The maximum functional

3.1. Random walk convergence 64 0 2 4 6 8 10 −4 −3 −2 −1 0 1 2

Figure 3.1: An example of a possible one-dimensional random walk plotted with time on the horizontal axis, with the two continuous-time trajectories we can create from it; the continuous interpolating Xn(t) in red and the piecewise constant process Xn0(t) in

black.

or we can consider the trajectory where the walk ‘stays still’ and makes small jumps when it reaches the next time indexing a new partial sum, in which case we have

X_n0(t) := 1

nSbntc. (3.1.2)

In order to study convergence of these trajectories, we need to specify the metric spaces in which they live. Conveniently, we have defined three such spaces in Section 1.3, the continuous trajectories endowed with the supremum norm (Cd, ρ∞) and the Skrorokhod

metric or Kolmogorov-Billingsley metric on Dd_{, (D}d_{, ρ}

S) or (Dd, ρ◦S). The first space

will be used to show the convergence of Xn(t) which is itself a continuous function,

and the latter two will be used when discussing X_n0(t). It would not be unreasonable to question why we have two metrics for X_n0(t), but the following result should make this a bit clearer.

Proposition 3.1.1 ([Kol56, Theorem 7]). The metrics ρ◦_S and ρS are equivalent. That

is, for a sequence of functions f, f1, f2, . . . on Dd, ρS(fn, f ) → 0 as n → ∞ if and only

3.1. Random walk convergence 65

The fact that the metrics are equivalent means we can use either one to prove continuity of a functional on Dd_{and the result will hold for the other; we will use the metric which}

is simplest for each application. Likewise, almost-sure statements using one metric carry over to the other. Note also that as equivalent metrics, ρS and ρ◦S generate

the same topology (open sets) on Dd_{, and hence also the same Borel sets. Further}

motivation behind ρ◦_S in particular is that, under this metric, Dd _{is both separable and}

complete, which is useful for the proofs of Donsker’s theorem below. Further details on this subject are left to the appendix.

We now can state the important results regarding the almost sure convergence of our trajectories.

Theorem 3.1.2 (Functional law of large numbers [Whi02, p. 26]). Consider the ran-

dom walk trajectories as defined at (3.1.1) and (3.1.2). Let Iµ ∈ Cd be the function

defined by Iµ(t) := µt for t ∈ [0, 1].

(a) We have Xn a.s.

−→ Iµ on (C0d, ρ∞).

(b) We have X_n0 −→ Ia.s. µ on (Dd0, ρ∞).

Remark 3.1.3. By Lemma 1.3.2, part (b) also shows that X_n0 −→ Ia.s. µ on (Dd0, ρS) and

Proposition 3.1.1 in turn shows that X_n0 −→ Ia.s. µ on (D0d, ρ ◦ S).

Alongside the convergence of the trajectories, we will need the following mapping the- orem in order to extend the results to further functionals of the trajectories including our convex hull properties. First, note that, given two metric measure spaces (S, S, ρ) and (S0, S0, ρ0) and a measurable function h : S → S0, the set Dh of discontinuities of

h satisfies Dh ∈ S: see [Bil99, p. 243], and hence P(X ∈ Dh) is well defined.

Theorem 3.1.4 (Continuous mapping theorem for almost-sure convergence [Gut05,

p. 244]). Let X, X1, X2, . . . be random variables on the probability space (Ω, F , P) taking

values in the metric measure space (S, S, ρ). Let (S0, S0, ρ0) be another metric measure

space, and let h : (S, S, ρ) → (S0, S0, ρ0) be measurable. If Xn a.s.

−→ X and P(X ∈ Dh) =

0, then h(Xn) a.s.

3.1. Random walk convergence 66

In the zero drift case, we need a new scaling, but still maintain the two different trajectories. Precisely, for n ∈ N and t ∈ [0, 1] we define

Yn(t) := 1 √ n  Sbntc+ (nt − bntc)ξbntc+1  ; (3.1.3) Y_n0(t) := √1 nSbntc. Here Yn ∈ C0d and Y 0

n ∈ D0d. Then, recalling bd is a standard d-dimensional Brownian

motion, we can now state the weak convergence result for our zero-drift trajectories. The one-dimensional case was first proven in [Don51], and further discussion can be found in, for example, [Bil99; Kal02]. We also point the reader to [EK09, §5] for a comprehensive discussion of both d-dimensional Brownian motion and the steps leading to this result.

Theorem 3.1.5 (Donsker’s theorem). Suppose that we have a random walk as defined

at (Wµ) with µ = 0 and satisfying (V).

(a) We have Yn⇒ Σ1/2bd in the sense of weak convergence on (C0d, ρ∞).

(b) We have Y_n0 ⇒ Σ1/2_b

d in the sense of weak convergence on (Dd0, ρS).

As with the almost-sure convergence result, we will need a mapping theorem in order to extend our results to functionals of the random walk.

Theorem 3.1.6 (Continuous mapping theorem for weak convergence [Bil99, p. 20]).

