We first develop a theory of conditional expectations for random variables with values in a complete metric space [Formula: see text] equipped with a contractive barycentric map [Formula: see text], and then give convergence theorems for martingales of [Formula: see text]-conditional expectations. We give the Birkhoff ergodic theorem for [Formula: see text]-values of ergodic empirical measures and provide a description of the ergodic limit function in terms of the [Formula: see text]-conditional expectation. Moreover, we prove the continuity property of the ergodic limit function by finding a complete metric between contractive barycentric maps on the Wasserstein space of Borel probability measures on [Formula: see text]. Finally, the large deviation property of [Formula: see text]-values of i.i.d. empirical measures is obtained by applying the Sanov large deviation principle.

中文翻译：

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重心映射的收敛定理

我们首先开发了一个随机变量的条件期望理论，其值在一个完整的度量空间 [公式：见文本] 配备了一个收缩重心映射 [公式：见文本]，然后给出了 [公式：见文本] 的鞅的收敛定理]-条件期望。我们给出了 [Formula: see text]-遍历经验测量值的 Birkhoff 遍历定理，并根据 [Formula: see text]-条件期望描述了遍历极限函数。此外，我们通过在 [公式：见文本] 上的 Borel 概率测度的 Wasserstein 空间上的收缩重心图之间找到一个完整的度量来证明遍历极限函数的连续性。最后，[公式：见正文]的大偏差性质-iid的值