Statistical Physics-Taylor expanding

Question

Could please someone explain why I can make Taylor expanding around s/N. What about states that s is almost equal to N?

Answer 1

So this is not really rigorous; $s$ actually ranges all the way from $-N$ to $N$ so that $s/N$ really ranges all the way from $-1$ to $1$. Near $s/N=0$, one can Taylor expand. Traditionally we expand the logarithm and then exponentiate, since the logarithm is (quantitatively) smoother. This shows that the profile looks like a Gaussian near there.

For certain purposes that's plenty. To be rigorous you would want to ask: how near is "near"? And if there are values of $s$ which are not "near", what happens there?

There are different ways to handle that. One way would be just brute force estimation (try to take more derivatives of $\ln(P(d))$ and use the Lagrange error). Another way would be a quantitative central limit theorem, such as the Berry-Esseen theorem. Indeed $d$ is a sum of $N$ independent identically distributed variables which are uniformly distributed on $\{ a,-a \}$. The mean of this is zero; the variance is $a^2$; the third absolute moment is $a^3$. So the ratio $\frac{\rho}{\sigma^3}$ (using Wikipedia's notation) in the Berry-Esseen theorem is $1$. So the uniform error in the CLT approximation is less than $\frac{1}{2\sqrt{N}}$...but probably not by a whole lot, maybe an order of magnitude or so. An interesting question for an analyst or probabilist (not so much for a physicist) is where the error is maximized.