Given a sequence $(P_n)_n$ of probability measures on a space satisfies a LDP with rate $r_n$ and rate function $I$ if both the following hold:
$$ \limsup_{n}\frac{1}{r_n}\log P_n [F] \le - \inf_{x \in F}I(x), \forall F: \text{ closed } $$
$$ \liminf_{n}\frac{1}{r_n}\log P_n [G] \ge - \inf_{x \in G}I(x), \forall G : \text{ open } $$
Consider the following sequences:
a) $P_n = Uniform([-n,n])$
b) $P_n = Uniform([-1/n,1/n])$
Determine if either of these satisfy the LDP with rate n, if so give the rate function $I$.
I already know that (a) doesn't satisfy the LDP but (b) does.
My biggest issue here is how to solve these problems, the way it is stated it feels like I should already have an idea of what this $I$ should be.
Do you first find a limiting probability?
In the case of (a) there is no uniform distribution on all of the real line so it seems immediate there is no LDP to be satisfied.
In (b) as $n \to \infty $ in the limit we get $\delta_0(x) = 1_{\{x = 0\}}$ so it makes sense that we would have $I(x) = 0$ if $x= 0$ otherwise $I = \infty$.
That approach works here but I'm not sure if that is the approach I should always take.