Extremes of normed empirical moment generating function processes

Michael Stewart^{∗}and John Robinson^{†} University of Sydney

March 16, 2004

Abstract

We present a method for deriving the limiting distribution of the maximum of a normed empirical moment generating function process indexed by one parameter. We first extend slightly the results of Csörgő et al. (1986b) to provide the rate of convergence for a Gaussian approximation to a non-Donsker empirical process. In cases we consider, the maximum tends to infinity in probability, but when appropriately scaled has a limiting Gumbel extreme value distribution.

Keywords: Empirical processes, extreme value distribution, limiting distribution, empirical moment generating function, extremes of Gaussian processes.

AMS 2000 subject classifications: 62E20, 60G70.

1 Introduction

Suppose X_{1}, X_{2}, …, X_{n} are independent and identically distributed random

variables with common distribution F and quantile function given by F^{−1}(u) =

inf { x | F (x) ≥ u }, and write F g = ∫ g dF if this exists. Let 𝒢 be a class of

functions satisfying F g^{2} < ∞ for all g ∈ 𝒢. The empirical process indexed by 𝒢

is S_{n}(g) = n^{−1/2}∑_{i=1}^{n}[g(X_{i}) − F g]. We are interested in the limiting behaviour

of the quantity M_{n}(𝒢) = sup_{g∈𝒢} S_{n}(g) as n → ∞. If 𝒢 is a Donsker class, then the empirical process { S_{n}(g) | g ∈ 𝒢 } converges weakly in the space of bounded functions on 𝒢 to a tight, mean-zero Gaussian process { S_{0}(g) | g ∈ 𝒢 } with covariance function F gh − F gF h, g, h ∈ 𝒢. In this case, M_{n}(𝒢) converges in distribution to M_{0}(𝒢) = sup_{g∈𝒢} S_{0}(g).

∗email: michaels@maths.usyd.edu.au

†email: johnr@maths.usyd.edu.au

In various statistical applications a statistic of the form M_{n}(𝒢) occurs but 𝒢 is not necessarily Donsker. What usually happens is that the class is restricted to some 𝒢_{0} ⊂ 𝒢 which is Donsker, so that the restricted maximum M_{n}(𝒢_{0}) converges in distribution to M_{0}(𝒢_{0}) (see Dacunha-Castelle & Gassiat (1999) for an example of this).

One particular example is the normed empirical moment generating function (emgf) process corresponding to the one parameter class of functions 𝒢 = { g_{θ} | θ ∈ Θ/2 } where Θ/2 = { θ/2 | θ ∈ Θ }, Θ = { θ ∈ ℝ | ∫ e^{xθ} dF (x) < ∞ }, g_{θ}(x) = exp{xθ − K(2θ)/2} and K(θ) = log ∫ e^{xθ} dF (x). Write S_{n}(θ) = S_{n}(g_{θ}) and for any I ⊂ Θ/2 write M_{n}(I) = sup_{θ∈I} S_{n}(θ). Several authors have noted that the unrestricted maximum M_{n}(Θ/2) tends to infinity in probability, mainly in connection with finite mixture models; see for instance Bickel & Chernoff (1993) and Hartigan (1985) for a similar problem. However if I is a closed interval of finite length, then results in Csörgő (1982) imply that M_{n}(I) converges weakly to M_{0}(I) = sup_{θ∈I} S_{0}(g_{θ}). The same conclusion can be reached using Theorems A and 2.1 from Csörgő et al. (1986a). Theorem A implies that

sup_{θ∈Θ/2} |∫_{0}^{1} g_{θ} ∘ F^{−1}(u) dα_{n}(u) − ∫_{1/n}^{1−1/n} g_{θ} ∘ F^{−1}(u) dB_{n}(u)| ⟶^{P} 0 ,  (1)

where α_{n}(u) = n^{−1/2}[G_{n}(u) − u] is the empirical process based on the empirical cumulative distribution function G_{n}(u) = n^{−1}∑_{i=1}^{n}1{U_{i} ≤ u} of a U(0, 1) random sample U_{1}, U_{2}, …, U_{n} and { B_{n}(u) | 0 ≤ u ≤ 1 } is a sequence of Brownian Bridge processes, all defined on the same probability space. Under a further metric entropy condition (similar to conditions in chapter 2.5 of van der Vaart & Wellner (1996)), Theorem 2.1 of Csörgő et al. (1986a) shows that the limits of integration in the second integral in (1) can be replaced by 0 and 1. This is shown to hold if the supremum in (1) is taken only over a closed interval of finite length I ⊂ Θ/2.

A different approach is taken in Bickel & Chernoff (1993); they show that when F is the standard normal distribution, writing M_{n} = M_{n}(Θ/2), for a certain sequence a_{n} = O(log n)^{1/2},

P {a_{n} [M_{n} − a_{n}] + log 2π ≤ x} → exp{−e^{−x}} .  (2)

We prove (2) for a general F, replacing M_{n} with M_{n}(Θ_{n}) where {Θ_{n}} is any sequence of subsets satisfying ∪_{n}Θ_{n} = Θ/2 and a certain technical condition (see remarks following Theorem 3.1 below). In many cases we can take Θ_{n} ≡ Θ/2. The sequence a_{n} → ∞ then depends on F and the sequence of indexing sets {Θ_{n}} chosen.

There are at least two applications of the work in this paper. Firstly, our Theorem 4.2 gives the limiting behaviour of test statistics for tests of homogeneity in two component exponential family mixture models, thus generalising work in Bickel & Chernoff (1993) and Liu et al. (2003). Secondly, Theorem 3.1 provides an improved approximation to the empirical moment generating function, which is relevant to various tests of goodness of fit and other diagnostic procedures; see for instance Epps et al. (1982) and Ghosh (1996).

