**Setup:$\quad$**

Suppose that $(X_n)$ is a stationary ergodic process with $E|X_1|<\infty$.

Given $X^{(n)}=(X_1, \dots, X_n)$, select a standard Efron bootstrap subsample $(X_{n,1}^*, \dots, X_{n,m(n)}^*)$ by pulling $m(n)$ times with replacement from a uniform distribution $U(\{X_1, \dots, X_n\})$, i.e. $$ X_{n,i}^* = X_{Z_{n,i}}, \quad Z_{n,i} \overset{\text{iid}}{\sim} U(\{1,\dots,n\}), \quad i=1,\dots,m(n), $$ where the $Z_{n,i}$ are independent of $(X_n)$ and form a triangular array with independent rows.

Let the * bootstrap mean* $\mu_{m(n)}^*$ be the sample mean of the bootstrap subsample, i.e.
$$
\mu_{m(n)}^* = \frac{1}{m(n)} \sum_{i=1}^{m(n)} X_{n,i}^*.
$$

**Question:**

In the case that $(X_n)$ is a stationary ergodic process, are there any known results about when the following WLLN holds? $$ \mu_{m(n)}^* \overset{P}{\longrightarrow} E[X] $$ as $n \to \infty$.

**What I've found:**

In the case that the $X_i$ are i.i.d. and $m(n) \to \infty$, it is known that the WLLN above holds for any $m(n) \to \infty$ (e.g. see p.2848 of this 2003 survey by Csörgő and Rosalsky).

Additionally, Einmal and Rosalsky later proved that $$ \mu_{m(n)}^* - \frac{1}{n} \sum_{i=1}^n X_i \overset{P}{\longrightarrow} 0 $$ holds for any $(X_n)$ (not necessarily independentent or identically distributed), provided $m(n) \uparrow \infty$ and $$ \frac{X_n}{\sqrt{m(n)}} \overset{\text{a.s.}}{\longrightarrow} 0. $$ This, however, doesn't cover all stationary ergodic processes with $E|X_1|<\infty$.