  We consider graphical models based on a recursive system of linear structural
equations. This implies that there is an ordering, $\sigma$, of the variables
such that each observed variable $Y_v$ is a linear function of a variable
specific error term and the other observed variables $Y_u$ with $\sigma(u) <
\sigma (v)$. The causal relationships, i.e., which other variables the linear
functions depend on, can be described using a directed graph. It has been
previously shown that when the variable specific error terms are non-Gaussian,
the exact causal graph, as opposed to a Markov equivalence class, can be
consistently estimated from observational data. We propose an algorithm that
yields consistent estimates of the graph also in high-dimensional settings in
which the number of variables may grow at a faster rate than the number of
observations, but in which the underlying causal structure features suitable
sparsity; specifically, the maximum in-degree of the graph is controlled. Our
theoretical analysis is couched in the setting of log-concave error
distributions.