In section 2 we present some preliminary results on boundary crossings by Gaussian processes. In section 3 we establish an approximation of the form (1). In section 4 we apply results from section 2 to derive our version of (2). In section 5 we give sufficient conditions and in section 6 examples are considered.

Proofs appear in section 7.

2 Some Gaussian preliminaries

Our main result in section 4 below relies on some results on boundary crossings by Gaussian processes, based on results of Hüsler (1990, 1995), who considers a single standardised Gaussian process { X(t) | t ∈ [0, T] }, by which is meant EX(t) = 0 and EX^{2}(t) = 1 for all t, with (0, ∞)-valued boundary functions { b_{T} (t) | t ∈ [0, T] } and computes lim_{T →∞} P {X(t) ≤ b_{T} (t), t ∈ [0, T]}.

We generalise this to a sequence of such processes { X_{n}(t) | t ∈ T_{n} } whose correlation functions may change with n, where the intervals {T_{n}} are possibly of infinite length and boundary functions { b_{n}(t) | t ∈ T_{n} } are possibly infinitevalued. We are particularly interested in the case where for each n the boundary is essentially flat in a central region and increases suddenly outside this region, and where the minimum boundary height and the lengths of the T_{n}’s tend to infinity with n. Our results are weaker than Hüsler’s though, in the sense that we demand more smoothness from the correlation functions than he does. “Note that for the rest of this section, all Gaussian processes are standardised”.

Condition A. The processes { X_{n}(t) | t ∈ T_{n} } are almost surely continous, the correlation functions r_{n}(s, t) = EX_{n}(s)X_{n}(t), have continous partial derivatives up to order 2, satisfying

r_{n}^{(1,1)}(t, t) = −r_{n}^{(0,2)}(t, t) = −r_{n}^{(2,0)}(t, t) ≡ 1 ,  (3)

for all t ∈ T_{n} and each n, where r_{n}^{(i,j)}(s, t) = ∂^{i+j}r_{n}(u, v)/∂u^{i}∂v^{j}|_{u=s,v=t}.

That (3) holds is not as restrictive as it might initially seem. It is possible to transform the scale of a standardised process with suitably smooth correlation function to obtain a new process whose correlation function satisfies (3), as we do in section 4. See also Hüsler (1995).

The next condition divides T_{n} into a region where the boundary functions are essentially flat and another where they increase suddenly. Henceforth, for any interval I, λ(I) denotes its Lebesgue measure.

Condition B. There is a sequence of intervals {T_{n1} ⊂ T_{n}} such that as n → ∞,

1. writing ℓ_{n1} = inf { b_{n}(t) | t ∈ T_{n1} } and m_{n1} = sup { b_{n}(t) | t ∈ T_{n1} }, ℓ_{n1} → ∞, m_{n1}/ℓ_{n1} → 1 and lim sup λ(T_{n1}) exp{−ℓ_{n1}^{2}/2} < ∞;

2. T_{n}\T_{n1} can be partitioned into a finite number of intervals ∪_{j=2}^{k}T_{nj}, where for each j = 2, 3, …, k, b_{n}(·) is bounded below on T_{nj} by a continuous monotone function B_{nj}(·) which satisfies inf { B_{nj}(t) | t ∈ T_{nj} } → ∞ and ∫_{T_{nj}} exp{−B_{nj}^{2}(t)/2} dt → 0.

The next condition also applies to a sequence of intervals {T_{n1}} and has two parts. The first is a stronger version of condition A which controls local dependence uniformly. It ensures that over T_{n1} the processes behave like locally stationary processes (see Hüsler (1990, 1995), Berman (1974)). The second is a decay condition that controls long-range dependence within T_{n1}.

Condition C. There is a sequence of intervals {T_{n1} ⊂ T_{n}}, such that as n → ∞,

1. for any h_{n} → 0,

sup_{{ t | t,t+h_{n}∈T_{n1} }} h_{n}^{−2}|r_{n}(t, t + h_{n}) − 1 + h_{n}^{2}/2| → 0  (4)

2. for any u_{n} → ∞ such that the set S(u_{n}) = { (s, t) | s, t ∈ T_{n1}, |s − t| ≥ u_{n} } remains non-empty,

sup { r_{n}(s, t) | (s, t) ∈ S(u_{n}) } log u_{n} → 0 .  (5)

Theorem 2.1. Suppose we have a sequence of standardised Gaussian processes { X_{n}(t) | t ∈ T_{n} } with correlation functions { r_{n}(s, t) | s, t ∈ T_{n} } for intervals {T_{n}} so that condition A holds. Suppose that there also exists a sequence of intervals {T_{n1} ⊂ T_{n}} and a sequence of (0, ∞]-valued boundary functions { b_{n}(t) | t ∈ T_{n} }, so that conditions B and C both hold. Then with ℓ_{n1} = inf { b_{n}(t) | t ∈ T_{n1} } and J_{n} = λ(T_{n1}) exp{−ℓ_{n1}^{2}/2} /(2π),

P {X_{n}(t) ≤ b_{n}(t), t ∈ T_{n1}} − exp{−J_{n}} → 0

and

P {X_{n}(t) ≤ b_{n}(t), t ∈ T_{n} \ T_{n1}} → 1

as n → ∞. Hence

P {X_{n}(t) ≤ b_{n}(t), t ∈ T_{n}} − exp{−J_{n}} → 0 .

The proof of Theorem 2.1 appears in section 7.

