The proposed research performs an analysis of the capacity higher-order statistics for a single-input multiple-output multiantenna wireless communication system equipped with a maximum-ratio combining scheme. It was assumed that the propagation multipath channel is described with the $κ-μ$ fading model with the correlated dominant components. Closed-form and asymptotic expressions were derived and applied to the problem of minimum capacity reliability (due to channel fluctuations, thus possible rate deterioration) and corresponding signal-to-noise ratio analysis. The performed computer simulation, verifying the correctness of the obtained expressions, along with the generalized $κ-μ$ fading channel with correlated shadowing, assumed several specific limiting simplified cases: Rayleigh, Rician and Nakagami-$m$. It was shown that the signal-to-noise ratio (at which minimum capacity reliability is attained) is achieved at greater values than that of simplified models, and the absolute value of this minimum can be smaller/higher than for the degenerate cases depending on the dominant components one-step correlation coefficient.