U-statistics: some old and new results with applications to patterns in random strings and permutations

U-statistics form a large class of random variables that appear in many contexts. I will focus on some simple and some less obvious applications to patterns in random permutations and random strings, and general results useful in these applications (and sometimes motivated by them). This will include some less common versions of U-statistics (asymmetric U-statistics and U-statistics based on an m-dependent sequence), and some new results on renewal theory for U-statistics.