3 Establishing the Gaussian approximation

For a fixed sequence {Θ_{n}} of subsets of Θ/2, define the envelope function ǧ_{n}(x) = sup_{θ∈Θ_{n}} g_{θ}(x). For any g write F_{n}g = ∫_{1/n}^{1−1/n} g ∘ F^{−1}(u) du whenever F g exists (note that F_{n}g → F g for each g as n → ∞).

Theorem 3.1. Suppose that, writing δ_{n} = n^{−1/2} log n, we have

sup_{F (x)≤δ_{n}} ǧ_{n}(x)F (x)^{1/2} = O(c_{n})  (6)

and

sup_{F (x)≥1−δ_{n}} ǧ_{n}(x)[1 − F (x)]^{1/2} = O(c_{n})  (7)

for some rate c_{n} → 0 as n → ∞. Then it is possible to construct a sequence of processes equal in distribution to { S_{n}(θ) | θ ∈ Θ_{n} } and a sequence of Gaussian processes { Z_{n}(θ) | θ ∈ Θ_{n} } with zero mean and covariance functions

C_{n}(η, θ) = F_{n}g_{η}g_{θ} − F_{n}g_{η}F_{n}g_{θ}  (8)

on a common probability space such that

sup_{θ∈Θ_{n}} |S_{n}(θ) − Z_{n}(θ)| = O_{p} (c_{n}) .  (9)

Remarks So long as Θ_{n} is an interval, the envelope functions are given by ǧ_{n}(x) = g(x; θ̃(x)), where g( ·; θ) = g_{θ} and θ̃(x) = (inf Θ_{n}) ∨ θ̂(x)/2 ∧ (sup Θ_{n}), where θ̂(x) satisfies K′(θ) = x (derivatives of all orders exist for θ in the interior of Θ, see Barndorff-Nielsen (1978, 1980)). In most cases we can take the sequence Θ_{n} ≡ Θ/2 to be maximal, in which case ǧ_{n} ≡ g( ·; θ̂(·)/2), but in some examples where the corresponding envelope functions ǧ_{n} do not satisfy both (6) and (7), it is possible to choose a sequence {Θ_{n}} such that Θ_{n} ⊂ Θ_{n+1} for each n, ∪_{n=1}^{∞}Θ_{n} = Θ/2 and (6) and (7) are both satisfied. For instance, it can be shown that when F is the standard normal distribution and Θ_{n} ≡ Θ/2 = ℝ, (6) and (7) both hold with c_{n} = (log n)^{−1/4}. However, if F is the gamma distribution on (0, ∞) with shape parameter α (and unit scale), and Θ_{n} ≡ Θ/2 = (−∞, 1/2), then (7) holds with c_{n} = (log n)^{−1/2}, but as x → 0, ǧ_{n}(x)F (x)^{1/2} → [(α/e)^{α}/Γ(α + 1)]^{1/2}, not even tending to zero, so (6) does not hold. However if Θ_{n} = (−θ_{n}, 1/2) for θ_{n} = [n^{1/2}(log n)^{−3/2}]^{1/α}, then (6) also holds with c_{n} = (log n)^{−1/2} (see Stewart (2002) for details).

In the case that each Θ_{n} ⊂ [0, ∞) (respectively (−∞, 0]) and each g_{θ} is thus non-decreasing (respectively non-increasing), only condition (7) (respectively (6)) is required. In the case Θ_{n} ≡ Θ_{0} is a fixed set of non-negative real numbers, it is of interest to compare the condition (7) with a sufficient condition that the class of non-decreasing functions 𝒢 = { g_{θ} | θ ∈ Θ_{0} } is Donsker, given in van der Vaart (1996). Write ǧ(x) = sup_{θ∈Θ_{0}} g_{θ}(x) for the envelope function. If ǧ(x)[1 − F (x)]^{1/2} → 0, or equivalently

ǧ(x)[1 − F (x)]^{−1/2}f(x) = o (f(x)/[1 − F (x)])  (10)

as F (x) → 1, then (7), and hence (9), holds for some c_{n} → 0. If

∫_{−∞}^{∞} ǧ(x)[1 − F (x)]^{−1/2}f(x) dx < ∞  (11)

then by Corollary 3.1 in van der Vaart (1996) 𝒢 is a Donsker class. That the condition (11) can be much stronger than (10) can be illustrated by considering what (10) reduces to for various F ’s. For instance, when F (x) = 1 − exp{−(x ∨ 0)}, the function required to be finitely integrable by (11) is only required to tend to zero as x → ∞ by (10). When F is the standard normal distribution, the function required to be finitely integrable in (11) is only required to be o(x) as x → ∞ by (10), so can even tend to infinity!

Theorem 3.2 below is a slight generalisation of Theorem 3.2 of Csörgő et al. (1986b) and is used to prove Theorem 3.1.

