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We look at decompositions of perpetuities and apply them to the study of the distributions of hitting times of Bessel processes of two types of square root boundaries. These distributions are linked giving a new proof of some Mellin transforms results obtained by DeLong  and Yor  . Several random factorizations and characterizations of the studied distributions are established.doi:10.1214/18-ecp139 fatcat:m2doedbr65fo7i6l5ieewzsbjm