On limiting theorems for conditional causation probabilities of multiple-run-rules

Causation probabilities of multiple-run-rules based on patterns of multi-state trials have been used and studied in various fields such as quality control, reliability of engineering system, biology, DNA sequence analysis and survival analysis. In this manuscript, we derive the limiting results for two types of conditional causation probabilities for multiple-run-rules by using the finite Markov chain imbedding technique. Extension and numerical examples are given to illustrate the theoretical results.