We investigate the convergence of higher order Cesàro means in Banach spaces. The main results of this paper are: (1) The proof of mean and Birkhoff-type ergodic theorems for higher order Cesàro means. (2) The existence of a one-to-one correspondence between convergent Cesàro means of different orders. (3) The proof of strong laws of large numbers for higher order sums of independent and identically distributed random elements. (4) A characterization of the ergodicity of measure preserving maps in terms of higher order mixing properties. To deal with higher order Cesàro means, one needs a notion of weighted mean more general than the one usually considered in the literature on weighted ergodic theorems. In this context, we also prove a characterization of generalized weighted means preserving Cesàro convergence.

我们研究了 Banach 空间中高阶 Cesàro 均值的收敛性。本文的主要成果是： (1) 高阶Cesàro均值的均值和Birkhoff型遍历定理的证明。(2) 不同阶的收敛切萨罗均值之间存在一一对应关系。(3) 独立同分布随机元素高阶和的强数定律的证明。(4) 根据高阶混合特性对测度保存图的遍历性进行表征。为了处理高阶 Cesàro 均值，需要一种比加权遍历定理文献中通常考虑的更一般的加权均值概念。在这种情况下，我们还证明了广义加权均值保持 Cesàro 收敛性的特征。