Let $X \sim p_X$ be a real-valued random variable with $\mathbb{E}[X] = \mu > c$ where $c \in \mathbb{R}.$
Assume you sample from $p_X$ and only accept samples such that the current sample mean is greater than $c$.
Formally, $$p_{Y_1}(y_1) = \frac{p_X(y_1)\mathbb{I}_{[y_1 > c]}}{\int p_X(y_1)\mathbb{I}_{[y_1 > c]}dy_1}, \dots$$ $$p_{Y_n\vert Y_{1:n-1}}(y_n\vert y_{1:n-1}) = \frac{p_X(y_n)\mathbb{I}_{[\overline{y}_{1:n} > c]}}{\int p_X(y_n)\mathbb{I}_{[\overline{y}_{1:n} > c]}dy_n},$$
where $\overline{y}_{1:n} = \frac{\sum^n_{i=1}y_i}{n}.$
Does it hold that $\overline{y}_{1:n} \overset{p}{\rightarrow} \mu$ for $n \rightarrow \infty$?