Diversity or complementarity of automatic speech recognition (ASR) systems is crucial for achieving a reduction in word error rate (WER) upon fusion using the ROVER algorithm. We present a theoretical proof explaining this often-observed link between ASR system diversity and ROVER performance. This is in contrast to many previous works that have only presented empirical evidence for this link or have focused on designing diverse ASR systems using intuitive algorithmic modifications. We prove that the WER of the ROVER output approximately decomposes into a difference of the average WER of the individual ASR systems and the average WER of the ASR systems with respect to the ROVER output. We refer to the latter quantity as the diversity of the ASR system ensemble because it measures the spread of the ASR hypotheses about the ROVER hypothesis. This result explains the trade-off between the WER of the individual systems and the diversity of the ensemble. We support this result through ROVER experiments using multiple ASR systems trained on standard data sets with the Kaldi toolkit. We use the proposed theorem to explain the lower WERs obtained by ASR confidence-weighted ROVER as compared to word frequency-based ROVER. We also quantify the reduction in ROVER WER with increasing diversity of the N-best list. We finally present a simple discriminative framework for jointly training multiple diverse acoustic models (AMs) based on the proposed theorem. Our framework generalizes and provides a theoretical basis for some recent intuitive modifications to well-known discriminative training criterion for training diverse AMs.