The Lindeberg-Feller and Lyapunov Conditions in Infinite Dimensions

TL;DR

This paper extends classical CLT conditions to infinite-dimensional Hilbert spaces, including cases with missing data, ensuring consistent statistical inference from partial functional data.

Contribution

It generalizes Lindeberg-Feller and Lyapunov CLTs to Hilbert spaces and missing data scenarios, providing simple boundedness conditions for universal inference consistency.

Findings

Generalized CLTs to Hilbert spaces using $L^2$

Extended results to spaces with missing data

Proved that Lindeberg-Feller condition ensures inference consistency

Abstract

This paper makes 3 contributions. First, it generalizes the Lindeberg\textendash Feller and Lyapunov Central Limit Theorems to Hilbert Spaces by way of $L^2$. Second, it generalizes these results to spaces in which sample failure and missingness can occur. Finally, it shows that satisfaction of the Lindeberg\textendash Feller Condition in such spaces guarantees the consistency of all inferences from the partial functional data with respect to the completely observed data. These latter two results are especially important given the increasing attention to statistical inference with partially observed functional data. This paper goes beyond previous research by providing simple boundedness conditions which guarantee that \textit{all} inferences, as opposed to some proper subset of them, will be consistently estimated. This is shown primarily by aggregating conditional expectations with respect to the space of missingness patterns. This paper appears to be the first to apply this technique.