- The central limit theorem relies on the mean and variance of your random variable existing; there are random variables for which mean and variance don't exist.
- The definition of "measure-preserving" in the ergodic theorem statement is missing the measure.
- dim (ran A)^perp = dim ker A^T can be strengthened by dropping the dim's. If v in ker A^T, then A^T v = 0. Pick any Ax in ran A; then <v, Ax> = <A^T v, x> = <0, x> = 0, so v is in (ran A)^perp.
- Others pointed out that 20 looks busted.
- The definition of Haar measure needs to fix the measure of some nonzero-measure set, or "unique" needs to be replaced with "unique up to scaling" otherwise the theorem isn't true.
- "Sounders Mac Lane"
- 25: x_0 came out of nowhere; it's unclear from the text which "invertible" is meant; the basin of convergence for Newton's method for finding a local continuation can be very small indeed.
The central limit theorem does hold for iid rv (independent identically distributed random variables) with finite mean and variance. Now, those assumptions can be relaxed (the rv need not be independent, but they can't be too dependent, and they need not have finite variance, but they can't be too "far out"), and some of the pertinent proofs are only a few decades old; but you can hardly expect a survey with 135 proofs to cover all the subtleties.
Some of the other points may be more egregious howlers, but, again, come on - this is not the definite reference for any one of those theorems.
EDIT to add: Having said that, the Lindeberg-Feller and the Lyuapunov formulation of the CLT do require finite variance, so maybe I was too quick in stating that that assumption can be relaxed.
In other words, a 130 page write up focused on the overall picture instead of lots of details has the usual number of typos without a couple passes from a skilled editor.
- The central limit theorem relies on the mean and variance of your random variable existing; there are random variables for which mean and variance don't exist.
- The definition of "measure-preserving" in the ergodic theorem statement is missing the measure.
- dim (ran A)^perp = dim ker A^T can be strengthened by dropping the dim's. If v in ker A^T, then A^T v = 0. Pick any Ax in ran A; then <v, Ax> = <A^T v, x> = <0, x> = 0, so v is in (ran A)^perp.
- Others pointed out that 20 looks busted.
- The definition of Haar measure needs to fix the measure of some nonzero-measure set, or "unique" needs to be replaced with "unique up to scaling" otherwise the theorem isn't true.
- "Sounders Mac Lane"
- 25: x_0 came out of nowhere; it's unclear from the text which "invertible" is meant; the basin of convergence for Newton's method for finding a local continuation can be very small indeed.
- 36: Why is the adjoint of A named T^*?
- 46: You need some assumptions about f.