# Local Asymptotic Normality of Infinite-Dimensional Concave Extended Linear Models

## TL;DR

This paper establishes local asymptotic normality for M-estimates in infinite-dimensional convex models using Mosco-convergence, enabling analysis despite non-differentiability and constraints.

## Contribution

It introduces a novel approach based on Mosco-convergence to prove local asymptotic normality in infinite-dimensional convex models with non-differentiable objectives.

## Key findings

- Proves local asymptotic normality for infinite-dimensional convex M-estimates.
- Develops a new technique using Mosco-convergence for non-uniform convergence issues.
- Derives asymptotic distribution of likelihood ratio test statistic in Hilbert spaces.

## Abstract

We study local asymptotic normality of M-estimates of convex minimization in an infinite dimensional parameter space. The objective function of M-estimates is not necessary differentiable and is possibly subject to convex constraints. In the above circumstance, narrow convergence with respect to uniform convergence fails to hold, because of the strength of it's topology. A new approach we propose to the lack-of-uniform-convergence is based on Mosco-convergence that is weaker topology than uniform convergence. By applying narrow convergence with respect to Mosco topology, we develop an infinite-dimensional version of the convexity argument and provide a proof of a local asymptotic normality. Our new technique also provides a proof of an asymptotic distribution of the likelihood ratio test statistic defined on real separable Hilbert spaces.