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## Main Question or Discussion Point

##X_i## is an independent and identically distributed random variable drawn from a non-negative discrete distribution with known mean ##0 < \mu < 1## and finite variance. No probability is assigned to ##\infty##.

Now, given ##1<M##, a sequence ##\{X_i\}## for ##i\in1...n## is said to meet ##conditions (1) and (2) for M## if it satisfies the following conditions: $$(1) S_n = X_1 + ... + X_n < 1 + n - M$$ $$(2) n>=M$$.

It is clear that the unconditional sample mean ##E_n[X_i]\xrightarrow{p}\mu##. But if we were to take ##M\to\infty## by considering sequences of ##\{X_i\}## for ##i\in1...n_k## that meet ##conditions (1) and (2) for M_k##, where ##\{M_k\}## is a sequence of successively larger numbers, does ##S_{n_k}/n_k\xrightarrow{p}\mu##?

Now, given ##1<M##, a sequence ##\{X_i\}## for ##i\in1...n## is said to meet ##conditions (1) and (2) for M## if it satisfies the following conditions: $$(1) S_n = X_1 + ... + X_n < 1 + n - M$$ $$(2) n>=M$$.

It is clear that the unconditional sample mean ##E_n[X_i]\xrightarrow{p}\mu##. But if we were to take ##M\to\infty## by considering sequences of ##\{X_i\}## for ##i\in1...n_k## that meet ##conditions (1) and (2) for M_k##, where ##\{M_k\}## is a sequence of successively larger numbers, does ##S_{n_k}/n_k\xrightarrow{p}\mu##?

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