Let $ X$ be a random variable on a manifold $M$. Is there a nice (intrinsic?) definition of the mean of $X$ and of its variance? This funny question comes from concrete motivations (imaging). What can be done with a chart? The problem here is that the mean is a global notion.
If $M$ has global coordinates or almost global, like stereographical projections for spheres, one may use them. This is ugly and non canonical. If $M$ is a Lie group, one may use the exponential map. When $M$ is equipped with a Riemannian metric $d:M\times M\to\mathbb{R}_+$ one may think about using a variational approach, and simply define the mean $m$ of $X$ as
$\displaystyle m:=\arg\min_{x\in M}\mathbb{E}(d(x,X)^2)$.
The value of the minimum is the variance of $X$. This definition does not always provide a unique point on $M$, as shown by the example of the uniform law on spheres for which every point is a mean! This is not a bug, it is a feature, a geometrical feature due to the invariance of the law by isometries in this example. One can ask about an empirical estimator of the mean, and its asymptotic fluctuations. For some answers, see e.g. the work of Pennec and Bhattacharya and Bhattacharya. The variational expression of $d$ in terms of geodesics is valid up to the cut-locus/injectivity-radius of the exponential map.
Beyond the mean, the law of $X$ may be viewed as the linear form
$f\in\mathcal{C}_b(M,\mathbb{R})\mapsto\mathbb{E}(f(X)).$
This does not rely on the manifold nature of $M$ since we only use the fact that $M$ is a topological space. Note that if $M$ is an open subset of $\mathbb{R}^d$ then we recover the usual $\mathbb{E}(X)$ by approximation via dominated convergence. The integrability of $X$ is the class of functions for which the map above is finite. In some sense, $\mathbb{E}(X)$ is an element of the bidual $M”$ of $M$, provided that we view $M’:=\mathcal{C}_b(M,\mathbb{R})$ as a sort of dual of $M$. Of course, $M\subset M”$ via the canonical injection but the converse does not hold in general.
If $(X_n)_{n\geq0}$ is an irreducible positive recurrent aperiodic Markov chain with state space $M$ and unique invariant law $\mu$ then the law of large numbers states that with probability one, and regardless of the initial law of the chain, we have
$$\frac{1}{n}\delta_{X_1}+\cdots+\frac{1}{n}\delta_{X_n} \underset{n\to\infty}{\overset{\mathcal{C}_b(M,\mathbb{R})}{\longrightarrow}}\mu.$$
The asymptotic fluctuations of this convergence are described by a central limit theorem, which involves the variance for $\mu$ of the solution of the Poisson equation associated to the dynamics.
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