Yu-Liang Wu

From finite to infinite dimension

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ProbabilityLarge Deviations
From finite to infinite dimension

1. The Large Deviation Principle in Infinite-Dimensional Spaces

When establishing the Large Deviation Principle (LDP) for random vectors in infinite-dimensional spaces, a common strategy is to first establish the LDP for projections onto finite-dimensional subspaces and then lift these results to the original space. This approach is codified in the following theorem.

Theorem 1.1. Suppose that satisfies a weak LDP with a convex rate function , and let be a locally convex Hausdorff topological vector space. Assume that for each , the limits exist as extended real numbers, and that is a lower semicontinuous function of . If for every and every ,

then , and consequently, controls a weak LDP associated with .

The ingredients for the proof are summarized in the following subsections.

1.1. Varadhan’s integral lemma

In this subsection we assume to be a regular topological space.

Theorem 1.2 (Varadhan). Suppose that satisfies the LDP with a good rate function , and let be any continuous function. Assume further that either the tail condition

𝟙(1.1)

or the following moment condition for some holds:

(1.2)

Then,

The theorem is a consequence of the following three lemmas.

Lemma 1.3. If is lower semicontinuous and the large deviations lower bound holds with , then

Lemma 1.4. If is an upper semicontinuous function for which the tail condition (1.1) holds, and the large deviations upper bound holds with the good rate function , then

Lemma 1.5. Condition (1.2) implies the tail condition (1.1).

Proof of Lemma 1.3. Fix . Since is lower semicontinuous, for every , there exists an open neighborhood of such that

Using Markov’s inequality and the lower bound of the LDP, we have

which proves the inequality since and are arbitrary.⁠ 

Proof of Lemma 1.4. Note that by the semicontinuity of and , and the regularity of , for each and each , there exists an open neighborhood such that

which yields the local estimate . For each , let be a finite cover of the compact set . Then,

By letting and then , we obtain

The proof is completed by using assumption (1.1), which implies

⁠ 

Proof of Lemma 1.5. The lemma follows from the observation that for any non-negative random variable and any ,

Hence, , which proves the lemma upon setting and .⁠ 

1.2. Fenchel-Legendre transform

Theorem 1.6. Let be a locally convex Hausdorff topological vector space. Assume that satisfies the LDP with a good rate function . Suppose in addition that

(1.3)

  1. For each , exists, is finite, and

  2. If is convex, then it is the Fenchel–Legendre transform of , namely,

  3. If is not convex, then is the affine regularization of , i.e.,

Proof. (1) Fix and . By assumption, Varadhan’s lemma (Theorem 1.2) asserts that

Since is convex and , it follows that .

(2) This follows from the duality lemma (Lemma D.1).

(3) From part (1), we have

By the Hahn-Banach separation theorem,

If , it is necessarily the case that , or equivalently,

from which the claim follows.⁠ 

1.3. Proof of the theorem

Proof of Theorem 1.1. By assumption,

Note that is convex with and is lower semicontinuous; therefore, it cannot attain the value . By the duality lemma (Lemma D.1), , proving

By Lemma 1.3, we have

Combining the above results gives . Applying the duality lemma (Lemma D.1) again yields .⁠ 

References