  Non-Poissonian bursty processes are ubiquitous in natural and social
phenomena, yet little is known about their effects on the large-scale spreading
dynamics. In order to characterize these effects we devise an analytically
solvable model of Susceptible-Infected (SI) spreading dynamics in infinite
systems for arbitrary inter-event time distributions and for the whole time
range. Our model is stationary from the beginning, and the role of lower bound
of inter-event times is explicitly considered. The exact solution shows that
for early and intermediate times the burstiness accelerates the spreading as
compared to a Poisson-like process with the same mean and same lower bound of
inter-event times. Such behavior is opposite for late time dynamics in finite
systems, where the power-law distribution of inter-event times results in a
slower and algebraic convergence to fully infected state in contrast to the
exponential decay of the Poisson-like process. We also provide an intuitive
argument for the exponent characterizing algebraic convergence.
