37.07.09 · probability / 07-large-deviations

Exponential Tightness, Exponential Approximation, and the Dawson–Gärtner Projective Limit

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Anchor (Master): Dembo & Zeitouni 1998 *Large Deviations Techniques and Applications* 2nd ed. (Springer) §4.1, §4.2.2 (Theorems 4.2.13, 4.2.16, exponential equivalence), §4.6 (Theorem 4.6.1 Dawson-Gärtner; §5.1 Mogulskii, §5.2 Schilder as projective-limit applications); Dawson & Gärtner 1987 *Large deviations from the McKean-Vlasov limit for weakly interacting diffusions* (Stochastics 20); Deuschel & Stroock 1989 *Large Deviations* (Academic Press) §3.3-§4.1

Intuition Beginner

A large deviation estimate comes in two halves that do not always fit together. One half is local: it tells you the exponential price of every rare value you can see inside any small, bounded window. The other half is global: it promises that the random quantity does not quietly slip away to infinity, dodging every window you set up. The local half is usually the easy part, because it follows from tilting and counting. The global half is the part that needs a separate guarantee, and that guarantee is exponential tightness: the promise that the chance of escaping every fixed bounded region is not just small, but exponentially small at any rate you demand.

Why insist on this? Because without it a good local picture can be a lie about the whole. Imagine a coin-flip average that almost always sits near one half, but with a tiny escaping spike of probability that races off to larger values as the experiment grows. Every fixed window sees the spike leave and reports the clean local story. Yet the spike carries real exponential mass off to infinity, so the global cost landscape is wrong. Exponential tightness forbids this: pin enough mass inside a compact region, at every rate, and the local picture becomes the true global one.

The second idea is exponential approximation. Often the quantity you care about is awkward, but it sits exponentially close to a friendlier one — close enough that the chance they ever differ by a noticeable amount decays faster than any rate. When two random quantities are that close, they must have the same cost landscape, because any difference in cost would show up as a difference in probability that the closeness forbids. So you may compute the rate function for the easy stand-in and hand the answer to the hard original. This is how rough objects — suprema of paths, integrals against jagged test functions — inherit clean rate functions from smooth approximations.

The third idea lifts everything to infinite dimensions. A random path, or a random function, is too big to handle whole, but you can look at it through finitely many readings at a time: its values at a few times, its averages over a few intervals. Each finite reading is a finite-dimensional random vector with its own cost landscape, and the readings are consistent — a coarser reading is a function of a finer one. The Dawson–Gärtner theorem says these consistent finite readings stitch together into a single cost landscape for the whole path. The cost of a path is the worst (largest) cost charged by any of its finite readings.

Put together, the three ideas are the plumbing of the whole subject. Tightness makes a local estimate global; approximation lets you swap a hard object for an easy one at no cost; and the projective limit builds a cost landscape for an infinite-dimensional object out of its finite shadows. Every sample-path large deviation theorem you will meet is assembled from exactly these three moves.

Visual Beginner

Figure: three panels. Left, exponential tightness — a number line with a central bump and a thin tail leaking right; a bracket marks a compact box capturing all but an exponentially-thin sliver. Middle, exponential approximation — two wiggly curves tracing nearly the same path, a shaded sliver between them labelled "the chance they ever separate decays faster than any rate." Right, the projective limit — a stack with an infinite-dimensional path at the top and arrows down to finite readings at the bottom, captioned "cost of the path = the largest cost charged by any finite reading."

 (1) exponential tightness        (2) exponential approximation
   prob
    |   __                          X_e -----~~~~~~~~~-----
    |  /  \                         Y_e -----~~~~~~~~~-----
    |_/    \__.--.____ tail              |  shaded sliver |
   -+----[ compact box ]----- x         | P(|X-Y|>d) ->   |
        capture all but an              | decays faster   |
        exp-thin sliver                 | than any rate   |
                                        => same cost I

 (3) Dawson-Gartner projective limit
        whole path  x  in  X = projective limit
                    |   \      \
            project |    \      \   (finitely many readings)
                    v     v      v
        reading p1(x)  p2(x)  p3(x) ...   each finite-dim, cost I_j
        ---------------------------------------------------------
        cost of the path :  I(x) = largest of  I_j( p_j(x) )

Worked example Beginner

We check exponential tightness for the simplest interesting family: the average of $n$ fair coin flips (heads $1$ , tails $0$ ), at size $n$ . We want to show all but an exponentially-thin sliver of probability stays inside a fixed bounded interval.

Step 1. Pick the box. The average $\overset{ˉ}{X}_{n}$ always lies in $[0, 1]$ , since each flip is $0$ or $1$ . So take the compact box to be $K = [0, 1]$ itself.

Step 2. Measure the outside mass. The chance that $\overset{ˉ}{X}_{n}$ lands outside $[0, 1]$ is exactly zero: there is no outside. So the "escape" probability is $0$ for every $n$ .

Step 3. Read off the exponential rate. The rate at which outside-the-box mass decays is governed by $\frac{1}{n} lo g (escape probability)$ . Here the escape probability is $0$ , and $\frac{1}{n} lo g 0 = - \infty$ , which beats every finite target rate $M$ .

Step 4. Conclude tightness. For any demanded rate $M$ , the single box $K = [0, 1]$ works: outside-mass decays at rate $- \infty \leq - M$ . The family is exponentially tight, with one box doing the job at every level.

What this tells us. When the random quantity is automatically confined to a bounded set, exponential tightness is free — the fence is already built. The idea earns its keep on unbounded spaces, like the average of unbounded variables or a random path of unbounded height, where you must actively show the escaping mass decays fast enough. There the box grows with the demanded rate $M$ , and proving the decay is the real work; the coin average is the warm-up that shows what "all but an exponentially-thin sliver stays inside" means.

Check your understanding Beginner

Exercise (easy, multiple choice).

A family of random quantities is exponentially tight when, for every demanded rate $M$ , there is a fixed bounded (compact) region such that:

A. the probability of being inside the region equals $1$ exactly B. the probability of being outside the region decays at least as fast as the rate $M$ on the exponential scale C. the region contains the typical value D. the average of the quantity stays inside the region

Hint

Tightness is a promise about the escaping mass, and the promise must hold at every demanded exponential rate.

Answer

B. the outside-the-region mass decays at least as fast as the rate $M$ .

Feedback-correct: correct — tightness controls the exponential rate of the escaping mass, and the region is allowed to grow as the demanded rate $M$ grows. Feedback-wrong: the inside probability need not be exactly $1$ , and merely containing the typical value is not enough; what matters is how fast the outside mass decays, at every rate $M$ .

Exercise (easy, multiple choice).

In the projective-limit (Dawson–Gärtner) picture, the cost of a whole infinite-dimensional path is built from the costs of its finitely-many readings as:

A. the sum of the reading costs B. the average of the reading costs C. the largest (supremum) of the reading costs D. the smallest (infimum) of the reading costs

Hint

A finer reading can only charge more cost than a coarser one; the whole-path cost must dominate every reading.

Answer

C. the largest of the reading costs.

Feedback-correct: correct — the path's cost is the supremum over all finite readings of the cost each reading charges, so refining the reading can only push the cost up toward the true whole-path cost. Feedback-wrong: it is not a sum or average; each finite reading gives a lower estimate of the true cost, and the supremum over readings recovers it.

