1. Existence and Uniqueness of the LDP

We begin by defining the $𝛿$ -rate function, which is useful for comparing rate functions.

Definition 1.1. The $𝛿$ -rate function $𝐼^{𝛿}$ associated with a rate function $𝐼$ is defined as

𝐼^{𝛿} (𝑥) = min {𝐼 (𝑥) - 𝛿, \frac{1}{𝛿}} .

Proposition 1.2 (Uniqueness). A family of probability measures

{𝜇_{𝜀}}

on a regular topological space can have at most one associated rate function.

Proof. Suppose $𝐼_{1}$ and $𝐼_{2}$ are two rate functions associated with the LDP of $𝜇_{𝜀}$ , and assume there exists a point $𝑥_{0}$ such that $𝐼_{1} (𝑥_{0}) < 𝐼_{2} (𝑥_{0})$ . By lower semicontinuity, there exists an open neighborhood $𝐴$ of $𝑥_{0}$ such that ${inf}_{𝑥 \in 𝐴} 𝐼_{2} (𝑥) \geq 𝐼_{2}^{𝛿} (𝑥_{0})$ , where $𝐼_{2}^{𝛿}$ is the $𝛿$ -rate function defined in Definition 1.1. By regularity, we can further assume ${inf}_{𝑥 \in \overset{─}{𝐴}} 𝐼_{2} (𝑥) \geq 𝐼_{2}^{𝛿} (𝑥_{0})$ . Consequently,

- 𝐼_{2}^{𝛿} (𝑥_{0}) \geq \underset{𝜀 \to 0}{lim sup} 𝜀 log 𝜇_{𝜀} (𝐴) \geq \underset{𝜀 \to 0}{lim inf} 𝜀 log 𝜇_{𝜀} (𝐴) \geq - 𝐼_{1} (𝑥_{0}),

which leads to a contradiction as

𝛿 \to 0

□

Remark 1.3.

All metric spaces are regular Hausdorff spaces.
If $𝒳︀$ is regular and locally compact, the LDP condition in Proposition 1.2 can be relaxed to a weak LDP.

Proposition 1.4 (Existence). Let $𝒜︀$ be a topological base for space $𝒳︀$ . For every $𝐴 \in 𝒜︀$ , define

{\underset{─}{ℒ︀}}_{𝐴} ≔ \underset{𝜀 \to 0}{lim inf} 𝜀 log 𝜇_{𝜀} (𝐴), {\overset{─}{ℒ︀}}_{𝐴} ≔ \underset{𝜀 \to 0}{lim sup} 𝜀 log 𝜇_{𝜀} (𝐴),

and

\underset{─}{𝐼} (𝑥) ≔ - inf_{𝐴 \in 𝒜︀ : 𝑥 \in 𝐴} {\underset{─}{ℒ︀}}_{𝐴}, \overset{─}{𝐼} (𝑥) ≔ - inf_{𝐴 \in 𝒜︀ : 𝑥 \in 𝐴} {\overset{─}{ℒ︀}}_{𝐴} .

\underset{─}{𝐼} = \overset{─}{𝐼}

, then

𝜇_{𝜀}

satisfies the weak LDP with the rate function

\underset{─}{𝐼}

Proof. That $\underset{─}{𝐼}$ and $\overset{─}{𝐼}$ are indeed rate functions follows naturally from their definitions.

To establish the lower bound of the LDP, observe that for every open set $𝐺$ and any $𝑥 \in 𝐺$ , there exists an $𝐴 \in 𝒜︀$ such that $𝑥 \in 𝐴 \subset 𝐺$ . This implies

{\underset{─}{ℒ︀}}_{𝐺} \geq {\underset{─}{ℒ︀}}_{𝐴} \geq - \underset{─}{𝐼} (𝑥),

from which the desired lower bound follows.

Regarding the upper bound, for any compact set $𝐹$ and every point $𝑥 \in 𝐹$ , we can find a neighborhood $𝐴_{𝑥}$ satisfying ${\overset{─}{ℒ︀}}_{𝐴_{𝑥}} \leq - {\overset{─}{𝐼}}^{𝛿} (𝑥)$ . Since $𝐹$ is compact, it admits a finite subcover ${𝐴_{𝑥_{𝑖}}}_{𝑖 = 1}^{𝑚}$ ; thus,

{\overset{─}{ℒ︀}}_{𝐹} \leq max_{1 \leq 𝑖 \leq 𝑚} {\overset{─}{ℒ︀}}_{𝐴_{𝑥_{𝑖}}} \leq max_{1 \leq 𝑖 \leq 𝑚} - {\overset{─}{𝐼}}^{𝛿} (𝑥_{𝑖}) \leq - inf_{𝑥 \in 𝐹} {\overset{─}{𝐼}}^{𝛿} (𝑥) .

Taking $𝛿 \to 0$ completes the proof. $□$

Remark 1.5. Proposition 1.4 holds for a parametrized family ${𝜇_{𝜀, 𝜎}}$ for any given $𝜎 \in Σ$ if we define:

{\underset{─}{ℒ︀}}_{𝐴} ≔ \underset{𝜀 \to 0}{lim inf} 𝜀 log [inf_{𝜎 \in Σ} 𝜇_{𝜀, 𝜎} (𝐴)], {\overset{─}{ℒ︀}}_{𝐴} ≔ \underset{𝜀 \to 0}{lim sup} 𝜀 log [sup_{𝜎 \in Σ} 𝜇_{𝜀, 𝜎} (𝐴)] .

This is useful for describing processes such as Markov chains conditioned on the initial state

𝜎

We also have a partial converse of Proposition 1.4.

Proposition 1.6. Suppose that

{𝜇_{𝜀}}

satisfies the LDP in a topological space

𝒳︀

with rate function

𝐼

. Then, for any topological base

𝒜︀

𝒳︀

𝐼 = \underset{─}{𝐼} = \overset{─}{𝐼}

, where these functions are defined as in Proposition 1.4.

Proof. Suppose $\underset{─}{𝐼} (𝑥) > \overset{─}{𝐼} (𝑥)$ for some $𝑥$ . By the lower semicontinuity of $𝐼$ and the regularity of $𝒳︀$ , for every $𝛿 > 0$ , there exists an open neighborhood $𝐴$ of $𝑥$ such that

- 𝐼^{𝛿} (𝑥) \geq - inf_{𝑦 \in \overset{─}{𝐴}} 𝐼 (𝑦) \geq {\overset{─}{ℒ︀}}_{𝐴} \geq - \overset{─}{𝐼} (𝑥) .

Conversely, by the definition of the LDP, for every $𝛿 > 0$ , there exists an open neighborhood $𝐴$ of $𝑥$ such that

- 𝐼 (𝑥) \leq - inf_{𝑦 \in \overset{˚}{𝐴}} 𝐼 (𝑦) \leq {\underset{─}{ℒ︀}}_{𝐴} \leq - {\underset{─}{𝐼}}^{𝛿} (𝑥) .

Combining these inequalities yields the desired result.

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Backgrounds in Large Deviation Principles

1. Existence and Uniqueness of the LDP