Theorem 3.2. Let {ℒ_{n}} be a sequence of classes of functions such that each ℓ ∈ ℒ_{n} can be written as ℓ = ℓ_{1}−ℓ_{2} where for i = 1, 2, ℓ_{i} is a left-continous nondecreasing function defined on (0,1). Define for any positive sequence δ_{n} → 0,

N_{n}(δ_{n}, ℒ_{n}) = sup_{\begin{matrix}0≤u≤δ_{n} \\ ℓ∈ℒ_{n} \end{matrix}} \{|ℓ_{1}(u)| + |ℓ_{2}(u)| + |ℓ_{1}(1 − u)| + |ℓ_{2}(1 − u)|\} u^{1/2} . (12)

If, for some rate c_{n} → 0 as n → ∞, we have

N_{n} (n^{−1/2} log n, ℒ_{n}) = O(c_{n}) ,  (13)

then there exists a probability space (Ω, 𝒜 , P) with a sequence of U(0, 1) empirical processes {α_{n}(·)} and a sequence of standard Brownian Bridges {B_{n}(·)} such that

sup_{ℓ∈ℒ_{n}} |∫_{0}^{1} ℓ(u) dα_{n}(u) − ∫_{1/n}^{1−1/n} ℓ(u) dB_{n}(u)| = O_{p} (c_{n}) .  (14)

Proof. This is proved exactly as Theorem 3.2 of Csörgő et al. (1986b), where the analogue of the left-hand side of (14) is bounded above by the sum of a finite number of terms of the form

N_{n} (n^{−1/2} log n, ℒ_{n}) O_{p} (1) .

This, together with (13), implies (14).

Proof of Theorem 3.1 In the language of Theorem 3.2, for each n, each element ℓ of the monotone, left-continuous class of functions defined by ℒ_{n} = { ℓ_{θ} = g_{θ} ∘ F_{0}^{−1} | θ ∈ Θ_{n} } satisfies |ℓ_{1}| + |ℓ_{2}| = |ℓ|. Conditions (6) and (7) then imply that N_{n}(δ_{n}, ℒ_{n}) = O(c_{n}) (see (12)). Equation (9) then follows from Theorem 3.2.

4 Main Result

In this section we derive the limiting distribution of the sequence of suprema of the processes { Z_{n}(θ) | θ ∈ Θ_{n} } appearing in (9). We introduce some definitions and present Theorem 4.1, the proof of which utilises Theorem 2.1. However the conditions of Theorem 4.1, while easy to formulate, are quite difficult to verify directly in examples. In section 5 we present conditions that imply Theorem 4.1 and are also easier to verify in examples.

We firstly introduce the modified process { Z̃_{n}(θ) = Z_{n}(θ) + X̃F_{n}g_{θ} | θ ∈ Θ_{n} }, where X̃ ∼ 𝒩 (0, 1) independently of Z_{n}, which is mean-zero Gaussian and has covariance function EZ̃_{n}(η)Z̃_{n}(θ) = C̃_{n}(η, θ) = F_{n}g_{η}g_{θ}. This is the generalisation of a trick used in Bickel & Chernoff (1993); the new covariance function is nicer, but the added term has a negligible effect on the maximum. Write v_{n}(θ) = VarZ̃_{n}(θ) = C̃_{n}(θ, θ). Next define the standardised processes { Y_{n}(θ) = v_{n}^{−1/2}(θ)Z̃_{n}(θ) | θ ∈ Θ_{n} }, which have correlation functions ρ_{n}(η, θ) = [v_{n}(η)v_{n}(θ)]^{−1/2} C̃_{n}(η, θ).

Writing partial derivatives as in the statement of condition A, since ρ_{n} is a correlation function, γ_{n}(θ) = ρ_{n}^{(1,1)}(θ, θ) = −ρ_{n}^{(2,0)}(θ, θ) = −ρ_{n}^{(0,2)}(θ, θ). Define the transformation θ ↦ τ_{n}(θ) = ∫_{0}^{θ} γ_{n}^{1/2}(η) dη, and write t ↦ θ_{n}(t) for its inverse. Define also T_{n} = { τ_{n}(θ) | θ ∈ Θ_{n} }. Then the standardised Gaussian processes given by { X_{n}(t) = Y_{n}(θ_{n}(t)) | t ∈ T_{n} } have correlation functions

r_{n}(s, t) = ρ_{n}(θ_{n}(s), θ_{n}(t))  (15)

which satisfy condition A in section 2, that is r_{n}^{(1,1)}(t, t) = −r_{n}^{(2,0)}(t, t) = −r_{n}^{(0,2)}(t, t) ≡ 1 for each n and t.

Theorem 4.1. Suppose that there exists a sequence of intervals {T_{n1} ⊂ T_{n}}, each containing 0, such that the correlation functions (15) satisfy condition C and if for each real x we define a_{n} = [2 log λ(T_{n1})]^{1/2} and u_{n}(x) = a_{n} + (x − log 2π)/a_{n} then the boundary functions { b_{n,x}(t) = v_{n}^{−1/2}[θ_{n}(t)]u_{n}(x) | t ∈ T_{n1} } satisfy condition B. Then for each real x,

P {a_{n} [ sup_{θ∈Θ_{n}} Z_{n}(θ) − a_{n}] + log 2π ≤ x} → exp{−e^{−x}} .  (16)

See section 7 for the proof.

Combining this with Theorem 3.1 gives our main result: Theorem 4.2. Suppose that Theorem 3.1 holds for some rate c_{n} → 0 and that Theorem 4.1 holds for some sequence of intervals {T_{n1}}. Suppose that c_{n} = o(a_{n}^{−1}), where a_{n} is as in the statement of Theorem 4.1. Then (2) holds for the same a_{n}, with M_{n} = M_{n}(Θ_{n}) = sup_{θ∈Θ_{n}} S_{n}(θ). Proof. If c_{n} = o(a_{n}^{−1}) then by (9), sup_{θ∈Θ_{n}} S_{n}(θ) = sup_{θ∈Θ_{n}} Z_{n}(θ) + o_{p} (a_{n})^{−1} and so

a_{n} [ sup_{θ∈Θ_{n}} S_{n}(θ) − a_{n}] = a_{n} [ sup_{θ∈Θ_{n}} Z_{n}(θ) − a_{n}] + o_{p} (1) ,

and so the final statement of Theorem 4.1 implies (2) with M_{n} replaced by M_{n}(Θ_{n}).

