Some extensions of Girko's limit theorems for permanents of random matrices (Q2722264)

The problem of finding the asymptotic distribution of a permanent of a random matrix has been studied over the last decade by several authors. For instance, \textit{V. S. Korolyuk} and \textit{Yu. V. Borovskikh} [Ukr. Math. J. 47, No. 7, 1058-1064 (1995); translation from Ukr. Mat. Zh. 47, No. 7, 922-927 (1995; Zbl 0941.60047)] as well as \textit{E. Yu. Kaneva} [Dopov. Akad. Nauk Ukr. 1992, No. 11, 34-37 (1992)] investigated permanents of the so-called \(k\)-dimensional projection random matrices. \textit{V. L. Girko} [``Theory of random determinants'' (1988; Zbl 0704.60003)] and \textit{G. Rempała} [Random Oper. Stoch. Equ. 4, No. 1, 33-42 (1996; Zbl 0849.60051)] obtained different results for permanents of matrices of independent, positive and bounded entries. Namely, in the latter paper it was shown that the centralized and normalized logarithm of the permanent of the sequence of random matrices converges weakly to the standard normal variable.NEWLINENEWLINENEWLINEIn this paper a different version of the latest result is derived, in a form similar to Yu. V. Borovskikh and V. S. Korolyuk. However, the approach taken in this paper differs considerably from that, as there are considered only matrices of independent entries. Namely, permanents of \(m\times n\) matrices \([X_{ij}]\) of i.i.d. entries satisfying \(P(c<X_{ij}<d)=1\) for some finite positive constants \(c, d\) and their limiting behavior as \(m,n\to\infty\) with \((m/n)\to\lambda>0.\) Examples of applications of the obtained results in asymptotic approximation of some random quantities arising in connection with the multiparameter multinomial distribution and order statistics are provided.