What is the optimal convergence rate for the strong law of large numbers for positive bounded iid random variables ?

]]>Let be iid random variables (such as random walk) and and and $latex (I guess this means you’ve “centered” the random variable.)

Then your empirical average approaches the theoretical average , i.e. almost surely.

Your use of Borel-Cantelli is already a sign that some measure-theoretic phenomena is going on. If only we knew the distribution of in practice. Or could genuinely assume these random variables were perfectly iid.

]]>I dont understand why everyone insists on comparing LLN and CLT without mentioning rate of convergnce and mode of convergence. To me it seams that we are talking about convergence in different sense and of sequences of different random varibales w.r.t $\frac{1}{n}$ and $\frac{1}{\sqrt{n}}$ . Noone seams to acknowledge this and I think it is really the essence of the ideas. Care to weight in?

]]>I am trying (for the sake of my own better understanding) to replace all of the big-O’s with explicit constants. I am having trouble with display (9), however. Is it being claimed with high probability for all n, for n large enough, almost surely for all n, etc?

Best regards,

-Aryeh