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The mean avalanche current impulse response in an avalanche photodiode exhibits an initial transient and then grows or decays, above or below breakdown, at exponential rates which depend only on the probability distributions of the electron and hole ionization events. The process continues while the electric field profile remains unchanged by the applied bias or the evolving space charge. Below breakdown the distribution in the avalanche duration also exhibits an initial transient and then decays exponentially at the same rate as the mean current. Below breakdown the standard deviation in current decays exponentially at one half of the rate of the mean current, while above breakdown it grows exponentially at the same rate as the mean. Consequently the jitter in a Geiger mode avalanche photodiode becomes independent of time after the initial transients have decayed away. This behavior is quite general and independent of the electric field profile or of the presence of heterojunctions in the multiplication region. Using simple models for carrier transport we find the predicted enhancement in the velocity to ionization of those carriers which ionise shortly after their ballistic dead space significantly speeds up the avalanche dynamics in short devices.