On the weak limits of elementary symmetric polynomials
The paper extends recent results of the reviewer and others [see e.g. the reviewer, Z. Wahrscheinlichkeitstheor. Verw. Geb. 59, 355-359 (1982; Zbl 0468.60029)] on the weak limits of elementary symmetric polynomials of i.i.d. rv's. The most interesting result is the following: Let \(X,X_ 1,X_ 2,..\). be i.i.d. rv's, \(P(X=1)=P(X=-1)=(1/2)P(X\neq 0)=p/2\) \((0<p<1)\), and let \((k_ n)\) be a sequence of integers with \(1\leq k_ n\leq n\) and \(k_ n/n\to c\) \((0<c<1)\). Let \(S_ n^{(k_ n)}\) be the elementary symmetric polynomial of order \(k_ n\) of \(X_ 1,X_ 2,...,X_ n\). If \(0<c<p\) and \(n^{1/2}(k_ n/n-p)\) converges then one can find a function L on the unit square such that \[ n^{1/2}(| S_ n^{(k_ n)}|^{1/k_ n}/\left( \begin{matrix} n\\ k_ n\end{matrix} \right)^{1/(2k_ n)}-L(p,k_ n/n)) \] converges in distribution to CN, where N denotes a standard normal random variable and C is a constant depending on c and p.
weak limits of elementary symmetric polynomials of i.i.d. rv's
scientific article; zbMATH DE number 3969765