I encountered an unusual problem and am struggling to prove the following result.

Let $T$ be a positive random variable, and $\{x_i\}$ be a sequence of i.i.d. strictly positive random variables with a finite mean $\mu$ and a finite variance $\sigma^2$. Let $N= \sup_{n} \{S_n \leq T\}$ where $S_n=\sum_{i=1}^n x_i$. I try to prove that: $$\lim_{\mu \rightarrow 0} N\mu \rightarrow T.$$ Intuitively, we are summing up $N$ terms of $x_i$s to get the quantity $T$, and by letting $\mu \rightarrow 0$, I can show that $N \rightarrow \infty$ and $S_n \rightarrow T$ (assuming some boundary conditions hold). And since $S_n$ is approximately $N\mu$ intuitively, the above relationship seems to be true.

I attempt to start from the SLLN for i.i.d. r.v.s: $$\lim_{N \rightarrow \infty} \frac{S_n}{N} \overset{a.s.}{\rightarrow} \mu $$ However $N \rightarrow \infty $ would imply $\mu \rightarrow 0$ and it does not seem to work unless I can multiply both side by $N$.

Any hints on where to start and the mode of convergence are highly appreciated.

  • $\begingroup$ It might be interesting to examine how your process applies to a lottery, where the vast majority of tickets give no prize and a single ticket gets a million. The expectation value is of the order $1$. It is then clear that for all values of $T$ up to 1 million you are basically checking a long string of zero-valued tickets until you hit the jackpot. $\endgroup$ – M. Wind Sep 18 '18 at 20:54
  • $\begingroup$ @M.Wind Thanks for the comment. I think it does not apply in this case since $x_i$s are strictly positive, so each $x_i$ adds something to the sum $S_n$. $\endgroup$ – user46666 Sep 19 '18 at 9:44
  • $\begingroup$ I don't think that is particularly relevant. You can imagine a lottery where each non-winning ticket receives a very small pay out (1 dollar cent), while there is one very large prize (1 million dollar). I think it is worthwhile to examine the mathematics of this case. $\endgroup$ – M. Wind Sep 19 '18 at 17:42

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