# Linear response and moderate deviations: hierarchical approach. II

## TL;DR

This paper extends the Moderate Deviations Principle to a new class of random fields using a hierarchical, inductive approach, improving understanding of deviations in complex stochastic systems.

## Contribution

It introduces a hierarchical, inductive method to establish upper bounds for moderate deviations in random fields, generalizing previous results beyond sums of independent variables.

## Key findings

- Upper bound for moderate deviations in a new class of random fields.
- Inductive approach applicable in multiple dimensions.
- Refined formulations removing stationarity assumptions.

## Abstract

The Moderate Deviations Principle (MDP) is well-understood for sums of independent random variables, worse understood for stationary random sequences, and scantily understood for random fields. An upper bound for a new class of random fields is obtained here by induction in dimension. Version 3. Sect 1. Stationarity, being not essential in the proofs, is removed from the definitions and the main result formulation. Sect. 2. $[C,2C]$ instead of $[C_1,2C_1]$ before Prop. 2.6; $ a\ge1$ instead of $a\ge C/C_1$ in the last proof; Remark 2.5 added; supremum over shifts in (2.2) (formerly (2.3)); "centered" instead of "CMS". Cosmetic changes: indexing of leaks; Remark 2.11. Sect. 3. Cosmetic change: semicolon after the second display of the proof of Lemma 3.9. References: "response", not "responce".