Trend in an approximately exponentially distributed random variable

[Ad VanderVen]

I have a series of variables X _i where ultimately the variables X_i 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 X_i depends on the outcome of X_i-1 so that in the long run the variable X_i becomes an exponential distribution with a constant rate parameter.

 

[FactChecker]

Just to be clear, is it $X_i$ or $X_i-X_{i-1}$ that is exponentially distributed?