Abstract
A positive random variable X with law L( X) and finite moment of order r > 0 has an induced length-biased law of order r, denoted by L( X r ). Let V ⩾ 0 be independent of X r . A characterization problem seeks solution pairs ( L( X), L( V)) for the “in-law” equation X ≅ VX r , where ≅ denotes equality in law. A renewal process interpretation asks when is the random rescaling of the stationary total lifetime VX 1 equal in law to a typical lifetime X? Solutions are known in special cases. A comprehensive existence/uniqueness theory is presented, and many consequences are explored. Unique solutions occur when − log X and − log V have spectrally positive infinitely divisible laws. Particular cases are explored. Connections with the stationary lifetime law of renewal theory also are investigated.
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