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Mixed binomial moments of overlapping r-runs in multicolour Hoppe–Pólya urns

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  • Aoudia, Djilali Ait

Abstract

We consider a multicolour Hoppe–Pólya urn with unit reinforcement and initial composition (a1,…,am,b). For each colour j and integer r≥2, let Sn,j(r) denote the number of consecutive r-runs of colour j among the first n draws. We derive a closed form for the joint mixed factorial (hence binomial) moments of (Sn,1(r),…,Sn,m(r)) in compact Pochhammer form. As a consequence, n−1(Sn,1(r),…,Sn,m(r))⟶n→∞L(U1r,…,Umr),(U1,…,Um)∼Dirichlet(a1,…,am,b).Finally, under disjoint Poissonisation, the total number of r–runs admits an exact Dirichlet–Poisson mixture law.

Suggested Citation

  • Aoudia, Djilali Ait, 2026. "Mixed binomial moments of overlapping r-runs in multicolour Hoppe–Pólya urns," Statistics & Probability Letters, Elsevier, vol. 233(C).
    • Handle: RePEc:eee:stapro:v:233:y:2026:i:c:s0167715226000283
    DOI: 10.1016/j.spl.2026.110664
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