Nonlinear Approximation in the Large Deviations Principle

The Markov random processes and their approximations are considered. The main object of study is the exponential generator of random processes with independent increments, which are the solution of the problems of large deviations. These processes satisfy the conditions that make it possible to consider the Poisson and Lévy approximation. Generators of random processes are normalized by nonlinear parameters. Found explicit form of normalization parameter estimation.


Introduction
The problem of large deviations was originated as a method of solving statistical problems associated with the estimation of probability of rare events.The first work in this direction was the article by Cramér [1], but ultimately the method was developed in the article by Chernoff [2].Publications [3,4] are also related to this problem.The purpose of solving the problem of large deviations is finding of action functional where x(s) is a Markov process, function L(x, u) defined by the exponential generator where p := φ ′ (u), H(v, φ ′ (u)) := H Γ φ(u).In the writing [5] in the scheme of Poisson approximation processes with independent increments without diffusion component were considered.Between jumps there were Markov processes with linear normalizing factor.
These processes are defined by the generator in the scale of time t/ε.In the articles [7]- [9] the generator of Markov process and their evolutions were considered in the scale of time t/g 1 (ε) and t/g 2 (ε) in the Poisson and Lévi approximation.

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The purpose of this work is to find functions that normalize the generator of random process with independent increments in the scale of time t/g 2 (ε) in the Large Deviations Principle.
These processes are defined by the generator Besides this, in the Poisson approximation the next condition take places and in the Lévy approximation where
We consider the problem of large deviations in the scheme of Poisson approximation in the case where the following conditions are satisfied: (P1) Approximation of the mean values: and The Poisson approximation condition for the intensity kernel The kernel Γ q has the following representation: In the scheme of Poisson approximation the solution of the problem of large deviations for these processes are defined by nonlinear exponential generator:

Lemma 1
The exponential generator in the scheme of Poisson approximation has the following asymptotical representation

Proof
The generator of Markov processes has the next form so, for the exponential generator where ) .We can write the generator in the form where the function e ∆εφ(u and since In addition, this function is continuous and bounded, because φ(u) is from C 2 0 (R).So, from conditions P1 and P2 we obtain: Applying the Taylor's formula for φ(u) and making use of the condition P2: From condition P3 and the condition (g 2 (ε) From this Lemma we have that the following Theorem holds true.

Theorem 1
The solution for the problem of large deviations for the process under conditions (P1-P5) and is determined by the limit generator H Γ of the form as in the article [6], but for more complex functions, such as

Lévy approximation
Now consider another normalization for the family of Markov processes with trajectories in D R [0, ∞) ) , t ≥ 0.
The conditions of Lévy approximation are the following: (L1) Approximation of the mean values: ). (L2) Lévy approximation condition for the intensity kernel This kernel has the following representation: (L5) The exponential boundedness ∫ R e p|v| Γ q (dv) < ∞.

Lemma 2
The exponential generator in the scheme of Lévy approximation has the next asymptotically representation and
Consider the example of a family of Markov processes α ε , for which