Thats a unique idea.You should consider the following idea to enhance your idea so that it would impact uniquely: The concentration of measure phenomenon may be summarized as follows: a function of many akly dependent random variables that is not too sensitive to any of its individual arguments will tend to take values very close to its expectation. This phenomenon is most completely understood when the arguments are mutually independent random variables, and there exist several porful complementary methods for proving concentration inequalities, such as the martingale method, the entropy method, and the method of transportation inequalities. The setting of dependent arguments is much less ll understood. This chapter focuses on the martingale method for deriving concentration inequalities without independence assumptions. In particular,  use the machinery of so-called Wasserstein matrices to show that the Azuma-Hoeffding concentration inequality for martingales with almost surely bounded differences, when applied in a sufficiently abstract setting, is porful enough to recover and sharpen several known concentration results for nonproduct measures. Wasserstein matrices provide a natural formalism for capturing the interplay beten the metric and the probabilistic structures, which is fundamental to the concentration phenomenon.
['Aryeh Kontorovich', 'Maxim Raginsky']
I am writing a research paper on a idea Concentration of measure without independence: a unified approach via the martingale method
[{"from": "human", "value": "I am writing a research paper on a idea Concentration of measure without independence: a unified approach via the martingale method"}, {"from": "gpt", "value": "Thats a unique idea.You should consider the following idea to enhance your idea so that it would impact uniquely: The concentration of measure phenomenon may be summarized as follows: a function of many akly dependent random variables that is not too sensitive to any of its individual arguments will tend to take values very close to its expectation. This phenomenon is most completely understood when the arguments are mutually independent random variables, and there exist several porful complementary methods for proving concentration inequalities, such as the martingale method, the entropy method, and the method of transportation inequalities. The setting of dependent arguments is much less ll understood. This chapter focuses on the martingale method for deriving concentration inequalities without independence assumptions. In particular,  use the machinery of so-called Wasserstein matrices to show that the Azuma-Hoeffding concentration inequality for martingales with almost surely bounded differences, when applied in a sufficiently abstract setting, is porful enough to recover and sharpen several known concentration results for nonproduct measures. Wasserstein matrices provide a natural formalism for capturing the interplay beten the metric and the probabilistic structures, which is fundamental to the concentration phenomenon."}]