Does the weak law of large numbers hold in a general Banach space?

Question

Let $B$ be a Banach space. Let $X_n$ be i.i.d. random variables on $B$ with expectation $\mu$.

Is it true that their empirical mean converges to $\mu$ in law? If not, are there additional assumptions?

A reference would also be welcome.

Answer 1

This follows from the scalar LLN. Let $X\in L^1(\Omega, B)$ be a random variable, with $\Omega$ being the probability space. Since $L_1(\Omega)\otimes B$ is dense in $L^1(\omega,B)$, for every $\epsilon>0$ there exist scalar random variables $Y_1,\dots,Y_N\in L^1(\Omega)$ and unit vectors $v_1,\dots,v_N\in B$ such that $$ E\left(\left\|X - \sum Y_k v_k\right\|\right) < \epsilon^2/2 $$ If $S_{n,k}$ is the average of $n$ independent instances of $Y_k$, then by the scalar LLN, $$ P(\|S_{n,k} - E(Y_k)\|>\epsilon/(2N))<\epsilon $$ for all sufficiently large $n$. Hence, $$ P\left(\left\|S_{n} - E(X)\right\|>\epsilon \right) \le P\left(\left\|S_{n} - \sum S_{n,k}v_k\right\|> \epsilon/2\right) + P\left(\left\|\sum S_{n,k}v_k - E(X)\right\|> \epsilon/2\right) \le (2/\epsilon) E\left(\left\|X - \sum Y_k v_k\right\|\right) + \max_k P(\|S_{n,k} - E(Y_k)\|>\epsilon/(2N)) \\ < \epsilon + \epsilon = 2\epsilon $$

Page 190 of the aforementioned book Probability in Banach spaces by Ledoux and Talagrand is relevant here, although the result is not exactly stated there for general Banach spaces.