▁well . ▁ ▁F ails ▁to ▁be ▁convolution - closed ▁ ▁An ▁important ▁point ▁about ▁infinitely ▁divis ible ▁distributions ▁is ▁their ▁connection ▁to ▁L év y ▁processes , ▁i . e . ▁at ▁any ▁point ▁in ▁time ▁a ▁L év y ▁process ▁is ▁infinitely ▁divis ible ▁distributed . ▁Many ▁families ▁of ▁well - known ▁infinitely ▁divis ible ▁distributions ▁are ▁so - called ▁convolution - closed , ▁ ▁i . e . ▁if ▁the ▁distribution ▁of ▁a ▁L év y ▁process ▁at ▁one ▁point ▁in ▁time ▁belongs ▁to ▁one ▁of ▁these ▁families , ▁then ▁the ▁distribution ▁of ▁the ▁L év y ▁process ▁at ▁all ▁points ▁in ▁time ▁belong ▁to ▁the ▁same ▁family ▁of ▁distributions . ▁For ▁example , ▁a ▁Poss ion ▁process ▁will ▁be ▁Poss ion ▁distributed ▁at ▁all ▁points ▁in ▁time , ▁or ▁a ▁Brown ian ▁motion ▁will ▁be ▁normal ▁distributed ▁at ▁all ▁points ▁in ▁time . ▁However , ▁a ▁L év y ▁process ▁that ▁is ▁general ised ▁hyper bol ic ▁at ▁one ▁point ▁in ▁time ▁might ▁fail ▁to ▁be ▁generalized ▁hyper bol ic ▁at ▁another ▁point ▁in ▁time . ▁In ▁fact , ▁the ▁generalized ▁Lap lace ▁distributions ▁and ▁the ▁normal ▁inverse ▁Gaussian ▁distributions ▁are ▁the ▁only ▁sub classes ▁of ▁the ▁generalized ▁hyper bol ic ▁distributions ▁that ▁are ▁closed ▁under ▁convolution . ▁ ▁Rel ated ▁distributions ▁▁ ▁As ▁the ▁name ▁suggests ▁it ▁is ▁of ▁a ▁very ▁general ▁form , ▁being ▁the ▁super class ▁of , ▁among ▁others , ▁the ▁Student ' s ▁t - distribution , ▁the ▁Lap lace ▁distribution , ▁the ▁hyper