  We consider the influence of active speed fluctuations on the dynamics of a
$d$-dimensional active Brownian particle performing a persistent stochastic
motion. We use the Laplace transform of the Fokker-Planck equation to obtain
exact expressions for time-dependent dynamical moments. Our results agree with
direct numerical simulations and show several dynamical crossovers determined
by the activity, persistence, and speed fluctuation.The persistence in the
motion leads to anisotropy, with the parallel component of displacement
fluctuation showing sub-diffusive behavior and non-monotonic variation. The
kurtosis remains positive at short times determined by the speed fluctuation,
crossing over to a negative minimum at intermediate times governed by the
persistence before vanishing asymptotically. The probability distribution of
particle displacement obtained from numerical simulations in two-dimension
shows two crossovers between contracted and expanded trajectories via two
bimodal distributions at intervening times. While the speed fluctuation
dominates the first crossover, the second crossover is controlled by
persistence like in the worm-like chain model of semiflexible polymers.
