Hi, thanks for the response! I apologize, the “Left as an exercise” line was mine, and written kind of tongue-in-cheek. The rough sketch of the proposition we had in the initial draft did not spell out sufficiently clearly what it was I want to demonstrate here and was also (as you point out correctly) wrong in the way it was stated. That wasted people’s time and I feel pretty bad about it. Mea culpa.
I think/hope the current version of the statement is more complete and less wrong. (Although I also wouldn’t be shocked if there are mistakes in there). Regarding your points:
The limit now shows up on both sides of the equation (as it should)! The dependence on B on the RHS does actually kind of drop away at some point, but I’m not showing that here. I’d previously just sloppily substituted “chose B as a large number” and then rewrite the proposition in the way indicated at the end of the Note for Proposition 2. That’s the way these large deviation principles are typically used.
Yeah, that should have been an ≈ rather than a ∼. Sorry, sloppy.
True. Thinking more about it now, perhaps framing the proposition in terms of “bridges” was a confusing choice; if I revisit this post again (in a month or so 🤦♂️) I will work on cleaning that up.
Hi, thanks for the response! I apologize, the “Left as an exercise” line was mine, and written kind of tongue-in-cheek. The rough sketch of the proposition we had in the initial draft did not spell out sufficiently clearly what it was I want to demonstrate here and was also (as you point out correctly) wrong in the way it was stated. That wasted people’s time and I feel pretty bad about it. Mea culpa.
I think/hope the current version of the statement is more complete and less wrong. (Although I also wouldn’t be shocked if there are mistakes in there). Regarding your points:
The limit now shows up on both sides of the equation (as it should)! The dependence on B on the RHS does actually kind of drop away at some point, but I’m not showing that here. I’d previously just sloppily substituted “chose B as a large number” and then rewrite the proposition in the way indicated at the end of the Note for Proposition 2. That’s the way these large deviation principles are typically used.
Yeah, that should have been an ≈ rather than a ∼. Sorry, sloppy.
True. Thinking more about it now, perhaps framing the proposition in terms of “bridges” was a confusing choice; if I revisit this post again (in a month or so 🤦♂️) I will work on cleaning that up.