We consider conditions for strict stationarity and ergodicity of a class of multivariate BEKK processes $(X_t : t=1,2,\ldots )$ and study the tail behavior of the associated stationary distributions. Specifically, we consider a class of BEKK-ARCH processes where the innovations are assumed to be Gaussian and a finite number of lagged $X_t$ ’s may load into the conditional covariance matrix of $X_t$ . By exploiting that the processes have multivariate stochastic recurrence equation representations, we show the existence of strictly stationary solutions under mild conditions, where only a fractional moment of $X_t$ may be finite. Moreover, we show that each component of the BEKK processes is regularly varying with some tail index. In general, the tail index differs along the components, which contrasts with most of the existing literature on the tail behavior of multivariate GARCH processes. Lastly, in an empirical illustration of our theoretical results, we quantify the model-implied tail index of the daily returns on two cryptocurrencies.

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一类 BEKK 过程的尾部行为的表征：随机递归方程方法

我们考虑了一类多元 BEKK 过程的严格平稳性和遍历性条件$(X_t : t=1,2,\ldots )$并研究相关平稳分布的尾部行为。具体来说，我们考虑一类 BEKK-ARCH 过程，其中假设创新是高斯的，并且有有限数量的滞后$X_t$的可能会加载到条件协方差矩阵中$X_t$. 通过利用过程具有多元随机递归方程表示，我们证明了在温和条件下存在严格平稳解，其中只有分数矩$X_t$可能是有限的。此外，我们展示了 BEKK 过程的每个组成部分都随着一些尾指数定期变化。一般来说，尾指数随着分量的不同而不同，这与大多数关于多元 GARCH 过程尾部行为的现有文献形成对比。最后，在我们的理论结果的实证说明中，我们量化了两种加密货币每日回报的模型隐含尾部指数。