ACTA issues

## Random multisets and bootstrap means

Sándor Csörgő, Benedek Valkó, Wei Biao Wu

Acta Sci. Math. (Szeged) 67:3-4(2001), 843-875
2822/2009

 Abstract. First the turning points $\alpha =1,2$ and $3$ in the asymptotic behavior, as $n\to\infty$, of the number of elements in the random integer-set ${\cal I}_n(\alpha ) = \{\lfloor\lfloor n^\alpha\rfloor U_{n,1}\rfloor,\ldots, \lfloor\lfloor n^\alpha\rfloor U_{n,n}\rfloor\}$ are delineated, where $\{U_{n,1}, \ldots, U_{n,n}\} _{n=1}^{\infty }$ is an array of independent Uniform$(0,1)$ random variables and $\alpha > 0$ is a fixed parameter. Then, proving some conjectures made in [6], for $\alpha >2$ the distribution of the number $N_n(\alpha )$ of elements in the random multiset ${\cal I}_n(\alpha )\cap[ \cup_{j=n+1}^{\infty } {\cal I}_j(\alpha )]$ is approximated in the variation distance by suitable binomial and Poisson distributions, with specified rates of approximation depending on $\alpha$, and an almost sure bound of the order of $n^{3-\alpha }$ is obtained for $N_n(\alpha )$. Finally, these results are used to extend necessary conditions concerning the strong convergence of some randomly rarefied and bootstrap means. AMS Subject Classification (1991): 60F05, 60F15, 62G09 Received December 20, 2000. (Registered under 2822/2009.)