  We present a heuristic derivation of the first passage time exponent for the
integral of a random walk [Y. G. Sinai, Theor. Math. Phys. {\bf 90}, 219
(1992)]. Building on this derivation, we construct an estimation scheme to
understand the first passage time exponent for the integral of the integral of
a random walk, which is numerically observed to be $0.220\pm0.001$. We discuss
the implications of this estimation scheme for the $n{\rm th}$ integral of a
random walk. For completeness, we also address the $n=\infty$ case. Finally, we
explore an application of these processes to an extended, elastic object being
pulled through a random potential by a uniform applied force. In so doing, we
demonstrate a time reparameterization freedom in the Langevin equation that
maps nonlinear stochastic processes into linear ones.
