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This paper proposes the use of a new generalized asymptotic paradigm in order to analyze the performance of subspace-based direction-of-arrival (DoA) estimation in array signal processing applications. Instead of assuming that the number of samples is high whereas the number of sensors/antennas remains fixed, the asymptotic situation analyzed herein assumes that both quantities tend to infinity at the same rate. This asymptotic situation provides a more accurate description of a potential situation where these two quantities are finite and hence comparable in magnitude. It is first shown that both MUSIC and SSMUSIC are inconsistent when the number of antennas/sensors increases without bound at the same rate as the sample size. This is done by analyzing and deriving closed-form expressions for the two corresponding asymptotic cost functions. By examining these asymptotic cost functions, one can establish the minimum number of samples per antenna needed to resolve closely spaced sources in this asymptotic regime. Next, two alternative estimators are constructed, that are strongly consistent in the new asymptotic situation, i.e., they provide consistent DoA estimates, not only when the number of snapshots goes to infinity, but also when the number of sensors/antennas increases without bound at the same rate. These estimators are inspired by the theory of G-estimation and are therefore referred to as G-MUSIC and G-SSMUSIC, respectively. Simulations show that the proposed algorithms outperform their traditional counterparts in finite sample-size situations, although they still present certain limitations.