Infinite Size-Biased Orders

The infinite random size-biased order with arbitrary positive size parameters is introduced and represented in terms of independent exponential random variables. We collect basic properties and constructions of the order, some of which belong to the folklore and some are new. Our central result shows how the isomorphism type of the order (e.g. Z>0, Q Z > 0 , Q or any other possible) depends on the parameters. In most cases, for a given set of parameters the type is the same almost surely.