5 Sufficient Conditions for Theorem 4.1

We identify in this section an exponential family of distributions in terms of which easier sufficient conditions may be formulated.

Define q_{n}^{−} = F^{−1}(1/n) and q_{n}^{+} = F^{−1}(1 − 1/n) as the lower and upper 1/n quantiles of F. Write A_{n} = { x | q_{n}^{−} ≤ x ≤ q_{n}^{+} }. Note that we can write C̃_{n}(θ, η) = F_{n}g_{θ}g_{η} = ρ(θ, η) ∫_{A_{n}} dF_{θ+η}, where F_{θ} is the distribution satisfying dF_{θ}(x) = e^{θx−K(θ)}dF (x) and ρ(θ, η) = exp{K(θ + η) − [K(2θ) + K(2η)]/2} is the pointwise limit of both C̃_{n}(θ, η) and ρ_{n}(θ, η) as n → ∞. Identifying the full, linear exponential family ℱ = { F_{θ} | θ ∈ Θ } provides nice interpretations of various quantities. For example, v_{n}(θ) = ∫_{A_{n}} dF_{2θ} is the mass put on the set A_{n} by F_{2θ}.

The first two derivatives K′(θ) and K′′(θ) give the mean and variance of F_{θ}. The function x ↦ θ̂(x) solving K′(θ) = x (already remarked upon below Theorem 3.2) gives the maximum likelihood estimate for the parameter θ within ℱ based on a single observation x. The mapping θ ↦ γ_{n}(θ) defined in the previous subsection can be rewritten as the conditional variance of X, given X ∈ A_{n}, when unconditionally X ∼ F_{2θ}. Hence the pointwise limit as n → ∞ of γ_{n}(θ) is K′′(2θ), the unconditional variance. Furthermore, the pointwise limit of τ_{n}(θ) defined in the previous subsection is τ_{0}(θ) = ∫_{0}^{θ} K′′(2η)^{1/2} dη = ϕ(2θ)/2, where ϕ(·) is the variance-stablising transformation of ℱ.

Condition D. There exists a monotone sequence of intervals { Θ_{n1} ⊂ Θ_{n} | n ∈ ℕ } with ∪_{n}Θ_{n1} = ∪_{n}Θ_{n} such that

sup_{\begin{matrix}a_{i}≥0, ∑_{i} a_{i}≤4 \\ θ_{i},η_{i}∈Θ_{n1} \end{matrix}} |∏_{i=1}^{4} ∫_{A_{n}} x_{i}^{a_{i}} dF_{θ_{i}+η_{i}}(x_{i}) − ∏_{i=1}^{4} ∫_{ℝ} x_{i}^{a_{i}} dF_{θ_{i}+η_{i}}(x_{i})| → 0 .

The pointwise limit of r_{n}(s, t) is given by r_{0}(s, t) = ρ(θ_{0}(s), θ_{o}(t)), where t ↦ θ_{0}(t) is the inverse of the transformation θ ↦ τ_{0}(θ) = ϕ(2θ)/2. In section 7 we show that condition D implies that r_{n}(s, t) = r_{0}(s, t)[1 + o(1)] and r_{n}^{(2,0)}(s, t) = r_{0}^{(2,0)}(s, t)[1 + o(1)], with both o(1) terms uniform over s, t ∈ T_{n1} = { τ_{n}(θ) | θ ∈ Θ_{n1} }. If r_{0}(·, ·) satisfies condition E below, the functions r_{n}(·, ·) can be shown to satisfy condition C in section 2 for this choice of T_{n1}. Write T_{0} = ∪_{n}T_{n1}.

Condition E. λ(T_{0}) = ∞, and the following two conditions are satisfied: sup_{t∈T_{0}} h^{−2} |r_{0}(t, t + h) + 1 − h^{2}/2| → 0 as h → 0 and sup { r_{0}(s, t) | s, t ∈ T_{0}, |s − t| ≥ u } log u → 0 as u → ∞.

The conditions D and E imply condition C. We still need more to imply B. Define θ_{n}^{−} = θ̂(q_{n}^{−})/2 and similarly θ_{n}^{+}. Condition F. There exist sequences θ_{n}^{−} < θ_{n1}^{−} ≤ 0 ≤ θ_{n1}^{+} < θ_{n}^{+}, θ_{n2}^{−} ≤ θ_{n1}^{−} and θ_{n1}^{+} ≤ θ_{n2}^{+} taking values in Θ/2 such that, defining Θ_{n1} = Θ_{n} ∩ [θ_{n1}^{−}, θ_{n1}^{+}], Θ_{n2}^{−} = Θ_{n} ∩ (θ_{n2}^{−}, θ_{n1}^{−}), Θ_{n2}^{+} = Θ_{n} ∩ (θ_{n1}^{+}, θ_{n2}^{+}), v_{n2}^{−} = inf_{θ∈Θ_{n2}^{−}} v_{n}(θ), similarly v_{n2}^{+}, Θ_{n3}^{−} = Θ_{n} ∩ (−∞, θ_{n2}^{−}], Θ_{n3}^{+} = Θ_{n} ∩ [θ_{n2}^{+}, ∞), L_{n1} = λ(τ_{0}(Θ_{n1})) L_{n2}^{−} = λ(τ_{0}(Θ_{n2}^{−})), similarly L_{n2}^{+}, we have that

1. either Θ_{n2}^{−} = ∅ or (v_{n2}^{−})^{−1/2}L_{n2}^{−}/L_{n1} → 0 and

2. either Θ_{n3}^{−} = ∅ or v_{n2}^{−} → 0 and

sup_{θ∈Θ_{n3}^{−}} v_{n}(θ)K′′(2θ)^{1/2}/|q_{n}^{−} − K′(2θ)| = O ([v_{n2}^{−}]^{1/2}L_{n1} log L_{n1}) (17)

and both statments also hold if − is replaced with + (here we interpret the “open interval” (x, x) as the empty set).

