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The Exponential Density
The Persistence of Bad Luck
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applies arbitrary argument assume bounded calculations called chapter characteristic function concentrated condition consider constant contains continuous convergence convolution corresponding defined definition denote density depends derived determined difference differential distribution F epoch equals equation event example exists expectation exponential fact finite fixed follows formula given hence holds identity implies independent inequality infinitely integral interval introduced ladder Laplace transform leads lemma limit matrix measure method monotone normal notation obtained obvious occurs operator origin Poisson positive possible present probability distribution problem proof prove random variables random walk relation remains renewal replaced represents requires respect result satisfies seen semi-group sequence shows side simple solution space stable stochastic stochastic kernel Suppose symmetric takes tends theorem theory transition true uniform uniformly unique variance vector zero