# Central limit theorems for the outputs of fully convolutional neural networks with time series input

arXiv: https://arxiv.org/abs/2603.29612 · HTML: https://mlsanity.com/paper/2603.29612/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 16 of its 40 references that are in the map.
- First posted on arXiv 31 Mar 2026.

## Builds on (most cited first, 16 of 16)

- [Deep Residual Learning for Image Recognition](https://mlsanity.com/paper/1512.03385/) (2015)
- [Learning Deep Features for Discriminative Localization](https://mlsanity.com/paper/1512.04150/) (2015)
- [Deep Neural Networks as Gaussian Processes](https://mlsanity.com/paper/1711.00165/) (2017)
- [Network In Network](https://mlsanity.com/paper/1312.4400/) (2013)
- [Deep learning for time series classification: a review](https://mlsanity.com/paper/1809.04356/) (2018)
- [The UCR Time Series Archive](https://mlsanity.com/paper/1810.07758/) (2018)
- [Time Series Classification from Scratch with Deep Neural Networks: A Strong Baseline](https://mlsanity.com/paper/1611.06455/) (2016)
- [ROCKET: Exceptionally fast and accurate time series classification using random convolutional kernels](https://mlsanity.com/paper/1910.13051/) (2019)
- [Gaussian Process Behaviour in Wide Deep Neural Networks](https://mlsanity.com/paper/1804.11271/) (2018)
- [Tensor Programs I: Wide Feedforward or Recurrent Neural Networks of Any Architecture are Gaussian Processes](https://mlsanity.com/paper/1910.12478/) (2019)
- [Deep Convolutional Networks as shallow Gaussian Processes](https://mlsanity.com/paper/1808.05587/) (2018)
- [Deep Learning for Time Series Classification and Extrinsic Regression: A Current Survey](https://mlsanity.com/paper/2302.02515/) (2023)
- [Transformers in Time-series Analysis: A Tutorial](https://mlsanity.com/paper/2205.01138/) (2022)
- [Quantitative Gaussian Approximation of Randomly Initialized Deep Neural Networks](https://mlsanity.com/paper/2203.07379/) (2022)
- [Learnable Dynamic Temporal Pooling for Time Series Classification](https://mlsanity.com/paper/2104.02577/) (2021)
- [Exact Upper and Lower Bounds for the Output Distribution of Neural Networks with Random Inputs](https://mlsanity.com/paper/2502.11672/) (2025)

## Abstract (arXiv)

Deep learning is widely deployed for time series learning tasks such as classification and forecasting. Despite the empirical successes, only little theory has been developed so far in the time series context. In this work, we prove that if the network inputs are generated from short-range dependent linear processes, the outputs of fully convolutional neural networks (FCNs) with global average pooling (GAP) are asymptotically Gaussian and the limit is attained if the length of the observed time series tends to infinity. The proof leverages existing tools from the theoretical time series literature. Based on our theory, we propose a generalization of the GAP layer by considering a global weighted pooling step with slowly varying, learnable coefficients.