We introduce a Markov product structure for multivariate tail dependence functions, building upon the well-known Markov product for copulas. We investigate algebraic and monotonicity properties of this new product as well as its role in describing the tail behaviour of the Markov product of copulas. For the bivariate case, we show additional smoothing properties and derive a characterization of idempotents together with the limiting behaviour of n-fold iterations. Finally, we establish a one-to-one correspondence between bivariate tail dependence functions and a class of positive, substochastic operators. These operators are contractions both on $L^1(\mathbb{R}_+)$ and $L^\infty(\mathbb{R}_+)$ and constitute a natural generalization of Markov operators.

中文翻译：

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尾依赖函数的马尔可夫积

我们为多变量尾依赖函数引入了马尔可夫乘积结构，建立在著名的联结马尔可夫乘积之上。我们研究了这个新产品的代数和单调性特性，以及它在描述 copula 的马尔可夫产品的尾部行为方面的作用。对于双变量情况，我们展示了额外的平滑属性，并推导出幂等特征以及 n 次迭代的限制行为。最后，我们建立了双变量尾部依赖函数和一类正的、亚随机算子之间的一一对应关系。这些算子都是 $L^1(\mathbb{R}_+)$ 和 $L^\infty(\mathbb{R}_+)$ 的收缩，构成了马尔可夫算子的自然推广。