Large deviations and applications. (English) Zbl 0661.60040

Varadhan, Srinivasa R. S.

Calcul des probabilités, Éc. d’Été, Saint-Flour/Fr. 1985-87, Lect. Notes Math. 1362, 1-49 (1988).

[For the entire collection see Zbl 0649.00016.]
The paper contains a cycle of 10 lectures devoted to the modern theory of large deviations and its applications. The main results presented, among others, are of the following type:
Let X be a complete separable metric space and let \(P_ n\) be Borel probability measures on X; then for every closed set \(C\subset X\), \[ (i)\quad \limsup_{n\to \infty}n^{-1}\log P_ n(C)\leq -\inf_{x\in C}I(x), \] and for every open set \(G\subset X,\) \((ii)\quad \liminf_{n\to \infty}n^{-1}\log P_ n(G)\geq -\inf_{x\in G}I(x),\)where I is the so-called rate function satisfying appropriate conditions. A theorem of this kind is given for distributions \(Q_ n\) of \[ R_{n,\omega}=n^{- 1}(\delta_{\omega}(n)+\delta_{T\omega}(n)+...+\delta_{T^{n- 1}\omega}(n)) \] on the probability space \(\Omega =R^{\infty}\) with \(P=the\) distribution of an ergodic stationary sequence \(...X_{-1},X_ 0,X_ 1,...\), where \(\omega^{(n)}\) is the periodic transformation of \(\omega =(...x_{-1},x_ 0,x_ 1,...)\), \(\omega_ i^{(n)}=x_ i\) for \(1\leq i\leq n\) and \(\omega_ i^{(n)}=\omega^{(n)}_{i+n}\) for all i,n, and T is the shift operator.
Under suitable hypotheses similar results for stationary Gaussian processes, for distributions \(Q_{n,x}\) induced by a Markov chain starting from x and for their continuous analogues in the case of a continuous-time Markov process are described. Moreover, if \(P_ n\) obey the principle of large deviations (i)-(ii), then asymptotic relations of the form \[ \lim_{n}n^{-1}\log \int_{X}\exp (n F(x)) dP_ n(x)=\sup_{x}[F(x)-I(x)] \] are proved for continuous and bounded functions F on X, and also F replaced by \(F_ n\). Results of this kind are applied to the Wiener sausage and the polaron problem.
Some of the proofs are rigorously carried out, the other are sketched or omitted. The details can be found in earlier papers by M. D. Donsker and S. R. S. Varadhan [see e.g. Commun. Pure Appl. Math. 28, 525-565 (1975; Zbl 0333.60077), ibid. 36, 505-528 (1983; Zbl 0538.60081) and Commun. Math. Phys. 97, 187-210 (1985; Zbl 0646.60030)].

Reviewer: A.M.Zapala

MSC:

60F10	Large deviations
60G10	Stationary stochastic processes
60J10	Markov chains (discrete-time Markov processes on discrete state spaces)
60J25	Continuous-time Markov processes on general state spaces
60G15	Gaussian processes

Keywords:

rate function; ergodic stationary sequence; periodic transformation; Wiener sausage; polaron problem

Citations:

Zbl 0649.00016; Zbl 0333.60077; Zbl 0538.60081; Zbl 0646.60030