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.

翻译：深度学习已广泛应用于时间序列学习任务，如分类与预测。尽管取得了实证成功，但在时间序列背景下，目前仅发展出少量理论。本研究证明，若网络输入由短程依赖线性过程生成，采用全局平均池化（GAP）的全卷积神经网络（FCNs）的输出渐近服从高斯分布，且当观测时间序列长度趋于无穷时该极限成立。证明过程借助了现有时间序列理论文献中的工具。基于这一理论，我们通过引入具有缓慢变化可学习系数的全局加权池化步骤，提出了GAP层的泛化方法。