# Statistical Properties of Deep Neural Networks with Dependent Data

arXiv: https://arxiv.org/abs/2410.11113 · HTML: https://mlsanity.com/paper/2410.11113/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 14 Oct 2024.

## Built on by (most cited first, 3 of 3)

- [Inference in Partially Linear Models under Dependent Data with Deep Neural Networks](https://mlsanity.com/paper/2410.22574/) (2024)
- [New Approximation Results and Optimal Estimation for Fully Connected Deep Neural Networks](https://mlsanity.com/paper/2512.09853/) (2025)
- [Prediction Inference of Time Series with Standard ReLU Deep Neural Networks](https://mlsanity.com/paper/2608.15362/) (2026)

## Abstract (arXiv)

This paper develops theory for deep neural network (DNN) estimators under dependent data. To provide theory applicable to a variety of DNN-based estimators, I first establish nonasymptotic probability bounds on the theoretical and empirical L2-errors of nonparametric sieve estimators for a general class of estimation problems under possibly nonstationary β-mixing data taking values in unbounded sets. I then apply the theory to fully connected and convolutional DNN estimators without bounds or sparsity restrictions on the DNN weights. For both DNN classes, I derive general results when the function to be estimated is H"older smooth and the data are nonstationary, subgaussian, and β-mixing with either exponential or polynomial decay. I then specialize these to nonparametric regression, logistic regression, and quantile regression settings. Under exponential β-mixing, the resulting estimators attain the nonparametric minimax rate of Stone (1982) up to logarithmic factors.