Formal definition Intermediate+

Throughout, $X$ is a Hausdorff topological space (regular where tightness is invoked) with Borel $σ$ -algebra, and ${μ_{ε}}_{ε > 0}$ are Borel probability measures at speed $a_{ε} \to 0$ , realised as laws of random variables $X_{ε}$ on a common space 37.01.01. We use the LDP, weak LDP, good rate function, and exponential tightness of 37.07.01 and the continuous transport (contraction) of 37.07.08.

Definition (exponential tightness, recalled and sharpened). The family ${μ_{ε}}$ is exponentially tight if for every $M < \infty$ there is a compact $K_{M} \subseteq X$ with $$ \limsup_{\varepsilon\to0} a_\varepsilon\log\mu_\varepsilon\big(K_M^{,c}\big) ;\le; -M. $$ On a metric space the criterion is checked through a coercive functional or a half-space cover: it suffices to produce, for each $M$ , a totally bounded closed set whose complement has exponential mass below $- M$ in each of finitely many coordinate directions.

Definition (exponentially equivalent families). Let $X$ be metric with metric $d$ . Two families ${X_{ε}}$ , ${Y_{ε}}$ of $X$ -valued random variables on a common probability space are exponentially equivalent (at speed $a_{ε}$ ) if for every $δ > 0$ $$ \limsup_{\varepsilon\to0} a_\varepsilon\log\mathbb{P}\big(d(X_\varepsilon,Y_\varepsilon)>\delta\big) ;=; -\infty. $$

Definition (exponentially good approximation). A family of families ${X_{ε}^{(m)}}_{m \in N}$ is an exponentially good approximation of ${X_{ε}}$ if each ${X_{ε}^{(m)}}$ is a random variable on the common space and $$ \lim_{m\to\infty}\ \limsup_{\varepsilon\to0}\ a_\varepsilon\log\mathbb{P}\big(d(X_\varepsilon^{(m)},X_\varepsilon)>\delta\big) ;=; -\infty \qquad\text{for every }\delta>0. $$ Exponential equivalence is the case where the approximating family is constant in $m$ and already exponentially close.

Definition (projective system). A projective (inverse) system of Hausdorff spaces indexed by a directed set $(J, \leq)$ is a family ${Y_{j}}_{j \in J}$ with continuous projections $p_{ij} : Y_{j} \to Y_{i}$ for $i \leq j$ satisfying $p_{ii} = id$ and $p_{ik} = p_{ij} \circ p_{j k}$ for $i \leq j \leq k$ . Its projective limit $$ \mathcal{X} ;=; \varprojlim_{j}\mathcal{Y}j ;:=; \Big{(y_j){j\in J}\in\textstyle\prod_j\mathcal{Y}j : p{ij}(y_j)=y_i\ \text{for all } i\le j\Big} $$ carries the subspace topology of the product (the projective-limit topology, the coarsest making every canonical projection $p_{j} : X \to Y_{j}$ continuous), and is Hausdorff. The maps $p_{j}$ satisfy $p_{i} = p_{ij} \circ p_{j}$ .

Definition (projective-limit rate function). Given a rate function $I_{j}$ on each $Y_{j}$ with $I_{i} (p_{ij} \cdot) \leq I_{j} (\cdot)$ — a compatible (monotone) family — the projective-limit rate function on $X = lim Y_{j}$ is $$ I(x) ;:=; \sup_{j\in J} I_j\big(p_j(x)\big). $$ The supremum is over a monotone net (refining $j$ only raises $I_{j} (p_{j} x)$ ), so $I$ is lower-semicontinuous as a supremum of the lsc maps $x \mapsto I_{j} (p_{j} x)$ .

This is the topological core of the chapter: tightness upgrades the local estimate (the weak LDP of 37.07.01) to a global one; exponential approximation makes the LDP stable under replacing $X_{ε}$ by an exponentially-close surrogate; and the projective limit assembles an infinite-dimensional rate function from the finite-dimensional marginals.

Counterexamples to common slips

Exponential tightness is strictly stronger than ordinary tightness. The escaping-atom family $μ_{ε} = (1 - e^{- 1/ ε}) δ_{0} + e^{- 1/ ε} δ_{1/ ε}$ on $R$ at speed $ε$ is ordinarily tight (the escaping mass $e^{- 1/ ε} \to 0$ ), so it is even weakly precompact; but it is not exponentially tight, since no compact $K_{M}$ with $M > 1$ captures the rate- $1$ leak to $+ \infty$ . Ordinary tightness controls the mass outside compacts; exponential tightness controls its rate, and the two diverge precisely when mass escapes at a finite positive rate.
Exponential equivalence is about a coupling, not about laws. Two families with the same marginal law need not be exponentially equivalent: equivalence is a statement about $d (X_{ε}, Y_{ε})$ on a common space. Conversely, families that are not even close in law can fail to be exponentially equivalent for an obvious reason, but the subtle error is to read "same LDP" as "exponentially equivalent" — sharing a rate function is necessary, not sufficient, for the coupling bound to hold.
The projective rate is a supremum, not a limit, and refinement matters. Replacing $I (x) = sup_{j} I_{j} (p_{j} x)$ by the value of a single fixed marginal $I_{j_{0}} (p_{j_{0}} x)$ undercounts: a path can be cheap in finitely many readings yet expensive once a finer reading resolves its true shape. Only the supremum over the whole directed family recovers the true cost; truncating the index set loses lower semicontinuity of the limit and breaks the upper bound.

Key theorem with proof Intermediate+

We prove the Dawson–Gärtner projective-limit theorem, the abstract route to infinite-dimensional LDPs, using the exponential-tightness upgrade of 37.07.01 and the continuous transport of 37.07.08 as inputs.

Theorem (Dawson–Gärtner). Let $X = lim_{j \in J} Y_{j}$ be the projective limit of a system of Hausdorff spaces with canonical projections $p_{j} : X \to Y_{j}$ . Suppose that for each $j \in J$ the pushforward family ${p_j{}_\mu_\varepsilon}={\mu_\varepsilon\circ p_j^{-1}} $o n$ \mathcal{Y}j $s a t i s f i es t h e L D P a t s p ee d$ a\varepsilon $w i t h g oo d r a t e f u n c t i o n$ I_j $. T h e n$ {\mu_\varepsilon} $s a t i s f i es t h e L D P o n$ \mathcal{X} $a t s p ee d$ a_\varepsilon$ with the good rate function* $$ I(x) ;=; \sup_{j\in J} I_j\big(p_j(x)\big). $$ ^{[Dembo & Zeitouni §4.6.1]}

Proof. First, the marginal rates are compatible: for $i \leq j$ , $p_{i} = p_{ij} \circ p_{j}$ , so $p_{i}_{*} μ_{ε} = (p_{ij})_{*} (p_{j}_{*} μ_{ε})$ , and the contraction principle 37.07.08 applied to the continuous $p_{ij}$ gives $I_{i} (y) = in f {I_{j} (z) : p_{ij} (z) = y} \leq I_{j}$ pulled back, i.e. $I_{i} (p_{ij} z) \leq I_{j} (z)$ . Hence $j \mapsto I_{j} (p_{j} x)$ is monotone non-decreasing along the directed set, and $I (x) = sup_{j} I_{j} (p_{j} x)$ is a supremum of lsc functions, so $I$ is lower-semicontinuous.

