusions.

Motivation and setting

The paper by A. A. Borovkov and K. A. Borovkov addresses the derivation of large deviation principles (l.d.p.'s) for a general univariate stochastic process $Z=\{Z(t): t\in\mathbb R\}$ with trajectories in the Skorokhod space $\mathbb D(\mathbb R)$, without assuming any specific structure such as independent increments or a compound renewal construction. The central object is the fundamental function

$$A(\mu):=\lim_{T\to\infty}\frac1T\ln \mathbf E e^{\mu Z(T)},$$

assumed to exist and to be convex and essentially smooth. The rate function is then the Legendre transform $$D(\alpha)=\mathcal L_A(\alpha)=\sup_\mu(\alpha\mu-A(\mu))$$. The authors deliberately work with classes of asymptotically equivalent processes rather than a fixed process, so that their results apply, for instance, when "noise" is added to a well-understood process. This contrasts with the classical G\"artner–Ellis theorem [10.1137/1122003], which requires $0\in(\operatorname{dom}A)$; notably, the paper's local results dispense with that requirement because they concern local rather than global bounds.

A key methodological device is the notion of $\varepsilon_T$ tending to zero slowly enough (denoted $\varepsilon_T=\overline{o}(1)$): a statement holds for all vanishing functions $\varepsilon_T$ that dominate some reference function $\widetilde\varepsilon_T$. The authors illustrate with the sample mean of i.i.d. variables, where $\mathbf P(|z(T)|<\varepsilon_T)\to 1$ requires $\mathbb D(\mathbb R)$0.

Conditional local l.d.p. under rough asymptotic independence

The main structural assumption, condition [A], postulates that for every interval $\mathbb D(\mathbb R)$1 the conditional cumulant generating function satisfies

$\mathbb D(\mathbb R)$2

uniformly over events where $\mathbb D(\mathbb R)$3 lies within $\mathbb D(\mathbb R)$4 of a prescribed level $\mathbb D(\mathbb R)$5. This is a form of uniform conditional exponential tightness/rough asymptotic independence of increments given the past filtration $\mathbb D(\mathbb R)$6. The authors note it covers processes of the form $\mathbb D(\mathbb R)$7, where $\mathbb D(\mathbb R)$8 is a compound renewal process whose driving sequence satisfies the moment Cramér condition and $\mathbb D(\mathbb R)$9 is an independent noise process with $$A(\mu):=\lim_{T\to\infty}\frac1T\ln \mathbf E e^{\mu Z(T)},$$0 — precisely the "noise robustness" application motivating the framework.

The first main result establishes a uniform conditional local l.d.p.: for any fixed $$A(\mu):=\lim_{T\to\infty}\frac1T\ln \mathbf E e^{\mu Z(T)},$$1, $$A(\mu):=\lim_{T\to\infty}\frac1T\ln \mathbf E e^{\mu Z(T)},$$2, and $$A(\mu):=\lim_{T\to\infty}\frac1T\ln \mathbf E e^{\mu Z(T)},$$3,

$$A(\mu):=\lim_{T\to\infty}\frac1T\ln \mathbf E e^{\mu Z(T)},$$4

uniformly on the conditioning event. The proof combines a Chernoff-type upper bound (valid for all $$A(\mu):=\lim_{T\to\infty}\frac1T\ln \mathbf E e^{\mu Z(T)},$$5, including boundary points $$A(\mu):=\lim_{T\to\infty}\frac1T\ln \mathbf E e^{\mu Z(T)},$$6 via a diagonal argument) with a change-of-measure lower bound in the spirit of Dembo–Zeitouni's proof of the G\"artner–Ellis theorem, where the exponentially tilted measure concentrates near $$A(\mu):=\lim_{T\to\infty}\frac1T\ln \mathbf E e^{\mu Z(T)},$$7 for suitably slowly shrinking $$A(\mu):=\lim_{T\to\infty}\frac1T\ln \mathbf E e^{\mu Z(T)},$$8 and $$A(\mu):=\lim_{T\to\infty}\frac1T\ln \mathbf E e^{\mu Z(T)},$$9. An implication worth noting: since only a local principle is asserted, the theorem does not require $$D(\alpha)=\mathcal L_A(\alpha)=\sup_\mu(\alpha\mu-A(\mu))$$0, unlike the classical theorem whose need for that condition stems from upper bounds over unbounded Borel sets.

The paper also records the equivalence of the l.d.p. and l.l.d.p. for general processes: the l.d.p. always implies the l.l.d.p., while the converse holds provided the rate function has compact level sets (a good rate function) together with an exponential tail bound $$D(\alpha)=\mathcal L_A(\alpha)=\sup_\mu(\alpha\mu-A(\mu))$$1 for arbitrarily large $$D(\alpha)=\mathcal L_A(\alpha)=\sup_\mu(\alpha\mu-A(\mu))$$2.

