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The convergence behavior of linear parallel interference cancellation is investigated. Especially the so-called ping-pong effect, where the bit-error rate performance is found to oscillate between two different convergence patterns is studied in detail. This effect is shown to be a direct consequence of the extreme eigenvalues of the correlation matrix, allowing for an analytical approach. Intervals for the dominating eigenvalues within which ping-pong effects can occur are specified and illustrated by examples. It is shown that the decision statistic for traditional parallel cancellation will always exhibit oscillating behavior with either short or long codes. Relaxation factors, leading to weighted cancellation, are shown to be effective for alleviating oscillations and ping-pong effects at the expense of convergence rate. Asymptotic analysis for large systems is applied to uncover the convergence behavior for long code systems.