{'fraction_non_alphanumeric': 0.03659976387249114, 'fraction_numerical': 0.004722550177095631, 'mean_word_length': 4.72972972972973, '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}}
On the probabilities of “large deviations” for multidimensional martingales

The central limit theorem gives an idea of the behaviour of the probabilities of the ^/«-deviations of a sum of independent random variables from its mean. An important step in this direction was Cramer's classical theorem on estimates of probabilities of deviations of order η ("large deviations"). Results of this type can naturally be regarded also as estimates of the speed of convergence in the law of large numbers (LLN). Very general results on the LLN for martingales have recently been obtained [1] —[2]. In the book [3] an analysis of the speed of convergence in the LLN is given for one-dimensional martingales. It is of interest to study this problem in the multidimensional case. In this connection, the paper [4] is the direct basis for our investigations.