In subsection 6 we verify these conditions for certain examples of interest. Theorem 5.1. Suppose there exist sequences θ_{n2}^{−} ≤ θ_{n1}^{−} < θ_{n1}^{+} ≤ θ_{n2}^{+} taking values in Θ/2 satisfying condition F and such that for Θ_{n1} = [θ_{n1}^{−}, θ_{n1}^{+}], conditions D and E hold. Then Theorem 4.1 holds taking T_{n1} = { τ_{n}(θ) | θ ∈ Θ_{n1} }.

6 Examples

In this subsection we show that conditions D, E and F hold for certain examples of interest.

6.1 Normal

Suppose F is the standard normal distribution. Then the first product appearing in condition D can be written as

∏_{i=1}^{4} [∫_{ℝ} x^{a_{i}}dF_{θ_{i}+η_{i}}(x) − ∫_{ℝ\A_{n}} x^{a_{i}}dF_{θ_{i}+η_{i}}(x)]

which when expanded gives a linear combination of sixteen terms of the form ∏_{i=1}^{4} ∫_{S_{i}} x^{a_{i}}dF_{θ_{i}+η_{i}}(x), where each S_{i} is either ℝ or ℝ \ A_{n}; the term corresponding to all S_{i}’s equalling ℝ is the second product in condition D. The remaining 15 terms constitute the remainder which we wish to show tends to zero, uniformly over θ_{i}, η_{i} in some Θ_{n1}. In this example each of these terms can be written as

∏_{i=1}^{4} \{∑_{j=0}^{a_{i}} \binom{a_{i}}{j}(θ_{i} + η_{i})^{a_{i}−j} ∫_{I_{i}} y^{j}f(y) dy\}  (18)

where f is the standard normal density, and at most 3 of the I_{i}’s are ℝ, the others being ℝ \ (−q_{n} − θ_{i} − η_{i}, q_{n} − θ_{i} − η_{i}). Suppose we take

Θ_{n1} = {θ: |θ| ≤ (q_{n} − Δ_{n})/2} ,  (19)

for some Δ_{n} → ∞ more slowly than q_{n} = O(log n)^{1/2}. When I_{i} = ℝ, each integral is O(1), so the biggest term in the sum in (18) is

sup_{θ,η∈Θ_{n1}} |θ + η|^{a_{i}} O(1) = O(q_{n}^{a_{i}}) .

Otherwise, the integral is bounded above by 2 ∫_{Δ_{n}}^{∞} y^{j}f(y) dy = O (Δ_{n}^{j−1} exp{−Δ_{n}^{2}/2}) and the biggest term in the sum is O(q_{n}^{a_{i}} exp{−Δ_{n}^{2}/2}. The biggest each product of the form (18) can be is when exactly 3 of the I_{i}’s are ℝ and only 1 is not. In this case the product is O (q_{n}^{∑_{i} a_{i}} exp{−Δ_{n}^{2}/2}) = O (q_{n}^{4} exp{−Δ_{n}^{2}/2}). Any Δ_{n} such that this last quantity goes to zero will suffice in the definition (19) above, e.g. Δ_{n} = [(4 + ε) log log n]^{1/2} or perhaps 2(log log n + \frac{1}{4} log log log n)^{1/2}. In any case we must have Δ_{n} = O(log log n)^{1/2}.

The verification of condition E is easier; the limiting correlation function is r_{0}(s, t) = exp{−(s − t)^{2}/2}, and since this is the correlation function of a stationary process, verifies the local stationarity condition (4). Also, since e^{−u^{2}/2} log u → 0 as u → ∞, (5) is also satisfied.

As for condition F, v_{n}(θ) = F (q_{n} − 2θ) − F (−q_{n} − 2θ). Define θ_{n1}^{+} = (q_{n} − Δ_{n})/2 and for some α_{n} → ∞, define θ_{n2}^{+} = (q_{n} + α_{n})/2. Also define θ_{nj}^{−} = −θ_{nj}^{+} for j = 1, 2. Then v_{n2}^{+} = sup_{θ_{n1}^{+}≤θ≤θ_{n2}^{+}} v_{n}(θ) = v_{n}(θ_{n2}^{+}) + O(n^{−1}) = O (F (−α_{n})) = O(exp{−α_{n}^{2}/2} /α_{n}). Using the notation of condition F, τ_{0}(θ) ≡ θ so L_{n1} = q_{n} − Δ_{n} = O(log n)^{1/2} and L_{n2}^{+} = [Δ_{n} + α_{n}]/2. The first statement of condition F is satisfied then if

exp{α_{n}^{2}/4} α_{n}^{1/2}[Δ_{n} + α_{n}]/q_{n} → 0 .

Take α_{n} = (log log n − 4 log log log n)^{1/2}. Then with q_{n} = O(log n)^{1/2}, Δ_{n} = O(log log n)^{1/2}, the previous display is

O(1) [(log n)^{1/4}(log log n)^{−1}] (log log n)^{1/2} [log log n/ log n]^{1/2} = O(log n)^{−1/4} → 0

so the first statement of condition F is satisfied. For the second statement, the supremum on the left hand side of the ‘+’ version of (17) is easily seen to occur at θ_{n2}^{+}, so the left hand side is O(v_{n2}^{+}/α_{n}). Dividing both sides then by v_{n2}^{+} gives that the left hand side tends to zero whereas the right-hand side tends to infinity, and so (17) is satisfied.

