{'fraction_non_alphanumeric': 0.021739130434782608, 'fraction_numerical': 0.0036231884057971015, 'mean_word_length': 4.921428571428572, 'pattern_counts': {'":': 0, '<': 0, '<?xml version=': 0, '>': 0, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 0, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
2 Discrete-Time Renewal Processes

Renewal processes (RPs) provide a theoretical framework for investigating the occurrence of patterns in repeated independent trials. Roughly speaking, the reason for using the term “renewal” comes from the basic assumption that when the pattern of interest occurs for the first time, the process starts anew, in the sense that the initial situation is reestablished. This means that, starting from this “renewal instant,” the waiting time for the second occurrence of the pattern has the same distribution as the time needed for the first occurrence. The process continues like this indefinitely. Renewal processes have known a huge success in the last 60 years and much work is still being carried out in the field. The reasons for this phenomenon are manifold and we give some of them below.