# Rate of convergence for SLLN

I am interested in writing a non-asymptotic rate of convergence for SLLN as a function of number of samples.

From the literature I've read so far, CLT provides an asymptotic convergence rate of $(1/\sqrt N)$ for SLLN.

Also, Berry-Esseen provides a non-asymptotic bound in terms of c.d.f's.

$$|F_N(x) - \Phi(x)| \le \frac{C\mathbb{E}(|X|^3)}{\sigma^3\sqrt N}$$

Is there a Berry-Esseen like statement to bound the difference between sample mean and the expected value of the underlying distribution as a function of N (number of samples)?

First, in the Berry-Esseen theorem, $F_N$ is not just any distributon, but a properly normalized convolutions of N identical distributions.
• @user90275: To use Berry-Esseen you correct the mean to be 0, then need to calculate the third absolute moment and the variance of X so you can get the upper bound on the maximum deviation between the two CDFs for a given N (i.e, $\Delta_N$). Now, for given N, you know that the maximum absolute deviation in the normalized distribution and the normal is less than the Berry-Esseen bound i.e, $\\max\{\Phi(x)-\Delta_N,0\}<P(\frac{\bar X_N-E[X]}{\sigma\sqrt{N}}\leq x)<min\{\Phi(x)+\Delta_N,1\}$ – user31668 Nov 11 '13 at 15:13
• (cont'd) Therefore, highly skewed distributions (i.e, higher third moment relative to their cubed standard deviation) will reduce the accuracy of the CLT for small N. Also, if you know that the $X_i$ are bounded, then you can truncate the normal $\Phi(x)$ at $(NX_{min},NX_{max})$ which probably won't do much, but at least you dont have bounds out to infinity for the maximum\minimum values. – user31668 Nov 11 '13 at 15:14