A Trend in an approximately exponentially distributed random variable

Ad VanderVen
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I have a series of variables X i where ultimately the variables Xi each follow approximately an exponential distribution with a constant rate. In the beginning, there is a certain long-term trend. Is there a probability model in which Xi depends on the outcome of Xi-1 so that in the long run the variable Xi becomes an exponential distribution with a constant rate parameter.
 
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Just to be clear, is it ##X_i## or ##X_i-X_{i-1}## that is exponentially distributed?
 
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