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Dongwei
Dongwei
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Limit Weak limit of pushforward measures with finite second moments is also a pushforward measure with finite second moment

Suppose $\mu \in P(\mathbb{R}^d)$ and for each $n$, $T_n:\mathbb{R^d} \rightarrow \mathbb{R^d}$ is such that the pushforward $T_n \# \mu$ has a finite second moment. If $\{T_n \# \mu\}$ converges weakly to $\nu \in P(\mathbb{R}^d)$, could we find $T:\mathbb{R^d} \rightarrow \mathbb{R^d}$ such that $\nu = T_\# \mu$ and the pushforward $T \# \mu$ also has a finite second moment. This problem is very similar to Limit of pushforward measures of random variables is "represented" by a random variable. It would be great if any reference book and paper could be given.

Limit of pushforward measures with finite second moments is also a pushforward measure with finite second moment

Suppose $\mu \in P(\mathbb{R}^d)$ and for each $n$, $T_n:\mathbb{R^d} \rightarrow \mathbb{R^d}$ is such that $T_n \# \mu$ has a finite second moment. If $\{T_n \# \mu\}$ converges weakly to $\nu \in P(\mathbb{R}^d)$, could we find $T:\mathbb{R^d} \rightarrow \mathbb{R^d}$ such that $\nu = T_\# \mu$ and $T \# \mu$ also has a finite second moment. This problem is very similar to Limit of pushforward measures of random variables is "represented" by a random variable. It would be great if any reference book and paper could be given.

Weak limit of pushforward measures with finite second moments is also a pushforward measure with finite second moment

Suppose $\mu \in P(\mathbb{R}^d)$ and for each $n$, $T_n:\mathbb{R^d} \rightarrow \mathbb{R^d}$ is such that the pushforward $T_n \# \mu$ has a finite second moment. If $\{T_n \# \mu\}$ converges weakly to $\nu \in P(\mathbb{R}^d)$, could we find $T:\mathbb{R^d} \rightarrow \mathbb{R^d}$ such that $\nu = T_\# \mu$ and the pushforward $T \# \mu$ also has a finite second moment. This problem is very similar to Limit of pushforward measures of random variables is "represented" by a random variable. It would be great if any reference book and paper could be given.

Source Link
Dongwei
Dongwei
  • 69
  • 4

Limit of pushforward measures with finite second moments is also a pushforward measure with finite second moment

Suppose $\mu \in P(\mathbb{R}^d)$ and for each $n$, $T_n:\mathbb{R^d} \rightarrow \mathbb{R^d}$ is such that $T_n \# \mu$ has a finite second moment. If $\{T_n \# \mu\}$ converges weakly to $\nu \in P(\mathbb{R}^d)$, could we find $T:\mathbb{R^d} \rightarrow \mathbb{R^d}$ such that $\nu = T_\# \mu$ and $T \# \mu$ also has a finite second moment. This problem is very similar to Limit of pushforward measures of random variables is "represented" by a random variable. It would be great if any reference book and paper could be given.