Subadditive Ergodic Theorems

Above is the famous Fekete’s lemma which demonstrates that the ratio of subadditive sequence (an) to n tends to a limit as n approaches infinity. This lemma is quite crucial in the field of subadditive ergodic theorems because it gives mathematicians some general ideas and guidelines in the non-random setting and leads to analogous discovery in the random setting. Kingman’s Subadditive Ergodic Theorem, for instance, is a perfect analogy in the random setting. This paper will introduce Kingman’s theorem and its variation in details as well as describing the concepts within the subadditive ergodic setting. In the latter section of the paper, the paper attempts to draw a parallel between the shape theorems and Kingman’s theorem so that random multidimensional distance functions satisfy similar properties like those of random single-dimensional subadditive sequences.