Upper bound on compact sets. Let $K \subseteq X$ be compact and fix $δ > 0$ . For each $x \in K$ pick $j (x) \in J$ with $I_{j (x)} (p_{j (x)} x) > min {I (x), 1/ δ} - δ$ . The cylinder neighbourhoods ${p_{j (x)}^{- 1} (G) : G ∋ p_{j (x)} x open}$ cover $K$ ; extract a finite subcover indexed by $x_{1}, \dots, x_{n}$ and choose $j^{*} \geq j (x_{1}), \dots, j (x_{n})$ by directedness. Then $K \subseteq p_{j^{*}}^{- 1} (B)$ where $B = p_{j^{*}} (K)$ is compact (continuous image), and by the marginal upper bound on the compact $B$ , $$ \limsup_\varepsilon a_\varepsilon\log\mu_\varepsilon(K) \le \limsup_\varepsilon a_\varepsilon\log(p_{j^}{}_\mu_\varepsilon)(B) \le -\inf_{B}I_{j^} \le -\inf_{x\in K}I_{j^}(p_{j^}x). $$ Since $I_{j^}(p_{j^}x)\le I(x) $p o in tw i se w o u l d g i v e t h e w r o n g d i r ec t i o n, r e f in e : t h e f ini t es u b co v er an d t h ec h o i ceo f$ j^\ge j(x_k) $mak e$ I_{j^}(p_{j^}x)\ge I_{j(x_k)}(p_{j(x_k)}x)>\min{I(x),1/\delta}-\delta $o n e a c h co v er p i ece, so$ \inf_K I_{j^}(p_{j^}\cdot)\ge \min{\inf_K I,1/\delta}-\delta $. L e tt in g$ \delta\downarrow0 $y i e l d s$ \limsup_\varepsilon a_\varepsilon\log\mu_\varepsilon(K)\le-\inf_K I$.

Upper bound on closed sets via tightness. Each ${p_{j}_{*} μ_{ε}}$ has a good rate, hence is exponentially tight on $Y_{j}$ (a full LDP with good rate is exponentially tight, 37.07.01); pulling the tightness compacts back through the projections and intersecting gives exponential tightness of ${μ_{ε}}$ on $X = lim Y_{j}$ in the projective-limit topology, because a subset of $X$ is compact iff it is closed and contained in a product of compacts $\prod_{j} C_{j}$ with $C_{j} \subseteq Y_{j}$ (Tychonoff). The compact-set upper bound just proved, together with exponential tightness, upgrades to the closed-set upper bound by the weak-to-full mechanism of 37.07.01.

Lower bound on open sets. Basic open sets of $X$ are cylinders $p_{j}^{- 1} (G)$ with $G \subseteq Y_{j}$ open. For such a set, $$ \liminf_\varepsilon a_\varepsilon\log\mu_\varepsilon(p_j^{-1}G) = \liminf_\varepsilon a_\varepsilon\log(p_j{}*\mu\varepsilon)(G) \ge -\inf_{G}I_j \ge -\inf_{p_j^{-1}G}I, $$ the last step because $I \geq I_{j} \circ p_{j}$ . A general open $U$ contains, around each of its points $x$ , a basic cylinder on which the bound holds with $in f$ no larger than $I (x) + δ$ ; taking the supremum of these local lower bounds over the cover and $δ ↓ 0$ gives $lim inf_{ε} a_{ε} lo g μ_{ε} (U) \geq - in f_{U} I$ . Goodness of $I$ follows from the tightness step: the sublevel set $Ψ_{I} (α) = {x : I_{j} (p_{j} x) \leq α \forall j} = ⋂_{j} p_{j}^{- 1} (Ψ_{I_{j}} (α))$ is an intersection of closed sets contained in the product $\prod_{j} Ψ_{I_{j}} (α)$ of compacts, hence compact. $□$

Bridge. This theorem builds toward every sample-path large deviation result — Mogulskii's theorem for random-walk trajectories and Schilder's theorem for small-noise Brownian motion — and appears again in the Freidlin–Wentzell theory of randomly perturbed dynamical systems, where the path-space rate is assembled from finite-time-marginal rates by exactly this supremum. This is exactly the mechanism that tames infinite dimensions: a function space is the projective limit of its finite-dimensional coordinate readings, each reading carries a finite-dimensional LDP (typically a Cramér or Gärtner-Ellis estimate), and the theorem stitches them. The foundational reason the limit rate is the supremum over readings is the compatibility $I_{i} (p_{ij} \cdot) \leq I_{j}$ forced by the contraction principle 37.07.08: each finer reading charges at least as much cost, so the supremum is the only consistent assignment and generalises the finite-dimensional rate to the limit. Putting these together with exponential tightness — which is dual to the projective compactness supplied by Tychonoff — shows that tightness in the limit is automatic once each marginal is good, and the closed-set upper bound is recovered exactly as in the weak-to-full upgrade of 37.07.01; the bridge is the identification of "infinite-dimensional LDP" with "compatible family of finite-dimensional LDPs", the projective dual of the contraction principle's pushforward.

Exercises Intermediate+

Exercise 1 (easy, symbolic).

Show that if ${X_{ε}}$ satisfies the LDP with good rate $I$ on a metric space and ${Y_{ε}}$ is exponentially equivalent to ${X_{ε}}$ , then ${Y_{ε}}$ satisfies the LDP with the same rate $I$ . (Prove the upper bound on closed sets; the lower bound is symmetric.)

Hint

For closed $F$ and $δ > 0$ , ${Y_{ε} \in F} \subseteq {X_{ε} \in F^{δ}} \cup {d (X_{ε}, Y_{ε}) > δ}$ , where $F^{δ}$ is the closed $δ$ -neighbourhood of $F$ .

Answer

Fix closed $F$ and $δ > 0$ . If $Y_{ε} \in F$ then either $X_{ε}$ lies within $δ$ of $F$ (so $X_{ε} \in F^{δ}$ , the closed $δ$ -neighbourhood) or $d (X_{ε}, Y_{ε}) > δ$ . By the finite-union max rule on the exponential scale, $$ \limsup_\varepsilon a_\varepsilon\log\mathbb{P}(Y_\varepsilon\in F)\le\max{\limsup_\varepsilon a_\varepsilon\log\mathbb{P}(X_\varepsilon\in F^\delta),\ \limsup_\varepsilon a_\varepsilon\log\mathbb{P}(d(X_\varepsilon,Y_\varepsilon)>\delta)}. $$ The second term is $- \infty$ by exponential equivalence; the first is $\leq - in f_{F^{δ}} I$ by the LDP for $X_{ε}$ . Hence $lim sup_{ε} a_{ε} lo g P (Y_{ε} \in F) \leq - in f_{F^{δ}} I$ . As $δ ↓ 0$ , $F^{δ} ↓ F$ (closed $F$ ) and lower semicontinuity with goodness give $in f_{F^{δ}} I ↑ in f_{F} I$ . Thus the closed-set upper bound holds for $Y_{ε}$ with rate $I$ .

Exercise 3 (medium, symbolic).

