  The aim of this paper is to address two related estimation problems arising
in the setup of hidden state linear time invariant (LTI) state space systems
when the dimension of the hidden state is unknown. Namely, the estimation of
any finite number of the system's Markov parameters and the estimation of a
minimal realization for the system, both from the partial observation of a
single trajectory. For both problems, we provide statistical guarantees in the
form of various estimation error upper bounds, $\rank$ recovery conditions, and
sample complexity estimates.
  Specifically, we first show that the low $\rank$ solution of the Hankel
penalized least square estimator satisfies an estimation error in $S_p$-norms
for $p \in [1,2]$ that captures the effect of the system order better than the
existing operator norm upper bound for the simple least square. We then provide
a stability analysis for an estimation procedure based on a variant of the
Ho-Kalman algorithm that improves both the dependence on the dimension and the
least singular value of the Hankel matrix of the Markov parameters. Finally, we
propose an estimation algorithm for the minimal realization that uses both the
Hankel penalized least square estimator and the Ho-Kalman based estimation
procedure and guarantees with high probability that we recover the correct
order of the system and satisfies a new fast rate in the $S_2$-norm with a
polynomial reduction in the dependence on the dimension and other parameters of
the problem.
