  The thermodynamic uncertainty relation is a prominent result in stochastic
thermodynamics that provides a bound on the fluctuations of any thermodynamic
flux, also known as current, in terms of the average rate of entropy
production. Such fluctuations are quantified by the second moment of the
probability distribution of the current. The role of higher order standardized
moments such as skewness and kurtosis remains largely unexplored. We analyze
the skewness and kurtosis associated with the first passage time of
thermodynamic currents within the framework of stochastic thermodynamics. We
develop a method to evaluate higher order standardized moments associated with
the first passage time of any current. For systems with a unicyclic network of
states, we conjecture upper and lower bounds on skewness and kurtosis
associated with entropy production. These bounds depend on the number of states
and the thermodynamic force that drives the system out of equilibrium. We show
that these bounds for skewness and kurtosis do not hold for multicyclic
networks. We discuss the application of our results to infer an underlying
network of states.