Prove the exponentially-good-approximation theorem in its lower-bound half: if ${X_{ε}^{(m)}}$ is an exponentially good approximation of ${X_{ε}}$ , each ${X_{ε}^{(m)}}$ satisfies the LDP with good rate $I_{m}$ , and $I (x) := sup_{δ > 0} lim inf_{m} in f_{z : d (z, x) \leq δ} I_{m} (z)$ is a good rate function, then the open-set lower bound holds for ${X_{ε}}$ with rate $I$ .

Hint

For open $G$ and $x \in G$ , pick a ball $B (x, 2 δ) \subseteq G$ ; bound $P (X_{ε} \in G)$ below by $P (X_{ε}^{(m)} \in B (x, δ)) - P (d (X_{ε}^{(m)}, X_{ε}) > δ)$ and use the max-rule lower estimate.

Answer

Fix open $G$ , a point $x \in G$ , and $δ > 0$ with $B (x, 2 δ) \subseteq G$ . Whenever $X_{ε}^{(m)} \in B (x, δ)$ and $d (X_{ε}^{(m)}, X_{ε}) \leq δ$ , the triangle inequality gives $X_{ε} \in B (x, 2 δ) \subseteq G$ . Hence $$ \mathbb{P}(X_\varepsilon\in G)\ge\mathbb{P}(X_\varepsilon^{(m)}\in B(x,\delta))-\mathbb{P}(d(X_\varepsilon^{(m)},X_\varepsilon)>\delta). $$ On the exponential scale, if the first term dominates the second then $lim inf_{ε} a_{ε} lo g P (X_{ε} \in G) \geq lim inf_{ε} a_{ε} lo g P (X_{ε}^{(m)} \in B (x, δ)) \geq - in f_{B (x, δ)} I_{m} \geq - I_{m} (x^{'})$ for $x^{'}$ near $x$ ; the open lower bound for $X_{ε}^{(m)}$ supplies $\geq - in f_{B (x, δ)} I_{m}$ . The approximation hypothesis sends the discrepancy term to $- \infty$ as $m \to \infty$ then $ε \to 0$ , so the subtraction does not degrade the rate. Taking $lim inf_{m}$ , then $sup_{δ > 0}$ , gives $lim inf_{ε} a_{ε} lo g P (X_{ε} \in G) \geq - I (x)$ . Optimising over $x \in G$ yields $\geq - in f_{G} I$ , the open-set lower bound.

Exercise 4 (medium, symbolic).

Let $X = R^{N}$ with the product topology, the projective limit of the coordinate spaces $Y_{n} = R^{n}$ under the truncation projections $p_{mn} (x_{1}, \dots, x_{n}) = (x_{1}, \dots, x_{m})$ . Suppose each finite marginal ${p_{n}_{*} μ_{ε}}$ on $R^{n}$ has the LDP with good rate $I_{n}$ . Write the Dawson–Gärtner rate and show it is consistent: $I_{m} (p_{m} x) = in f {I_{n} (z) : p_{mn} z = p_{m} x}$ .

Hint

The contraction principle along the continuous truncation $p_{mn}$ gives the marginal $I_{m}$ as a pushforward of $I_{n}$ ; consistency is exactly that pushforward identity.

Answer

The Dawson–Gärtner rate is $I (x) = sup_{n} I_{n} (x_{1}, \dots, x_{n})$ , the increasing supremum over truncation depth. Consistency follows from the contraction principle 37.07.08: $p_{m} = p_{mn} \circ p_{n}$ , so $p_{m}_{*} μ_{ε} = (p_{mn})_{*} (p_{n}_{*} μ_{ε})$ , and contracting the $R^{n}$ -LDP through the continuous linear truncation $p_{mn}$ gives the LDP for the $R^{m}$ -marginal with rate $(p_{mn})_{*} I_{n} (y) = in f {I_{n} (z) : p_{mn} z = y}$ . By uniqueness of the rate function on the Hausdorff space $R^{m}$ , this equals $I_{m}$ , so $I_{m} (p_{m} x) = in f {I_{n} (z) : p_{mn} z = p_{m} x}$ . The monotonicity $I_{m} (p_{m} x) \leq I_{n} (p_{n} x)$ for $m \leq n$ is immediate (the infimum is over a set containing $p_{n} x$ ), confirming the supremum is over an increasing net.

Exercise 5 (medium, symbolic).

Establish the half-space criterion for exponential tightness on $R^{d}$ : if for each coordinate direction $e_{k}$ and each $M$ there is $R_{M, k}$ with $lim sup_{ε} a_{ε} lo g μ_{ε} ({∣ x_{k} ∣ > R_{M, k}}) \leq - M$ , then ${μ_{ε}}$ is exponentially tight.

Hint

A box $\prod_{k} [- R_{k}, R_{k}]$ is compact; its complement is a finite union of slabs ${∣ x_{k} ∣ > R_{k}}$ , and use the finite-union max rule.

Answer

Fix $M$ and set $M^{'} = M + a$ where $a$ absorbs the $lo g d$ correction below. For each $k$ choose $R_{M^{'}, k}$ from the hypothesis and let $K_{M} = \prod_{k = 1}^{d} [- R_{M^{'}, k}, R_{M^{'}, k}]$ , a compact box. Its complement is $K_{M}^{c} = ⋃_{k = 1}^{d} {∣ x_{k} ∣ > R_{M^{'}, k}}$ , a finite union of $d$ slabs. By the finite-union max rule (the principle of the largest term), $$ \limsup_\varepsilon a_\varepsilon\log\mu_\varepsilon(K_M^c)=\max_{1\le k\le d}\limsup_\varepsilon a_\varepsilon\log\mu_\varepsilon({|x_k|>R_{M',k}})\le-M'\le-M. $$ The finite union costs only a $max$ , not a sum, on the exponential scale, so finitely many directional controls assemble into a single compact-box control. Hence ${μ_{ε}}$ is exponentially tight. This is the criterion used to verify the closure hypothesis of the Gärtner-Ellis theorem 37.07.04 when $Λ$ is finite near the origin.

Exercise 6 (hard, symbolic).

Derive Mogulskii's theorem schematically: for the polygonal interpolation $Z_{ε}$ of a random walk with i.i.d. increments having cumulant generating function $Λ$ , viewed in $C_{0} ([0, 1])$ , use the projective limit over finite time-grids to produce the rate $I (f) = \int_{0}^{1} Λ^{*} (\dot{f} (t)) d t$ on absolutely continuous $f$ , $+ \infty$ otherwise. Identify the role of the projective limit and of exponential approximation.

Hint

The finite-dimensional marginals (values at grid times) obey a Cramér LDP whose rate is a sum of $Λ^{*}$ over increments; the supremum over grids gives the action integral; the polygonal-vs-uniform gap is exponentially negligible.

