We investigate recent claims in the literature for the existence of a low velocity dispersion population of galaxies in the cores of rich clusters. The suggested population, which has been generally interpreted as gravitationally bound to the potential well of the first-ranked cluster galaxy (often morphologically identifiable as a D/cD galaxy), is taken as evidence in support of the galactic cannibalism model for D/cD galaxy formation. The statistical tests of kinematic evidence which have been used to support this claim are, however, fundamentally flawed. In particular, tests for bound populations which rely on the binning of small numbers of galaxy redshifts may lead to spurious results. We examine five clusters which have been suggested in the literature to possess bound populations of galaxies associated with the D/cD. We develop an improved statistical test, the Indicator test, which compares the number of galaxies with relative velocities (measured with respect to the D/cD) less than a given fraction of the cluster scale ("dispersion") to the number expected for an assumed parent velocity distribution (for convenience taken to be a Gaussian). We find strong supporting kinematic evidence for the existence of a bound population in only one of the five clusters investigated, Klemola 44. Marginal kinematic evidence in support of the bound population hypothesis is found for two clusters, A1991 and A2589. We find no strong supporting kinematic evidence for a bound population in two clusters, A2107 and A2593. To avoid difficulties with small number statistics, some authors have employed tests for bound populations which rely on a summation of redshift distributions from many different clusters. Detection of a bound population is claimed if the summed distribution cannot be adequately fit with a single Gaussian. We show, however, that the null hypothesis for such a procedure is incorrect. A finite mixture of velocities drawn from clusters with nonidentical dispersions is not a Gaussian, but rather a distribution which is generally peakier than Gaussian in the center, lighter than Gaussian in the middle quantiles, and heavier than Gaussian in the outer tails. This realization, along with sample-dependent selection effects, can explain differences in the proposed mixture model parameters between Cowie & Hu and Bothun & Schombert, and can also account for the anomalously large second-component dispersion in the mixture models by both sets of authors. Simulations of the expected pooled samples when draws are made from clusters with a range of dispersions are compared with the observed distributions of Cowie & Hu and Bothn & Schombert. A one-parameter functional form which matches such mixture populations well, at least in the central portion, is the Logistic function. No strong kinematic evidence is found for the existence of a low-dispersion "bound" population of galaxies in the vicinity of the D/cD for most clusters, when compared to Logistic fits to the expected pooled samples. We conclude, in agreement with the analysis of Lauer, that the high multiplicity of first-ranked galaxies in clusters is best explained as a result of the central structure of galaxy clusters. We next consider claims for the existence of significant velocity offsets of a number of D/cD galaxies with respect to the central location in velocity space ("mean") of the remaining cluster galaxies. We point out that, particularly in clusters with small numbers of available redshifts, confidence intervals on central location are often asymmetric. Estimates of (symmetric) confidence intervals on the mean velocity obtained via canonical procedures are potentially misleading and may lead to false claims of significance. We employ a bootstrap resampling technique to obtain realistic confidence intervals on cluster central locations and incorporate observational errors in the measurement of the D/cD velocity in a consistent manner. Of 14 clusters (15 D/cD galaxies) noted in the literature with discrepant D/cD velocities, we confirm only four (A1795, A1809, A2670, and Shapley 8). Three of the four clusters for which we confirm suspected velocity offsets have more than 40 redshifts available, enough to reduce the error in central velocity location of the samples to the point that velocity offsets of the D/cD galaxies on the order of 250-400 km s-1 become statistically significant. Three clusters (A85, A2634, and Klemola 44) have only marginally significant D/cD velocity offsets. The remaining seven clusters, which generally have only 10-30 redshifts available, are shown not to exhibit statistically significant D/cD velocity offsets. Many available redshifts per cluster are required before apparently large D/cD velocity offsets can be accepted in most clusters.
