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The objects of our interest are the so-called $A$-permutations, which arepermutations whose cycle length lie in a fixed set $A$. They have beenextensively studied with respect to the uniform or the Ewens measure. In thispaper, we extend some classical results to a more general weighted probabilitymeasure which is a natural extension of the Ewens measure and which inparticular allows to consider sets $A_n$ depending on the degree $n$ of thepermutation. By means of complex analysis arguments and under reasonableconditions on generating functions we study the asymptotic behaviour ofclassical statistics. More precisely, we generalize results concerning largecycles of random permutations by Vershik, Shmidt and Kingman, namely the weakconvergence of the size ordered cycle length to a Poisson-Dirichletdistribution. Furthermore, we apply our tools to the cycle counts and obtain aBrownian motion central limit theorem which extends results by DeLaurentis,Pittel and Hansen.

Large cycles and a functional central limit theorem for generalized weighted random permutations