I have a question that repeats the question I found here:
If X and Y are real valued random variables with infinitely divisible distributions, does aX+bY also have an infinitely distribution (a,b∈R). I've seen this stated in several places as obvious, but I have only seen the proof in the case that X and Y are independent. Does anyone know where I can find the proof if we don't assume independence?
The answer was: I doubt that is true in general. Counter example: Let W1 be a standard normal random variable, and W2=W1, if |W1|≤1 W2=−W1, otherwise Then W2 is also a standard normal r.v., but W1+W2 has a finite support, and is not constant, therefore not infinitely divisible, according to https://www.google.com/url?sa=t&rct=j&q=&esrc=s&source=web&cd=4&ved=0ahUKEwiowYfs3srKAhXEMyYKHaM1DskQFggzMAM&url=http%3A%2F%2Fweb.abo.fi%2Ffak%2Fmnf%2Fmate%2Fgradschool%2Fsummer_school%2Ftammerfors2011%2Fslides_rosinski.pdf&usg=AFQjCNE_2G83w4nq7gqDM5xvP3pt8c281A&cad=rja
However,
According to
The book "Random and Vector Measures (Series on Multivariate Analysis)" p.66
aX+bY does be ID
Can you pls help me regarding this issue?
If $X$ has standard normal distribution and $Y=X$ when $|X| \leq 1$ and $-X$ when $|X|>1$ then $Y$ also has standard normal distribution and $X+Y$ does not have an ID distribution. This is because it is a bounded random variable and it is not a constant.