Answer

Take the grids $π = {0 = t_{0} < t_{1} < \dots < t_{n} = 1}$ directed by refinement. The marginal of $Z_{ε}$ at grid $π$ , the vector $(Z_{ε} (t_{1}), \dots, Z_{ε} (t_{n}))$ , is a sum of independent scaled increments, so by Cramér's theorem its LDP rate is $I_{π} (y) = \sum_{i = 1}^{n} (t_{i} - t_{i - 1}) Λ^{*} (\frac{y _{i} - y _{i - 1}}{t _{i} - t _{i - 1}})$ , the increment form of the action on the piecewise-linear interpolant through $y$ . These marginal rates are compatible under coarsening (Jensen's inequality makes a coarser grid charge no more, since $Λ^{*}$ is convex), so Dawson–Gärtner gives the projective rate $$ I(f)=\sup_\pi I_\pi(p_\pi f)=\sup_\pi\sum_i(t_i-t_{i-1})\Lambda^!\Big(\tfrac{f(t_i)-f(t_{i-1})}{t_i-t_{i-1}}\Big)=\int_0^1\Lambda^(\dot f(t)),dt $$ for absolutely continuous $f$ (the supremum of Riemann-type sums of the convex integrand converges up to the action integral, and is $+ \infty$ for non-absolutely-continuous $f$ ). The projective limit assembles the path rate from finite-grid Cramér rates; exponential approximation handles the topology, showing the polygonal interpolation $Z_{ε}$ and the genuinely-sampled process are exponentially equivalent in the uniform topology, so the grid-based rate is the rate on $C_{0} ([0, 1])$ . Schilder's theorem is the Gaussian special case $Λ^{*} (v) = v^{2} /2$ , giving $I (f) = \frac{1}{2} \int_{0}^{1} ∣ \dot{f} ∣^{2}$ .

Exercise 7 (hard, symbolic).

Show that a full LDP with good rate function $I$ on a metric space implies exponential tightness is equivalent to goodness in the following precise sense: if ${μ_{ε}}$ satisfies the LDP with rate $I$ , then $I$ is good if and only if ${μ_{ε}}$ is exponentially tight — under the additional hypothesis that the space is such that the upper bound holds on all closed sets. Prove the direction "exponentially tight $\Rightarrow$ $I$ good", and give the standard counterexample showing the converse can fail without metric/topological regularity.

Hint

The forward direction is the goodness clause of the weak-to-full upgrade 37.07.01. For the converse failure, recall that on a non-locally-compact or irregular space a good rate function need not produce exponentially tight measures.

Answer

Exponentially tight $\Rightarrow$ $I$ good. Fix $α$ and choose $M > α$ with compact $K_{M}$ . For $x \in / K_{M}$ , regularity gives an open $G ∋ x$ with $\overline{G} \subseteq K_{M}^{c}$ , and the closed-set upper bound on $\overline{G}$ yields $- in f_{\overline{G}} I \leq lim sup_{ε} a_{ε} lo g μ_{ε} (K_{M}^{c}) \leq - M < - α$ , so $I (x) \geq in f_{\overline{G}} I > α$ . Contraposing, $Ψ_{I} (α) \subseteq K_{M}$ ; being closed (lsc) inside a compact, it is compact, so $I$ is good. This is exactly the goodness clause of the upgrade theorem of 37.07.01. Converse failure. On a non-locally-compact space such as an infinite-dimensional Banach space with the norm topology, a good rate function $I$ (compact sublevel sets) need not force exponential tightness of the measures: the measures can place mass on the non-compact unit sphere in a way that escapes every compact at a bounded rate, so goodness of $I$ alone does not capture the global mass. Exponential tightness is an extra hypothesis there, not a consequence; it must be verified separately (e.g. via a coercive functional), which is why the Dawson–Gärtner route, building tightness from compact marginals and Tychonoff, is so useful in infinite dimensions.

Exercise 8 (hard, symbolic).

Prove that exponential tightness on a metric space implies the partial LDP / subsequence-extraction property: every subsequence $ε_{n} \to 0$ has a further subsequence along which ${μ_{ε_{n}}}$ satisfies a full LDP with some good rate function. (This is the Pukhalskii compactness theorem; it is the engine behind the inverse contraction principle 37.07.08.)

Hint

Define candidate rates via $I (x) = sup_{G ∋ x} - lim inf a_{ε} lo g μ_{ε} (G)$ along a diagonal subsequence making all relevant limits exist; use a countable base and tightness to get goodness and the upper bound.

Answer

Work on a separable metric space with a countable base ${G_{k}}$ of open sets. Along any subsequence, a diagonal extraction produces a further subsequence (still denoted $ε$ ) for which $L (G_{k}) := lim_{ε} a_{ε} lo g μ_{ε} (G_{k}) \in [- \infty, 0]$ exists for every $k$ . Define $$ I(x):=-\inf{L(G_k)\in G_k}=\sup{-L(G_k)\in G_k}. $$ Then $I$ is lsc (a supremum of functions constant on basic opens, each lsc), and the open-set lower bound holds by construction: for open $G$ and $x \in G$ pick $G_{k}$ with $x \in G_{k} \subseteq G$ , so $lim inf_{ε} a_{ε} lo g μ_{ε} (G) \geq L (G_{k}) \geq - I (x)$ , optimise over $x$ . The compact upper bound follows by covering a compact $K$ by finitely many basic opens with controlled $L$ -values and the finite-union max rule, giving the weak LDP. Exponential tightness then upgrades the weak LDP to a full LDP and forces $I$ good, by the weak-to-full mechanism of 37.07.01. Thus every subsequence has a further subsequence with a full good-rate LDP. This extraction is precisely what the inverse contraction principle 37.07.08 invokes: tightness manufactures subsequential good-rate limits, and a known downstairs LDP pins them all to one rate.

Advanced results Master

Exponential tightness as the closure of every weak LDP

The structural place of exponential tightness is fixed by a single equivalence: on a regular Hausdorff space, a family satisfying a weak LDP with rate $I$ satisfies the full LDP with the same $I$ , and $I$ is good, if and only if it is exponentially tight ^{[Dembo & Zeitouni §4.1.10]}. The weak LDP is the local, tilting-driven content that the Chernoff bound and the change-of-measure lower bound naturally produce; exponential tightness is the orthogonal, compactness content. The Gärtner-Ellis theorem 37.07.04 is the paradigm: differentiability and steepness of the limiting cumulant generating function $Λ$ deliver the weak LDP with rate $Λ^{*}$ , and finiteness of $Λ$ in a neighbourhood of the origin delivers exponential tightness through the half-space criterion of Exercise 5; the two combine to the full LDP. Verifying tightness through a coercive functional $U$ with compact sublevel sets and $lim sup_{ε} a_{ε} lo g \int e^{U / a_{ε}} d μ_{ε} < \infty$ is the universal recipe — Markov's inequality on $e^{U / a_{ε}}$ bounds $μ_{ε} ({U > c}^{c}^{c})$ at the demanded rate.

Exponential equivalence and the metric of the LDP

Exponential equivalence is a genuine equivalence relation on families of $X$ -valued random variables, and the rate function is an invariant of its classes ^{[Dembo & Zeitouni §4.2.2]}. The slogan is that the LDP sees random variables only up to exponentially-negligible perturbation: replacing $X_{ε}$ by $X_{ε} + o_{e x p} (1)$ — a perturbation whose size exceeds any $δ$ only with super-exponentially small probability — changes no rate. This is the large-deviation analogue of the fact that weak convergence sees laws only up to convergence in probability of a coupling. The exponentially-good-approximation theorem extends this to a sequence of stand-ins $X_{ε}^{(m)}$ that need only be close in the iterated limit $m \to \infty$ after $ε \to 0$ , together with a $Γ$ -type convergence of the approximating rates $I_{m}$ to a limit $I$ ; the conclusion is the LDP for $X_{ε}$ with rate $I$ . This is the device that transports an LDP through functionals that are merely uniform-on-compact-sublevel-sets limits of continuous functionals, the same machinery underlying the approximate contraction principle 37.07.08 and Varadhan's lemma 37.07.07.