6.2 Poisson

Now suppose that F is the Poisson distribution with mean μ. Note then that F_{θ} is the Poisson distribution with mean μe^{θ}.

Define q_{n} = F^{−1}(1 − 1/n), x_{n} = q_{n} + 1 and θ_{nj}, j = 1, 2 via

μe^{2θ_{n1}} = x_{n} − (8x_{n} log x_{n})^{1/2}
μe^{2θ_{n2}} = x_{n} + (x_{n} log x_{n})^{1/2} .

Exploiting the form of the Poisson density, it is sufficient to show the condition D holds with x^{a} replaced by 1{x ≥ a_{i}}x!/(x − a_{i})!. The first product there becomes

∏_{i=1}^{4} ∑_{x=a_{i}}^{q_{n}} \frac{exp\{−μe^{θ_{i}+η_{i}}\} (μe^{θ_{i}+η_{i}})^{x}}{(x − a_{i})!}
= ∏_{i=1}^{4} \{(μe^{θ_{i}+η_{i}})^{a_{i}} ∑_{x−a_{i}=0}^{q_{n}−a_{i}} \frac{exp\{−μe^{θ_{i}+η_{i}}\} (μe^{θ_{i}+η_{i}})^{x−a_{i}}}{(x − a_{i})!} \}
= ∏_{i=1}^{4} \{(μe^{θ_{i}+η_{i}})^{a_{i}} [1 − ∑_{y=q_{n}−a_{i}+1}^{∞} \frac{exp\{−μe^{θ_{i}+η_{i}}\} (μe^{θ_{i}+η_{i}})^{y}}{y!} ]\} ,

which is ∏_{i=1}^{4}(μe^{θ_{i}+η_{i}})^{a_{i}}, the second product appearing in condition D, plus 15 other terms, the biggest of which is bounded above by

(μe^{2θ_{n1}})^{4} ∑_{y=x_{n}−4}^{∞} \frac{exp\{−μe^{2θ_{n1}}\} (μe^{2θ_{n1}})^{y}}{y!}

∼ (μe^{2θ_{n1}})^{4} \frac{exp\{−μe^{2θ_{n1}}\} (μe^{2θ_{n1}})^{x_{n}−4}}{(x_{n} − 4)!}
∼ \frac{exp\{−μe^{2θ_{n1}}\} (μe^{2θ_{n1}})^{x_{n}}}{\sqrt{2π(x_{n} − 4)}} (\frac{e}{x_{n} − 4})^{x_{n}−4}

using Stirling’s approximation. Upon inserting the defintion of θ_{n1} given above, this becomes (2π)^{−1/2} times

exp{(8x_{n} log x_{n})^{1/2} − x_{n}} [x_{n} − (8x_{n} log x_{n})^{1/2}]^{x_{n}} (e/x_{n})^{x_{n}}(x_{n} − 4)^{3.5}

= exp\{(8x_{n} log x_{n})^{1/2}\} exp\{x_{n} log [1 − (\frac{8 log x_{n}}{x_{n}} )^{1/2}]\} (x_{n} − 4)^{3.5}
∼ exp{(8x_{n} log x_{n})^{1/2}} exp{−(8x_{n} log x_{n})^{1/2} − 4 log x_{n}} (x_{n} − 4)^{3.5}
∼ x_{n}^{−1/2} → 0

as n → ∞, establishing condition D.

Condition E is easily shown to be satisfied, since the limiting correlation function r_{0}(s, t) is again exp{(s − t)^{2}/2}, and so the same argument applies as in the previous example.

Next, note that in this case τ(θ) = μ^{1/2} [e^{θ} − 1], so L_{n1} = O (x_{n}^{1/2}). Also

L_{n2} = O (e^{θ_{n2}} − e^{θ_{n1}})
= O(1) [x_{n} + (x_{n} log x_{n})^{1/2}]^{1/2} − [x_{n} − (8x_{n} log x_{n})^{1/2}]^{1/2}
= x_{n}^{1/2} \{[1 + (\frac{log x_{n}}{x_{n}} )^{1/2}]^{1/2} − [1 − (\frac{8 log x_{n}}{x_{n}} )^{1/2}]^{1/2}\}
= O(log x_{n})^{1/2} .

Using the well-known relation between tail probabilities for the Poisson and Gamma distributions,

v_{n2} = P (Pois(μe^{2θ_{n2}}) < x_{n}) = P (Gam(x_{n}, 1) > μe^{2θ_{n2}})  (20)
∼ \frac{(μe^{2θ_{n2}})^{x_{n}} exp\{−μe^{2θ_{n2}}\}}{Γ(x_{n})(μe^{2θ_{n2}} − x_{n})} ,

using an integration-by-parts approximation for the upper tail of the Gamma distribution (see also relation (101) in Stewart (2002)). Using Stirlings approximation for the Gamma function, and inserting the definition of θ_{n2}, we obtain an expression which is asymptotic to a constant times

\frac{[x_{n} + (x_{n} log x_{n})^{1/2}]^{x_{n}} exp\{−(x_{n} log x_{n})^{1/2}\}}{(x_{n} − 1)^{x_{n}−1/2}(x_{n} log x_{n})^{1/2}}

∼ (log x_{n})^{−1/2} exp\{−(x_{n} log x_{n})^{1/2} + x_{n} log [1 + (\frac{log x_{n}}{x_{n}} )^{1/2}]\}
∼ (log x_{n})^{−1/2} exp\{−\frac{1}{2} log x_{n}\} = (x_{n} log x_{n})^{−1/2} .

