 Fixation of leadership in non-Markovian growth processesT. Iyer Stochastic Processes and their Applications, 2026, vol. 199, issue C Abstract: Consider a model where N equal agents possess ‘values’, belonging to N0, that are subject to incremental growth over time. More precisely, the values of the agents are represented by N independent, increasing N0 valued processes with random, independent waiting times between jumps. We show that the event that a single agent possesses the maximum value for all sufficiently large times (called ‘leadership’) occurs with probability zero or one, and provide necessary and sufficient conditions for this to occur. Under mild conditions we also provide criteria for a single agent to become the unique agent of maximum value for all sufficiently large times, and also conditions for the emergence of a unique agent having value that tends to infinity before ‘explosion’ occurs (i.e. conditions for ‘strict leadership’ or ‘monopoly’ to occur almost surely). The novelty of this model lies in allowing non-exponentially distributed waiting times between jumps in value. In the particular case when waiting times are mixtures of exponential distributions, we improve a well-established result on the ‘balls in bins’ model with feedback, removing the requirement that the feedback function be bounded from below and also allowing random feedback functions. As part of the proofs we derive necessary and sufficient conditions for the distribution of a convergent series of independent random variables to have an atom on the real line, a result which we believe may be of interest in its own right. Keywords: Growth processes; Birth processes; Balls in bins processes with feedback; Generalised Pólya urns; Non-linear urns; Atoms of random series; Convergence of random series; Reinforced processes (search for similar items in EconPapers) Downloads: (external link) Related works: Export reference: BibTeX RIS (EndNote, ProCite, RefMan) HTML/Text Persistent link: https://EconPapers.repec.org/RePEc:eee:spapps:v:199:y:2026:i:c:s0304414926001006 Ordering information: This journal article can be ordered from DOI: 10.1016/j.spa.2026.104968 Access Statistics for this article Stochastic Processes and their Applications is currently edited by T. Mikosch More articles in Stochastic Processes and their Applications from Elsevier |
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