The Dawson–Gärtner theorem and the inverse-limit philosophy

The Dawson–Gärtner theorem ^{[Dawson & Gärtner 1987]} is the categorical statement that the LDP is a limit-preserving functor on the appropriate diagram: a compatible family of LDPs on the objects $Y_{j}$ of a projective system induces an LDP on the limit object $lim Y_{j}$ , with the limit rate the supremum $I = sup_{j} I_{j} \circ p_{j}$ . The compatibility $I_{i} (p_{ij} \cdot) \leq I_{j}$ is forced by the contraction principle 37.07.08 applied to the projection maps, so the input is automatically a monotone net and the supremum is the unique consistent rate. The theorem's power is that it converts the infinite-dimensional problem into a family of finite-dimensional ones plus a topological gluing: the function space is the projective limit of its finite-dimensional coordinate readings, each reading carries a Cramér or Gärtner-Ellis LDP, and the theorem produces the path-space rate without any direct infinite-dimensional estimate. Exponential tightness in the limit is supplied for free by Tychonoff: a product of the compact marginal sublevel sets is compact, and the limit sublevel set is a closed subset of such a product.

Mogulskii, Schilder, and the action integral

The two canonical applications are Mogulskii's theorem ^{[Mogulskii 1976]} and Schilder's theorem ^{[Schilder 1966]}. For a random walk with increment cumulant generating function $Λ$ , the polygonal sample paths $Z_{ε}$ obey, on $C_{0} ([0, 1])$ , the LDP with rate $$ I(f)=\int_0^1\Lambda^(\dot f(t)),dt $$ on absolutely continuous $f$ with $f (0) = 0$ , and $+ \infty$ otherwise — the time-integral of the Cramér rate of the velocity. Schilder's theorem is the Gaussian instance $\Lambda^(v)=\tfrac12 v^2 $, g i v in g t h e k in e t i c - e n er g y a c t i o n$ I(f)=\tfrac12\int_0^1|\dot f|^2 $f or s ma l l - n o i se B r o w nianm o t i o n$ \sqrt\varepsilon,W $. B o t ha r eo b t ain e d b y t h e p r o j ec t i v er o u t e : t h e f ini t e - t im e ma r g ina l s a r es u m so f in d e p e n d e n t in cr e m e n t s (C r am \overset{e}{ˊ} r), t h e ma r g ina l r a t es a r e t h e d i scr e t e a c t i o n s, t h e i r s u p r e m u m o v er r e f inin g g r i d s i s t h eco n t in u o u s a c t i o nb y co n v e x i t y o f$ \Lambda^*$ and the Jensen monotonicity that makes coarsening cheap, and exponential approximation closes the polygonal-to-uniform gap. The same action integral reappears as the Onsager-Machlup functional in the path-integral treatment of fluctuations and as the cost in Freidlin-Wentzell theory.

Projective limits in measure space: the McKean–Vlasov origin

Dawson and Gärtner introduced the theorem to handle large deviations for the empirical measure of weakly interacting diffusions, where the state space is the space of probability measures on path space — doubly infinite-dimensional ^{[Dawson & Gärtner 1987]}. There the projective system runs over finite collections of bounded continuous test functions, each reading being the finite-dimensional vector of integrals $⟨ μ, ϕ_{1} ⟩, \dots, ⟨ μ, ϕ_{k} ⟩$ , and the limit rate is a relative-entropy-type functional on measure space. This is the genealogy of modern mean-field large deviations: the abstract projective theorem is what makes the empirical-measure rate (a Donsker-Varadhan-type entropy) accessible from finite test-function marginals, exactly as Sanov's theorem 37.07.05 is the i.i.d. prototype.

Synthesis. The central insight is that the three operations of this unit are one toolkit for moving an LDP across the boundary between finite and infinite dimensions: tightness makes a local estimate global, approximation makes the LDP a property of exponential-equivalence classes rather than of individual variables, and the projective limit assembles an infinite-dimensional rate from finite marginals. This is exactly why every sample-path theorem factors the same way — Mogulskii and Schilder are the Cramér finite-marginal LDP putting these together with the Dawson–Gärtner supremum and an exponential-approximation closure of the topology. The foundational reason the limit rate is the supremum $sup_{j} I_{j} \circ p_{j}$ is the contraction-forced compatibility $I_{i} (p_{ij} \cdot) \leq I_{j}$ , so the projective construction is dual to the contraction principle's pushforward 37.07.08: contraction lowers an LDP through a single continuous map by fibrewise infimum, while the projective limit raises a compatible family through all the projections by supremum. The central insight sharpens to a slogan: exponential tightness is exactly the orthogonal complement of the weak LDP, the global compactness datum that the local tilting datum lacks, and it generalises the Prokhorov tightness of weak-convergence theory to the exponential scale. The bridge is the recognition that "infinite-dimensional LDP" means "compatible family of finite-dimensional LDPs plus tightness", a statement that appears again in Freidlin-Wentzell theory, mean-field/McKean-Vlasov large deviations, and the hydrodynamic-limit large deviations that all sit downstream of this plumbing.

Full proof set Master

Proposition 1 (exponential equivalence preserves the LDP). Let $X$ be metric, ${X_{ε}}$ satisfy the LDP with good rate $I$ , and ${Y_{ε}}$ be exponentially equivalent to ${X_{ε}}$ . Then ${Y_{ε}}$ satisfies the LDP with the same rate $I$ .

Proof. For the upper bound, let $F$ be closed and $δ > 0$ ; with $F^{δ}$ the closed $δ$ -neighbourhood, ${Y_{ε} \in F} \subseteq {X_{ε} \in F^{δ}} \cup {d (X_{ε}, Y_{ε}) > δ}$ . The max rule gives $lim sup_{ε} a_{ε} lo g P (Y_{ε} \in F) \leq max {- in f_{F^{δ}} I, - \infty} = - in f_{F^{δ}} I$ . As $δ ↓ 0$ , $in f_{F^{δ}} I ↑ in f_{F} I$ by lower semicontinuity and goodness (the sublevel sets are compact, so the infimum over shrinking closed neighbourhoods of a closed set converges to the infimum over the set). For the lower bound, let $G$ be open and $x \in G$ with $B (x, 2 δ) \subseteq G$ ; then ${X_{ε} \in B (x, δ)} \subseteq {Y_{ε} \in G} \cup {d (X_{ε}, Y_{ε}) > δ}$ , so $P (Y_{ε} \in G) \geq P (X_{ε} \in B (x, δ)) - P (d > δ)$ , and the exponential negligibility of the second term gives $lim inf_{ε} a_{ε} lo g P (Y_{ε} \in G) \geq lim inf_{ε} a_{ε} lo g P (X_{ε} \in B (x, δ)) \geq - in f_{B (x, δ)} I \geq - I (x)$ in the limit $δ ↓ 0$ using lower semicontinuity. Optimising over $x \in G$ gives the open lower bound. $□$

Proposition 2 (the projective-limit rate is a good rate function). Let ${I_{j}}$ be good rate functions on the spaces $Y_{j}$ of a projective system, compatible in the sense $I_{i} (p_{ij} y) \leq I_{j} (y)$ . Then $I (x) = sup_{j} I_{j} (p_{j} x)$ is a good rate function on $X = lim Y_{j}$ .