Let P, P1, P2, . . . be a sequence of probability measures on a metric measure space

(S, S, ρ). Let (S0, S0, ρ0) be another metric measure space, and let h : (S, S, ρ) → (S0, S0, ρ0) be measurable. For each n, we define Pnh−1, a probability measure on

(S0, S0, ρ0) by Pnh−1(A) = Pn(h−1(A)) for A ∈ S0. If Pn ⇒ P and P (Dh) = 0,

then Pnh−1 ⇒ P h−1.

Corollary 3.1.7. If Xn ⇒ X and P(X ∈ Dh) = 0, then h(Xn) ⇒ h(X).

Remark 3.1.8. Part (b) of Theorem 3.1.5 is stated for the space (Dd₀, ρS), but weak

convergence on (Dd

0, ρS) is equivalent to weak convergence on (Dd0, ρ ◦

S). To see this,

3.1. Random walk convergence 67

measure discussion above there, and note Proposition 3.1.1 which tells us that a con- tinuous function f under one metric is continuous under the other. Thus, the set of bounded continuous functions is the same in both metric spaces and so weak conver- gence must be equivalent.

3.1.1

The maximum functional

As a first example of the theory developed above, we consider a d-dimensional version of the maximum functional M : Md _{→ R defined by M(f) := sup0≤t≤1}kf (t)k, where we recall Md _{is the set of trajectories, see Section 1.3. Note that |M (f )| ≤ kf k}

∞. The

next result shows that M is a continuous map from (Md, ρ∞) to (R, ρE) and also a

continuous map from (Md_{, ρ}

S) to (R, ρE).

Theorem 3.1.9. For any f, g ∈ Md _{we have |M (f ) − M (g)| ≤ ρ}

S(f, g) ≤ ρ∞(f, g).

Proof. Take f, g ∈ Md, and suppose without loss of generality that sup_s∈[0,1]kf (s)k ≥ sup_t∈[0,1]kg(t)k. Recall Λ is the set of λ : [0, 1] → [0, 1] that are strictly increasing and surjective. For any λ ∈ Λ0,

|M (f ) − M (g)| = sup s∈[0,1] kf (s)k − sup t∈[0,1] kg(t)k = sup s∈[0,1] kf (s)k − sup t∈[0,1] kg ◦ λ(t)k, since λ[0, 1] = [0, 1]. Hence |M (f ) − M (g)| = sup s∈[0,1] kf (s)k − sup t∈[0,1] kg ◦ λ(t)k ! ≤ sup s∈[0,1] (kf (s)k − kg ◦ λ(s)k) ≤ sup s∈[0,1] kf (s) − g ◦ λ(s)k = kf − g ◦ λk∞ ≤ kλ − Ik∞∨ kf − g ◦ λk∞.

We therefore have that

|M (f ) − M (g)| ≤ inf

3.1. Random walk convergence 68

Lemma 1.3.2 completes the proof.

Since we have shown the maximum functional is continuous, we can also apply the mapping theorem to the functional law of large numbers, to obtain the following result.

Theorem 3.1.10. Consider the random walk defined at (Wµ). Then, as n → ∞,

1

n0≤k≤nmax kSkk a.s.

−→ kµk.

Proof. Let X_n0(t) be as defined at (3.1.2). The functional strong law of large numbers, Theorem 3.1.2, says that X_n0 −→ Ia.s. µ on (Dd, ρ∞), while Theorem 3.1.9 says that M

is continuous. Thus the mapping theorem, Theorem 3.1.4, implies that M (X_n0) −→a.s.

M (Iµ) on (R, ρE). But M (Xn0) = n−1max0≤k≤nkSkk and M (Iµ) = kµk, giving the

result.

Further to the law of large numbers scaling result, we can apply the mapping theorem to the functional central limit theorem to establish the following result in the case of zero drift.

Theorem 3.1.11. Suppose that we have a random walk as defined at (W_µ) with µ = 0,

and satisfying (V). Then as n → ∞,

1 √ n0≤k≤nmax kSkk d −→ sup t∈[0,1] kΣ1/2bd(t)k.

Proof. Donsker’s theorem, Theorem 3.1.5, together with the mapping theorem, Corol-

lary 3.1.7, and continuity of the function M : (Dd_{, ρ}

S) → (R+, ρE), Theorem 3.1.9, shows that M (Y_n0) = sup t∈[0,1] kY_n0(t)k−→ M (Σd 1/2bd) = sup t∈[0,1] kΣ1/2bd(t)k.

But we have that sup_t∈[0,1]kY0

n(t)k = n−1/2max{kS0k, kS1k, . . . , kSnk}, completing the

proof.

Remark 3.1.12. In the case Σ = Id, the d-dimensional identity matrix, the right

hand side of Theorem 3.1.11 is concerned with the maximum of a d-dimensional Bessel process. In the case d = 1, the maximum functional would more naturally be M (f ) := sup_0≤t≤1f (t), which would give different results. This functional was presented in

3.1. Random walk convergence 69

[LMW18]. In this case, the distribution of sup_t∈[0,1]b(t) can be determined by the

reflection principle for Brownian motion, and so Theorem 3.1.11 would give us the limiting distribution for max1≤k≤nSk/

√

n: see [Bil99, pp. 91–93].