I have the independent and identically distributed random variables $X_1,X_2,\ldots$ with a finite expectation $\mu$. I also have defined $S_n = X_1 + \cdots + X_n$.
According to the law of large numbers, I already know that
$S_n/n \to \mu$ almost surely as $n\to\infty$.
However, my question is: How does the convergence take place? What is the “shape” of this convergence? For example, for a given $N>0$, how close is $S_N/N$ to $\mu$? How does the difference, or remainder/residual, look like?
Are there any results, or theorems, available that tell me these kinds of things?