# Finite Sample Analysis of Stochastic System Identification

## TL;DR

This paper provides a finite sample analysis for stochastic system identification, establishing non-asymptotic error bounds that decrease with more data and are valid even for marginally stable systems.

## Contribution

It introduces a non-asymptotic analysis framework for system identification using modern probabilistic tools, extending classical results to finite samples and marginally stable systems.

## Key findings

- Estimation errors decrease at a rate of 1/√N with high probability.
- Non-asymptotic bounds align with classical asymptotic results.
- Results are valid even for marginally stable systems.

## Abstract

In this paper, we analyze the finite sample complexity of stochastic system identification using modern tools from machine learning and statistics. An unknown discrete-time linear system evolves over time under Gaussian noise without external inputs. The objective is to recover the system parameters as well as the Kalman filter gain, given a single trajectory of output measurements over a finite horizon of length $N$. Based on a subspace identification algorithm and a finite number of $N$ output samples, we provide non-asymptotic high-probability upper bounds for the system parameter estimation errors. Our analysis uses recent results from random matrix theory, self-normalized martingales and SVD robustness, in order to show that with high probability the estimation errors decrease with a rate of $1/\sqrt{N}$. Our non-asymptotic bounds not only agree with classical asymptotic results, but are also valid even when the system is marginally stable.