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What is the relationship between a random variable obeying the subexponential distribution defined here and a random variable $X$ satisfying $P\left(\left|X\right|>t\right)\le\alpha e^{-\beta t}$ for all $t>0$ and some $\alpha,\beta>0$?

Thanks a lot for any helpful answers.

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    There is no such thing as THE subexponential distribution. You might wish to explain which distribution you have in mind.2012-09-15
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    @did is it true that when $\lim_{t\rightarrow\infty}$, $P(t>|x|)\sim 1-e^{-\beta t}$?2012-09-15
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    @SeyhmusGüngören I do not understand your question.2012-09-15
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    @did can we say that any $X$ satisfying $P\left(\left|X\right|>t\right)\le\alpha e^{-\beta t}$ has an exponential distribution? then it will be also subexponential.2012-09-15
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    @SeyhmusGüngören The usual meaning of the term *subexponential distribution* for distributions on $[0,+\infty)$ (Teugels, Pitman, others) is that $P(X+X'\gt x)\sim2P(X\gt x)$ when $x\to+\infty$, for $X$ and $X'$ i.i.d. following this distribution. In particular, $e^{cx}P(X\gt x)\to\infty$ for every $c\gt0$. Hence, no, exponential distributions are not subexponential.2012-09-15

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