[en] The utility of operator-moments or traces in the various applications of spectral-distribution theory is well documented. To take full advantage of the powerful entree that spectral distribution theory offers in nuclear physics, it may be generally necessary to have many moments beyond th first two hamiltonian moments < H> and < H²>. In order to calculate, for one example, level densities reliably in the excitation-energy regions of physical interest, it may be necessary to have the moments < J/sub z/²H/sup n/> and of course < H/sup n/> with n ranging as high as 8 or so. The subject of this paper is a new method for obtaining these higher moments which is based on the use of random multiparticle vectors, called random representative vectors (RRV), in conjunction with an appropriate shell-model space and hamiltonian. With this method it is possible to calculate average properties of very large spaces with well-defined symmetries by averaging the results over a relatively few RRV. The demonstration of the statistical formulas of the RRV method is given in the next section. In the following section numerical results are given for the test case of 5 nucleons in the single-particle orbits 1d/sub 5/2/, 2s/sub 1/2/; and 1d/sub 3/2/ (A = 21), as well as some results for 12 nucleons in the same orbits plus the 1f/sub 7/2/ orbit (A = 28). With the representative-vector method one can evaluate average moments of any operator O that commutes with H so long as the application of O to the RRV does not produce a new vector which exceeds the memory capacity of the computer. 3 figures, 5 tables