  In causal discovery, non-Gaussianity has been used to characterize the
complete configuration of a Linear Non-Gaussian Acyclic Model (LiNGAM),
encompassing both the causal ordering of variables and their respective
connection strengths. However, LiNGAM can only deal with the finite-dimensional
case. To expand this concept, we extend the notion of variables to encompass
vectors and even functions, leading to the Functional Linear Non-Gaussian
Acyclic Model (Func-LiNGAM). Our motivation stems from the desire to identify
causal relationships in brain-effective connectivity tasks involving, for
example, fMRI and EEG datasets. We demonstrate why the original LiNGAM fails to
handle these inherently infinite-dimensional datasets and explain the
availability of functional data analysis from both empirical and theoretical
perspectives. {We establish theoretical guarantees of the identifiability of
the causal relationship among non-Gaussian random vectors and even random
functions in infinite-dimensional Hilbert spaces.} To address the issue of
sparsity in discrete time points within intrinsic infinite-dimensional
functional data, we propose optimizing the coordinates of the vectors using
functional principal component analysis. Experimental results on synthetic data
verify the ability of the proposed framework to identify causal relationships
among multivariate functions using the observed samples. For real data, we
focus on analyzing the brain connectivity patterns derived from fMRI data.