Proof. Each $x \mapsto I_{j} (p_{j} x)$ is lsc ( $I_{j}$ lsc, $p_{j}$ continuous), so the supremum $I$ is lsc. For sublevel sets, $$ \Psi_I(\alpha)={x:\sup_j I_j(p_j x)\le\alpha}=\bigcap_j{x(p_j x)\le\alpha}=\bigcap_j p_j^{-1}\big(\Psi_{I_j}(\alpha)\big). $$ Each $p_{j}^{- 1} (Ψ_{I_{j}} (α))$ is closed (continuity, $Ψ_{I_{j}} (α)$ closed), so $Ψ_{I} (α)$ is closed. Moreover $Ψ_{I} (α) \subseteq \prod_{j} Ψ_{I_{j}} (α) \cap X$ , a closed subset of the product of the compacts $Ψ_{I_{j}} (α)$ , which is compact by Tychonoff; a closed subset of a compact set is compact, so $Ψ_{I} (α)$ is compact. Properness: compatibility makes $j \mapsto I_{j} (p_{j} x)$ monotone, and since some marginal rate is finite somewhere (each $I_{j} \neq \equiv \infty$ and the system is consistent), $I \neq \equiv + \infty$ . Hence $I$ is good. $□$

Proposition 3 (compact upper bound in the projective limit). Under the hypotheses of the Dawson–Gärtner theorem, $lim sup_{ε} a_{ε} lo g μ_{ε} (K) \leq - in f_{K} I$ for every compact $K \subseteq X$ .

Proof. Fix $δ > 0$ . For each $x \in K$ , choose $j (x)$ with $I_{j (x)} (p_{j (x)} x) > min {I (x), 1/ δ} - δ$ , possible since $I (x) = sup_{j} I_{j} (p_{j} x)$ . By lower semicontinuity of $I_{j (x)} \circ p_{j (x)}$ there is an open cylinder $U_{x} = p_{j (x)}^{- 1} (G_{x}) ∋ x$ on which $I_{j (x)} \circ p_{j (x)} > min {I (x), 1/ δ} - 2 δ$ . Compactness of $K$ extracts a finite subcover $U_{x_{1}}, \dots, U_{x_{n}}$ ; pick $j^{*} \geq j (x_{1}), \dots, j (x_{n})$ . The image $B := p_{j^{*}} (K)$ is compact, and the marginal compact upper bound gives $lim sup_{ε} a_{ε} lo g μ_{ε} (K) \leq lim sup_{ε} a_{ε} lo g (p_{j^{*}}_{*} μ_{ε}) (B) \leq - in f_{B} I_{j^{*}} = - in f_{x \in K} I_{j^{*}} (p_{j^{*}} x)$ . On $U_{x_{k}}$ , monotonicity $I_{j^{*}} \circ p_{j^{*}} \geq I_{j (x_{k})} \circ p_{j (x_{k})}$ gives $I_{j^{*}} (p_{j^{*}} x) > min {I (x), 1/ δ} - 2 δ$ , hence $in f_{K} I_{j^{*}} \circ p_{j^{*}} \geq min {in f_{K} I, 1/ δ} - 2 δ$ . Therefore $lim sup_{ε} a_{ε} lo g μ_{ε} (K) \leq - min {in f_{K} I, 1/ δ} + 2 δ$ , and $δ ↓ 0$ yields $\leq - in f_{K} I$ . $□$

Proposition 4 (Schilder rate from the projective construction). Let $μ_{ε} = Law (ε W)$ on $C_{0} ([0, 1])$ . The finite-time marginals satisfy the Gaussian LDP with rate $I_{π} (y) = \frac{1}{2} \sum_{i} (y_{i} - y_{i - 1})^{2} / (t_{i} - t_{i - 1})$ , and $sup_{π} I_{π} (p_{π} f) = \frac{1}{2} \int_{0}^{1} ∣ \dot{f} ∣^{2} d t$ for absolutely continuous $f$ , $+ \infty$ otherwise.

Proof. The increments $W (t_{i}) - W (t_{i - 1})$ are independent $N (0, t_{i} - t_{i - 1})$ , so $ε W$ has marginal increments $N (0, ε (t_{i} - t_{i - 1}))$ ; the Cramér rate of a centred Gaussian of variance $σ^{2}$ at value $v$ is $v^{2} / (2 σ^{2})$ , and independence adds rates, giving $I_{π} (y) = \frac{1}{2} \sum_{i} (y_{i} - y_{i - 1})^{2} / (t_{i} - t_{i - 1})$ . For the supremum, fix absolutely continuous $f$ with $f = \int_{0}^{\cdot} g$ , $g \in L^{2}$ . By Cauchy-Schwarz on each subinterval, $(f (t_{i}) - f (t_{i - 1}))^{2} = (\int_{t_{i - 1}}^{t_{i}} g)^{2} \leq (t_{i} - t_{i - 1}) \int_{t_{i - 1}}^{t_{i}} g^{2}$ , so $I_{π} (p_{π} f) \leq \frac{1}{2} \int_{0}^{1} g^{2} = \frac{1}{2} \int_{0}^{1} ∣ \dot{f} ∣^{2}$ , giving the supremum $\leq \frac{1}{2} \int ∣ \dot{f} ∣^{2}$ . Conversely, the piecewise-constant approximations $g_{π} = \sum_{i} \frac{f ( t _{i} ) - f ( t _{i - 1} )}{t _{i} - t _{i - 1}} 1_{(t_{i - 1}, t_{i}]}$ are the $L^{2}$ -projections of $g$ onto grid-step functions, and $\frac{1}{2} ∥ g_{π} ∥_{2}^{2} = I_{π} (p_{π} f) ↑ \frac{1}{2} ∥ g ∥_{2}^{2}$ as the grid refines (martingale/projection convergence in $L^{2}$ ). Hence the supremum equals $\frac{1}{2} \int_{0}^{1} ∣ \dot{f} ∣^{2}$ . If $f$ is not absolutely continuous the increment sums are unbounded along refining grids, so the supremum is $+ \infty$ . $□$

Connections Master

The weak-to-full upgrade and the goodness of the rate function are the abstract machinery of 37.07.01: exponential tightness is defined there as the bridge from a weak LDP to a full LDP, and this unit organises the criteria (coercive functional, half-space cover, Tychonoff in the projective limit) by which that bridge is verified in practice, so the closure clause of 37.07.01 is the load-bearing input here.
The Dawson–Gärtner projective rate $I = sup_{j} I_{j} \circ p_{j}$ is the categorical dual of the contraction principle 37.07.08: contraction pushes an LDP down a single continuous map with the rate transforming by fibrewise infimum, while the projective limit lifts a compatible family up all the projections with the rate the supremum, and the contraction principle is exactly what forces the compatibility $I_{i} (p_{ij} \cdot) \leq I_{j}$ that makes the supremum the unique consistent assignment.
The exponential-tightness criteria assembled here are the closure step of the Gärtner-Ellis theorem 37.07.04: that theorem produces a weak LDP from the steepness of the limiting cumulant generating function $Λ$ and finishes it to a full LDP exactly by the half-space tightness check (finiteness of $Λ$ near the origin), so the tightness plumbing of this unit is what makes Gärtner-Ellis deliver a full principle.
The exponentially-good-approximation theorem is the device behind Varadhan's integral lemma and the Laplace principle 37.07.07, which transport an LDP through a functional that is only a uniform limit of continuous functionals; exponential equivalence is the same relation that lets the approximate contraction principle of 37.07.08 handle limits of continuous maps, and the path-space rate of Mogulskii and Schilder is assembled from the empirical-measure and increment LDPs of 37.07.05 through this approximation step.

