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In some places, I find the following definition of stable distribution:

A distribution is said to be stable if a linear combination of two independent random variables with this distribution has the same distribution, up to location and scale parameters.

And in some places,

Let X1 and X2 be independent copies of a random variable X. Then X is said to be stable if for any constants a>0 and b>0 the random variable aX1 + bX2 has the same distribution as cX + d for some constants c>0 and d.

In the first definition, we don't have any constraints on $a$, $b$ and $c$; they need not to be strictly positive. So, are these two definitions equivalent?

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    $\begingroup$ hey, maybe the paper of the second definition fs2.american.edu/jpnolan/www/stable/chap1.pdf helps you. He discusses the different definitions of stable functions very well. $\endgroup$ – mischva11 Jul 16 '18 at 9:26
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    $\begingroup$ I believe your question is easily answered if you adopt the understanding, in the first quotation, that "has the same distribution" means "have identical distribution functions up to location and scale parameters." $\endgroup$ – whuber Jul 16 '18 at 13:15

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