It is known that if independent random variables X and Y have common subexponential distribution F, then distribution of their minimum min{X,Y} is also subexponential. The proof can be found in Theorem 1 of Geluk "Some closure properties for subexponential distributions" Statistics & Probability Letters 79 (2009). https://reader.elsevier.com/reader/sd/pii/S0167715209000029?token=FA3000965369626F157C77F71E95CFE090C2FC14EF7CBC7C254BACE773568EA19E0B88E882002E55F3269E431B0049A6&originRegion=eu-west-1&originCreation=20230216101708
I tried to prove that the inverse statement is wrong:
Assume that we know that the distribution of minimum of i.i.d. random variables X and Y is subexponential. I need a counter-example showing that it not necessarily follows that the distribution of X is subexponential.
Since P(min(X,Y)>x)=(\bar F(x))^2, one should to prove that 1-(\bar F(x))^2 \in S \nRightarrow F\in S. Here \bar F:=1-F.
