# A Large Deviation Inequality for $\beta$-mixing Time Series and its Applications to the Functional Kernel Regression Model

## TL;DR

This paper establishes a new large deviation inequality for $eta$-mixing time series, enabling improved concentration bounds and demonstrating its application in the consistency analysis of functional kernel regression models for dynamic forecasting.

## Contribution

It introduces a novel large deviation inequality for $eta$-mixing processes and applies it to prove consistency in a functional kernel regression setting.

## Key findings

- New large deviation inequality for $eta$-mixing time series
- Application to prove consistency of functional kernel regression forecasts
- Enhanced concentration inequalities for dependent data

## Abstract

We give a new large deviation inequality for sums of random variables of the form $Z_k = f(X_k,X_t)$ for $k,t\in \mathbb{N}$, $t$ fixed, where the underlying process $X$ is $\beta$-mixing. The inequality can be used to derive concentration inequalities. We demonstrate its usefulness in the functional kernel regression model of Ferraty et al. (2007) where we study the consistency of dynamic forecasts.