Historical & philosophical context Master

Exponential tightness was isolated as the right closure condition for large deviations in the work systematised by Deuschel and Stroock ^{[Deuschel & Stroock §3.3]} and by Dembo and Zeitouni ^{[Dembo & Zeitouni §4.1.10]}, reflecting the recognition — visible already in the Prokhorov theory of weak convergence — that the local content of a limit estimate and its global compactness content are logically separable, the latter supplied by tightness. The exponential scale demands more than ordinary tightness: it is the rate of the escaping mass, not merely its size, that must be controlled, and the distinction is sharp on non-locally-compact spaces where a good rate function alone does not guarantee it.

The projective-limit theorem is due to Donald Dawson and Jürgen Gärtner in 1987 ^{[Dawson & Gärtner 1987]}, who proved it to obtain large deviations for the empirical measure of weakly interacting diffusions — the McKean-Vlasov limit — where the state space is the space of probability measures on path space and no direct infinite-dimensional estimate is available. Their theorem reduced the problem to the finite-dimensional marginals indexed by finite collections of test functions, gluing them by the supremum formula. The sample-path large deviation theorems it subsumes are older: Mogulskii's trajectory LDP for multidimensional random walks dates to 1976 ^{[Mogulskii 1976]}, and Schilder's small-noise Brownian asymptotics to 1966 ^{[Schilder 1966]}, the same year Varadhan gave the abstract LDP formulation. The exponential-approximation machinery, codified by Dembo and Zeitouni ^{[Dembo & Zeitouni §4.2.2]}, made precise the long-used practice of replacing an awkward functional by an exponentially-close smooth surrogate, placing the action-integral rate functions of Mogulskii, Schilder, and Freidlin-Wentzell on a uniform footing.

Bibliography Master

@book{dembozeitouni1998ldp,
  author    = {Dembo, Amir and Zeitouni, Ofer},
  title     = {Large Deviations Techniques and Applications},
  edition   = {2nd},
  series    = {Applications of Mathematics},
  number    = {38},
  publisher = {Springer},
  year      = {1998}
}

@article{dawsongartner1987mckean,
  author  = {Dawson, Donald A. and G\"artner, J\"urgen},
  title   = {Large deviations from the {M}c{K}ean-{V}lasov limit for weakly interacting diffusions},
  journal = {Stochastics},
  volume  = {20},
  pages   = {247--308},
  year    = {1987}
}

@article{mogulskii1976large,
  author  = {Mogulskii, A. A.},
  title   = {Large deviations for trajectories of multidimensional random walks},
  journal = {Theory of Probability and its Applications},
  volume  = {21},
  pages   = {300--315},
  year    = {1976}
}

@article{schilder1966asymptotic,
  author  = {Schilder, M.},
  title   = {Some asymptotic formulas for {W}iener integrals},
  journal = {Transactions of the American Mathematical Society},
  volume  = {125},
  pages   = {63--85},
  year    = {1966}
}

@book{deuschelstroock1989large,
  author    = {Deuschel, Jean-Dominique and Stroock, Daniel W.},
  title     = {Large Deviations},
  series    = {Pure and Applied Mathematics},
  number    = {137},
  publisher = {Academic Press},
  year      = {1989}
}

@book{denhollander2000large,
  author    = {den Hollander, Frank},
  title     = {Large Deviations},
  series    = {Fields Institute Monographs},
  number    = {14},
  publisher = {American Mathematical Society},
  year      = {2000}
}

@book{bogachev2007measure,
  author    = {Bogachev, Vladimir I.},
  title     = {Measure Theory, Volume 2},
  publisher = {Springer},
  year      = {2007}
}

Prerequisites

37.07.01
37.07.08

Tier anchors

beginner: Touchette 2009 *The large deviation approach to statistical mechanics* (Physics Reports 478) §3-§4 (tightening rare-event estimates and lifting to processes); den Hollander 2000 *Large Deviations* (AMS Fields Institute Monographs) §III (combining and lifting LDPs, informal projective picture)
intermediate: Dembo & Zeitouni 1998 *Large Deviations Techniques and Applications* 2nd ed. (Springer) §4.1.10-§4.1.11 (exponential tightness, the weak-to-full upgrade), §4.2.2 (Theorem 4.2.13 exponentially good approximations / exponential equivalence), §4.6.1 (Theorem 4.6.1 Dawson-Gärtner); den Hollander 2000 §III.5-§III.7
master: Dembo & Zeitouni 1998 *Large Deviations Techniques and Applications* 2nd ed. (Springer) §4.1, §4.2.2 (Theorems 4.2.13, 4.2.16, exponential equivalence), §4.6 (Theorem 4.6.1 Dawson-Gärtner; §5.1 Mogulskii, §5.2 Schilder as projective-limit applications); Dawson & Gärtner 1987 *Large deviations from the McKean-Vlasov limit for weakly interacting diffusions* (Stochastics 20); Deuschel & Stroock 1989 *Large Deviations* (Academic Press) §3.3-§4.1

References

Dembo, A. & Zeitouni, O. — Large Deviations Techniques and Applications, 2nd ed. (Springer, 1998) · §4.1.10-§4.1.11 (Lemma 4.1.23 exponential tightness; weak-to-full upgrade); §4.2.2 (Theorem 4.2.13 exponentially good approximations; Theorem 4.2.16 exponential equivalence); §4.6.1 (Theorem 4.6.1 Dawson-Gärtner)
Dawson, D. A. & Gärtner, J. — Large deviations from the McKean-Vlasov limit for weakly interacting diffusions · Stochastics 20 (1987), 247-308 (the projective-limit theorem, applied to measure-valued diffusions)
den Hollander, F. — Large Deviations (AMS Fields Institute Monographs 14, 2000) · §III.5-§III.7 (combining, lifting, and approximating LDPs)
Deuschel, J.-D. & Stroock, D. W. — Large Deviations (Academic Press, 1989) · §3.3-§4.1 (exponential tightness, projective limits, Mogulskii and Schilder)
Mogulskii, A. A. — Large deviations for trajectories of multidimensional random walks · Theory of Probability and its Applications 21 (1976), 300-315 (the sample-path LDP obtained by the projective route)
Schilder, M. — Some asymptotic formulas for Wiener integrals · Transactions of the AMS 125 (1966), 63-85 (small-noise Brownian LDP, the canonical projective-limit application)
Bogachev, V. I. — Measure Theory, Vol. 2 (Springer, 2007) · §7.7 (projective limits of measures and the Prokhorov projective-limit theorem, the weak-convergence template for Dawson-Gärtner)

Estimated time

beginner: 18m
intermediate: 44m
master: 78m