Comparison of extreme-value tails of a lognormal and an exponential

by Isambard Kingdom   Last Updated September 11, 2019 18:19 PM

For an exponential distribution:

$F(x) = \exp(-x)$

we have

$\frac{F(x+t)}{F(x)} = F(t)$ for all $t>0$.

Does this result then demonstrate that an exponential process is "memoryless"? Or is it the converse, if the process is memoryless, then this result holds?

On the other hand, for a lognormal distribution

$L(x) = \frac{1}{x}\ \exp(-(\ln(x) - \mu)^2)$

we have

$\lim_{x\rightarrow \infty}\frac{L(x+t)}{L(x)} = 1$,

(calculated using Wolfram Alpha).

From this can I safely conclude that the lognormal distribution does not have an extreme-value tail like an exponential? And, furthermore, can I conclude that a lognormal process is not "memoryless"?

Tags : extreme-value

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