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This paper reports a novel recurrence theory that enables us to calculate the exact joint probability density function (pdf) of the random gain and the random avalanche buildup time in avalanche photodiodes (APDs) including the effect of dead space. Such calculations reveal a strong statistical correlation between the gain and the buildup time for all widths of the multiplication region. To facilitate the calculation of the photocurrent statistics in the presence of this correlation, the impulse-response function of the APD is approximately modeled by a function of time whose prespecified shape is appropriately parameterized by two random variables: the gain and the buildup time. The evaluation of the variance of the photocurrent under this model leads to the definition of the shot-noise-equivalent bandwidth of the APD, which captures the statistical correlation between the gain and the buildup time. It is shown that the shot-noise-equivalent bandwidth in GaAs APDs is greater, by approximately 30%, than the traditional buildup-time-limited 3-dB bandwidth, which is calculated from the mean of the impulse-response function. A thorough analysis of the performance of APD-based integrate-and-dump digital receivers reveals that the strong correlation between the gain and the buildup time accentuates intersymbol interference (ISI) noise, and thus, adversely affects receiver sensitivity at high transmission rates beyond previously known limits.