Balls in Constrained Urns and Cantor-Like Sets

  • Günther J. Wirsching

    Katholische Universität Eichstätt, Germany


Let An(k)A_n(k) denote the number of different ways to distribute kk indistinguishable balls into nn constrained urns, with capacities c1,,cnc_1,\cdots, c_n. We consider the normalized counting functions φn(x)=γnAn(ϱx)normalized \ counting \ functions \ \varphi_n(x) = \gamma_nA_n(|\varrho x|), where φn,ϱn>0\varphi_n, \varrho_n > 0 are appropriate constants such that supp(φn)=[0,1](\varphi_n) = [0,1] and 01φn(x)dx=1\int^1_0 \varphi_n (x)dx = 1. It is shown here that, if (cn)nN(c_n)_{n \in \mathbb N} is asymptotically geometric with weight q>32asymptotically \ geometric \ with \ weight \ q > \frac{3}{2}, i.e. if qncnq^{–n}c_n converges to some positive real number, then the functions φn\varphi_n converge to some CC^{\infty}-function φ\varphi on R\mathbb R. This function φ\varphi is the unique solution of the integral equation φ(x)=qq1qxqxq+1φ(t)dt\varphi(x) = \frac{q}{q–1} \int^{qx}{qx–q+1} \varphi (t)dt satisfying supp φ[0,1]\varphi \in [0,1] and 01φ(t)dt=1\int^1_0 \varphi(t)dt = 1. Moreover, if q>2q > 2, it is shown that φ\varphi is a polynomial on each interval outside a Cantor-like set in the interval [0, 1].

Cite this article

Günther J. Wirsching, Balls in Constrained Urns and Cantor-Like Sets. Z. Anal. Anwend. 17 (1998), no. 4, pp. 979–996

DOI 10.4171/ZAA/862