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A new look at independence. (English) Zbl 0858.60019
Let $$X$$ be the product of $$N$$ probability spaces $$(\Omega, \Sigma, \mu)$$ and let $$P$$ be the product measure $$\mu^{\otimes N}$$ on $$X$$. The distance $$d$$ on $$X$$ is defined by $$d(x,y) = \text{card}\{1 \leq i \leq N: x_i \neq y_i\}$$, where $$x = (x_i; 1\leq i \leq N)$$ and $$y = (y_i; 1\leq i \leq N)$$. Given a set $$A \subset X$$, consider its enlargement $$A_t :=\{x\in X: d(x,A)\leq t\}$$ for $$t>0$$. Then the bound $$1-P(A_t)\leq 2\text{exp} \{-t^2/N\}$$ holds whenever $$P(A) \geq 1/2$$. It is an example of what is called as the concentration of measure phenomenon. According to the author: “The present paper represents my best attempt to explain in the simplest way I can achieve what this is all about, without ever doing anything technical”.

##### MSC:
 60E15 Inequalities; stochastic orderings 28A35 Measures and integrals in product spaces
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