AI 中文总结
研究马尔可夫切换仿射递归平稳协方差矩阵存在性，核心方法是通过马尔可夫切换克罗内克算子及佩龙激励条件判断，贡献是将均方谱条件转化为平稳分布协方差存在准则，并举例说明相关特性。
AI 中文摘要
本文给出了有限状态马尔可夫切换仿射递归的平稳协方差矩阵存在时的有限维谱特征。对于形如\[Z_{n + 1}=A_{\Theta_{n + 1}}Z_n+\zeta_{n + 1}\]的随机递归，严格平稳性通常由随机矩阵乘积的顶部李雅普诺夫指数的负性决定。但该条件不能确保平稳律具有有限方差、协方差和皮尔逊相关性。我们表明这些二阶对象由马尔可夫切换克罗内克算子决定。若\[T_{ji}:=p_{ij}(A_j\otimes A_j)\]，在创新协方差的自然佩龙激励条件下，\(\rho(T)<1\)是平稳解平方可积的充要条件。当此条件成立时，状态加权二阶矩求解有限维线性系统，可明确得到协方差矩阵。示例说明了李雅普诺夫指数为负时二阶谱半径可能超过1，即严格平稳性存在时方差、协方差和皮尔逊相关性可能不是有限的。
英文摘要
This paper provides a finite-dimensional spectral characterisation for when the stationary covariance matrix of a finite-state Markov-switching affine recurrence exists. For stochastic recurrences of the form \[ Z_{n+1}=A_{Θ_{n+1}}Z_n+ζ_{n+1}, \] strict stationarity is typically governed by negativity of the top Lyapunov exponent of the random matrix products. This condition, however, does not ensure that the stationary law has finite variances, covariances, and Pearson correlations. We show that these second-order objects are governed instead by a Markov-switching Kronecker operator. If \[ T_{ji}:=p_{ij}(A_j\otimes A_j), \] then, under a natural Perron-excitation condition on the innovation covariance, \(ρ(T)<1\) is necessary and sufficient for the stationary solution to be square-integrable. When this condition holds, the regime-weighted second moments solve a finite-dimensional linear system, which yields the covariance matrix explicitly. Hence, this paper translates the mean-square spectral condition into an exact covariance-existence criterion for the stationary distribution. Examples illustrate the separation: the Lyapunov exponent may be negative while the second-order spectral radius exceeds one, so strict stationarity can persist even though variances, covariances, and Pearson correlations are not finite objects.
