Let {sij: i, j=1, 2, …} consist of i.i.d. random variables in ℂ with $\mathsf{E}s_{11}=0$ , $\mathsf{E}|s_{11}|^{2}=1$ . For each positive integer N, let sk=sk(N)=(s1k, s2k, …, sNk)T, 1≤k≤K, with K=K(N) and K/N→c>0 as N→∞. Assume for fixed positive integer L, for each N and k≤K, αk=(αk(1), …, αk(L))T is random, independent of the sij, and the empirical distribution of (α1, …, αK), with probability one converging weakly to a probability distribution H on ℂL. Let βk=βk(N)=(αk(1)skT, …, αk(L)skT)T and set C=C(N)=(1/N)∑k=2K βk βk*. Let σ2>0 be arbitrary. Then define SIR1=(1/N)β1*(C+σ2I)−1 β1, which represents the best signal-to-interference ratio for user 1 with respect to the other K−1 users in a direct-sequence code-division multiple-access system in wireless communications. In this paper it is proven that, with probability 1, SIR1 tends, as N→∞, to the limit ∑ℓ,ℓ'=1Lα̅1(ℓ)α1(ℓ')aℓ,ℓ', where A=(aℓ,ℓ') is nonrandom, Hermitian positive definite, and is the unique matrix of such type satisfying $A=\bigl(c\,\mathsf{E}\frac{\bolds{\alpha}\bolds{\alpha}^{*}}{1+\bolds{\alpha}^{*}A\bolds{\alpha}}+\sigma^{2}I_{L}\bigr)^{-1}$ , where α∈ℂL has distribution H. The result generalizes those previously derived under more restricted assumptions.