  We propose an analytical approach to study non-Markov random walks by
employing an exact enumeration method. Using the method, we derive an exact
expansion for the first-passage time (FPT) distribution for any continuous,
differentiable non-Markov random walk with Gaussian or non-Gaussian
multivariate distribution. As an example, we study the FPT distribution of a
fractional Brownian motion with a Hurst exponent $H\in(1/2,1)$ that describes
numerous non-Markov stochastic phenomena in physics, biology and geology, and
for which the limit $H=1/2$ represents a Markov process.
