Define the sample mean $S_n=(X_1+...+X_n)/n$ of a sequence $\{X_1,...,X_n,...\}$ of i.i.d. random variables with non-negative discrete support, known mean $0<\mu<1$, finite standard deviation $\sigma$, and no probability on $\infty$. Results like the law of large numbers tell us that for a sequence of indices $n_1< ... <n_j<...$, where for example $n_1=1,n_2=2,...,n_j=j,...$ , that the sequence of sample means ${S_{n_1},...,S_{n_j},...}\xrightarrow{p}\mu$.
Now consider the subsequence of indices $n_{j_1}< ... <n_{j_k}<...$, consisting of all $n_{j_k}$ in the original sequence such that $n_{j_k}$'s associated set $\{X_1,X_2,...,X_{n_{j_k}}\}$ meets the following condition: $X_1+X_2+...+X_{n_{j_k}} < n_{j_k}$. Does ${S_{n_{j_1}},...,S_{n_{j_k}},...}\xrightarrow{p}\mu$?