Pair Reduction #

The goal of this file is to prove the theorem pair_reduction. This is essentially Lemma 6.1 in [kratschmer_urusov2023] which is an extension of Lemma B.2.7. in [talagrand2014]. Given pseudometric spaces T and E, c ≥ 0, and a finite subset J of T such that |J| ≤ aⁿ for some a ≥ 0 and n : ℕ, pair_reduction states that there exists a set K ⊆ J² such that for any function f : T → E:

|K| ≤ a|J|
∀ (s, t) ∈ K, d(s, t) ≤ cn
sup_{s, t ∈ J : d(s, t) ≤ c} d(f(s), f(t)) ≤ 2 sup_{(s, t) ∈ K} d(f(s), f(t))

When applying the chaining technique for bounding the supremum of the incremements of stochastic processes, pair_reduction is used to reduce the order of the dependence of the bound on the covering numbers of the pseudometric space. As a simple example of how it could be used, suppose T has an ε-covering number N and suppose J is an ε-covering of T with |J| = N. Let f : Ω → T → E be any stochastic process such that 𝔼 d(f(s), f(t)) ≤ d (s, t) for all s, t ∈ T. Then naively

  𝔼[sup_{(s, t) ∈ J} : d(s, t) ≤ c} d(f(s), f(t))]
    ≤ ∑_{(s, t) ∈ J² : d(s, t) ≤ c} 𝔼[d(f(s), f(t))]
    ≤ |J|² c
    = c N²

but applying pair_reduction with n = log |J| we get

  𝔼[sup_{(s, t) ∈ J : d(s, t) ≤ c} d(f(s), f(t))]
    ≤ 2 𝔼[sup_{(s, t) ∈ K} d(f(s), f(t))]
    ≤ 2 ∑_{(s, t) ∈ K} 𝔼[d(f(s), f(t))]
    ≤ 2 a |J| c log |J|
    ≤ 2 a c N log N

pair_reduction is used in [kratschmer_urusov2023] to prove a form of the Kolmogorov-Chentsov theorem that applies to stochastic processses which satisfy the Kolmogorov condition but works on very general metric spaces.

Implementation #

In this section we sketch a proof of pair_reduction with references to the corresponding steps in the lean code.

For any V : Finset T and t : T we define the log-size radius of t in V to be the smallest natural number k greater than zero such that |{x ∈ V | d(t, x) ≤ kc}| ≤ aᵏ. (see logSizeRadius)

We construct a sequence Vᵢ of subsets of J, a sequence tᵢ ∈ Vᵢ and a sequence of rᵢ : ℕ inductively as follows (see logSizeBallSeq):

V₀ = J, tₒ is chosen arbitarily in J, r₀ is the log-size radius of t₀ in V₀
Vᵢ₊ᵢ = Vᵢ \ Bᵢ where Bᵢ := {x ∈ V | d(t, x) ≤ (rᵢ - 1) c}, tᵢ₊₁ is chosen arbitarily in Vᵢ₊₁ (if it is nonempty), rᵢ₊₁ is the log-size radius of tᵢ₊₁ in Vᵢ₊ᵢ.

Then Vᵢ is a strictly decreasing sequence (see card_finset_logSizeBallSeq_add_one_lt) until Vᵢ is empty. In particular Vᵢ = ∅ for i ≥ |J| (see card_finset_logSizeBallSeq_card_eq_zero).

We will show that K = ⋃_{i=1}^|J| {tᵢ} × {x ∈ Vᵢ | d(tᵢ, x) ≤ crᵢ} suffices (see pairSet and pairSetSeq).

To prove (1) we have that

  |K| ≤ ∑_{i=0}^|J| |{x ∈ Vᵢ : d(t, x) ≤ crᵢ}|
      ≤ ∑_{i=0}^|J| a ^ rᵢ  (by definition of `rᵢ`)
      = a ∑_{i=0}^|J| a ^ (rᵢ - 1)
      ≤ a ∑_{i=0}^|J| |Bᵢ| (by definition of `rᵢ`)
      ≤ a |J| (since the `Bᵢ` are disjoint (see `disjoint_smallBall_logSizeBallSeq`))

(see card_pairSet_le).

(2) follows easily from the definition of K and the fact that rᵢ ≤ n for each i (see edist_le_of_mem_pairSet and radius_logSizeBallSeq_le)

Finally we prove (3). Let s, t ∈ J such that d(s, t) ≤ c. Let i be the largest integer such that both s, t ∈ Vᵢ. WLOG suppose s ∉ Vᵢ₊₁ so that in particular s ∈ Bᵢ which means by definition that d(tᵢ, s) ≤ (rᵢ - 1)c. Then we also have

d(tᵢ, t) ≤ d(tᵢ, s) + d(s, t) ≤ (rᵢ - 1)c + c = rᵢc

hence (tᵢ, s), (tᵢ, t) ∈ K. Furthermore

d(f(s), f(t)) ≤ d(f(tᵢ), f(s)) + d(f(tᵢ), f(t))

taking supremums completes the proof (see iSup_edist_pairSet).

References #

[V. Krätschmer, M. Urusov, A Kolmogorov–Chentsov Type Theorem on General Metric Spaces with Applications to Limit Theorems for Banach-Valued Processes][kratschmer_urusov2023]
[M. Talagrand, Upper and Lower Bounds for Stochastic Processes][talagrand2014]