Graduate Trend in an approximately exponentially distributed random variable

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The discussion centers on modeling a series of variables Xi that follow an approximately exponential distribution with a constant rate. The main inquiry is whether a probability model can be established where each Xi depends on the previous variable Xi-1, leading to a long-term exponential distribution. Clarification is sought on whether it is Xi or the difference Xi - Xi-1 that exhibits the exponential distribution. The conversation explores the implications of these dependencies on the distribution's characteristics. Ultimately, the goal is to understand how to maintain an exponential distribution in the presence of a long-term trend.
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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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Thread 'How do E[X] and E[|X|] relate?'

Greetings, I am studying probability theory [non-measure theory] from a textbook. I stumbled to the topic stating that Cauchy Distribution has no moments. It was not proved, and I tried working it via direct calculation of the improper integral of E[X^n] for the case n=1. Anyhow, I wanted to generalize this without success. I stumbled upon this thread here: https://www.physicsforums.com/threads/how-to-prove-the-cauchy-distribution-has-no-moments.992416/ I really enjoyed the proof...
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