From finite-dimensional distributions to a functional l.d.p.

Iterating the conditional l.l.d.p. across a partition $$D(\alpha)=\mathcal L_A(\alpha)=\sup_\mu(\alpha\mu-A(\mu))$$3 of $$D(\alpha)=\mathcal L_A(\alpha)=\sup_\mu(\alpha\mu-A(\mu))$$4 yields a conditional l.l.d.p. for finite-dimensional distributions: jointly, the normalized increments satisfy

$$D(\alpha)=\mathcal L_A(\alpha)=\sup_\mu(\alpha\mu-A(\mu))$$5

uniformly on $$D(\alpha)=\mathcal L_A(\alpha)=\sup_\mu(\alpha\mu-A(\mu))$$6. The upper bound proceeds by successive conditioning backwards from $$D(\alpha)=\mathcal L_A(\alpha)=\sup_\mu(\alpha\mu-A(\mu))$$7 to $$D(\alpha)=\mathcal L_A(\alpha)=\sup_\mu(\alpha\mu-A(\mu))$$8; the lower bound uses per-interval reference widths $$D(\alpha)=\mathcal L_A(\alpha)=\sup_\mu(\alpha\mu-A(\mu))$$9 and takes their maximum, exploiting the $0\in(\operatorname{dom}A)$0 property to pass to arbitrary slowly vanishing $0\in(\operatorname{dom}A)$1.

To lift this to trajectories, the paper employs the deviation integral $0\in(\operatorname{dom}A)$2, defined as the partition-independent Darboux integral of the interval function $0\in(\operatorname{dom}A)$3. Citing earlier work of Borovkov and Mogul'skii, the authors use the facts that $0\in(\operatorname{dom}A)$4 always exists for $0\in(\operatorname{dom}A)$5, equals $0\in(\operatorname{dom}A)$6 over piecewise-linear interpolants, and satisfies $0\in(\operatorname{dom}A)$7 for absolutely continuous $0\in(\operatorname{dom}A)$8 whenever $0\in(\operatorname{dom}A)$9; moreover $\varepsilon_T$0 off $\varepsilon_T$1 in that case.

An upper bound for $\varepsilon_T$2 holds under condition [A] alone, giving $\varepsilon_T$3 uniformly on $\varepsilon_T$4. The matching lower bound, however, requires control of trajectory oscillations at short scales: condition [B] posits a.s. bounds of the form $\varepsilon_T$5 with $\varepsilon_T$6, $\varepsilon_T$7. Under [A] and [B] jointly, the full conditional functional l.l.d.p. holds:

$\varepsilon_T$8

with the same uniformity. Condition [B] is satisfied by processes with almost Lipschitz trajectories, e.g. partial sums of bounded jumps. Note that [A] + [B] force $\varepsilon_T$9, so the functional result is confined to the full-domain case; the authors state plainly that without oscillation control of this kind, no asymptotics for $\varepsilon_T=\overline{o}(1)$0 can be obtained for a process of completely general form.

Limitations and open questions

Several restrictions are explicit. First, the exposition is confined to the univariate, non-triangular-array case; the authors assert that all results extend, with obvious modifications, to multivariate processes $\varepsilon_T=\overline{o}(1)$1 depending on $\varepsilon_T=\overline{o}(1)$2, but do not carry out the extension, and remark that in the triangular scheme condition [B] would more naturally be formulated via probability bounds on oscillation exceedances rather than a.s. constraints — variants they omit as too complex. Second, the functional lower bound relies on the deterministic a.s. oscillation bound [B], so processes with unbounded or heavy-tailed jump sizes fall outside its scope even if [A] holds. Third, the approach bypasses the topological G\"artner–Ellis theorem because verifying existence of an essentially smooth fundamental functional on the dual of $\varepsilon_T=\overline{o}(1)$3 is deemed infeasible; whether a direct dual-space formulation could weaken condition [B] remains unaddressed. Finally, the equivalence of the l.d.p. and l.l.d.p. is established only under a good-rate-function assumption plus exponential tightness, leaving the general converse open.

Conclusion

The paper provides a self-contained route from a uniform conditional moment assumption ([A]) to a conditional local l.d.p., a finite-dimensional conditional l.l.d.p., and — under an additional short-scale oscillation bound ([B]) — a conditional functional l.l.d.p. in $\varepsilon_T=\overline{o}(1)$4 with rate given by the deviation integral of the Legendre transform of the fundamental function. Its principal value lies in requiring no structural knowledge of the process beyond the fundamental function, thereby covering perturbed compound renewal processes and providing criteria for noise that preserves large deviation asymptotics, while the scope of the functional result is delimited by the Lipschitz-type restriction inherent in condition [B].