Thus v_{n2}L_{n2}/L_{n1} = O (x_{n}^{−1/4}[log x_{n}]^{3/4}) → 0, verifying the first statement of condition F.

Finally, note that K′(2θ) = K′′(2θ) = μe^{2θ}, and so using a similar approximation to that used in (20) above, we have

sup_{θ≥θ_{n2}} \frac{v_{n}(θ)K′′(2θ)}{K′(2θ) − q_{n}} ∼ sup_{θ≥θ_{n2}} \frac{(μe^{2θ})^{x_{n}} exp\{−μe^{2θ}\}}{Γ(x_{n})(μe^{2θ} − x_{n})} \frac{(μe^{2θ})^{1/2}}{(μe^{2θ} − x_{n} + 1)}
≤ sup_{θ≥θ_{n2}} \frac{(μe^{2θ})^{x_{n}+1/2} exp\{−μe^{2θ}\}}{Γ(x_{n})(μe^{2θ_{n2}} − x_{n})^{2}} .

The numerator is maximised over all θ at μe^{2θ} = x_{n} + 1/2, and this, together with Stirlings approximation again, gives that the left-hand side of equation (17) is at most asymptotic to a constant times

\frac{(x_{n} + \frac{1}{2})^{x_{n}+ \frac{1}{2}} e^{−x_{n}− \frac{1}{2}} e^{x_{n}−1}}{(x_{n} − 1)^{x_{n}− \frac{1}{2}} x_{n} log x_{n}} = O(log x_{n})^{−1} .

Note that v_{n2}^{1/2}L_{n1} log L_{n1} has exact rate x_{n}^{1/4}(log x_{n})^{3/4} → ∞, and so (17), and thus condition F, is easily satisfied.

6.3 Gamma

Now suppose F is the standard Gamma distribution on (0, ∞) with shape parameter α. We take Θ_{n} = (−θ_{n}, 1/2), where θ_{n} = [n^{1/2}(log n)^{−3/2}]^{1/α}, as remarked on below Theorem 3.1. We claim that for j = 1, 2, with θ_{nj}^{−} = −θ_{n}, θ_{nj}^{+} = [1 − αΔ_{n}^{3−2j}/c_{n}]/2 and Δ_{n} = (5/α) log log n, conditions D, E and F are satisfied. We use the same approach as in subsection 6.1. For convenience write b_{n} = q_{n}^{−} and c_{n} = q_{n}^{+}. Again we have a linear combination of 15 terms of the form

∏_{i=1}^{4} ∫_{S_{i}} x^{a_{i}}dF_{θ_{i}+η_{i}}(x) ,  (21)

where this time S_{i} is either (0, ∞) or (0, b_{n}) ∪ (c_{n}, ∞). Standard tail approximations for the gamma distribution (see, for example, Stewart (2002)) give that (note dF_{θ}(x) = x^{α−1}(1 − θ)^{α}e^{−x(1−θ)}/Γ(α) dx)

∫_{I} x^{a}dF_{θ+η}(x)

= \begin{cases}O(1 − θ − η)^{−a} & if I = (0, ∞) , \\ O(1)b_{n}^{a}\{b_{n}(1 − θ − η)\}^{α} & if I = (0, b_{n}) and \\ O(1)c_{n}^{a}\{c_{n}(1 − θ − η)\}^{α−1} exp\{−c_{n}(1 − θ − η)\} & if I = (c_{n}, ∞) . \end{cases}

Now for θ, η ∈ Θ_{n1}, (1 − θ − η) ≥ 1 − 2θ_{n1}^{+} = c_{n}^{−1}αΔ_{n}. So the above three bounds respectively reduce to

O(1)c_{n}^{a}(log log n)^{−a}
O(1)b_{n}^{a}n^{−1/2}(log n)^{−3/2} = o(1)c_{n}^{a}n^{−1/2}(log n)^{−3/2}  (22)
O(1)c_{n}^{a}Δ_{n}^{a} exp{−αΔ_{n}} = O(1)c_{n}^{a}(log log n)^{α−1}(log n)^{−5} .

The largest a product of the form (21) can be is where three of the S_{i}’s are (0, ∞) and one is not. In this case, the product can be written as

O(1)c_{n}^{4}(log log n)^{α−1}(log n)^{−5} = O(1)(log log n)^{α−1}(log n)^{−1} = o(1) (23)

since c_{n} = O(log n) for any α > 0. This proves that condition D is satisfied.

The limiting correlation function is r_{0}(s, t) = cosh(|s − t|/α^{1/2})^{−α}, which again is the correlation function of a stationary process. Also, r_{0}(s, t) = O(e^{−α^{−1/2}|s−t|}) as |s − t| → ∞, and so Condition E is satisfied.

Note that, using the notation of condition F that Θ_{nj}^{−} = ∅ for j = 2, 3, so we only need show the ‘+’-versions of the two statements appearing there. The transformation τ_{0}(·) in this case is given by θ ↦ −(α^{1/2}/2) log(1 − 2θ). So

L_{n1} = τ_{0}(θ_{n1}^{+}) − τ_{0}(−θ_{n}) = O(log n) ;
L_{n2}^{+} = τ_{0}(θ_{n2}^{+}) − τ_{0}(θ_{n1}^{+}) = O(log log log n) .

Also, since v_{n2}^{+} = ∫_{b_{n}}^{c_{n}} dF_{2θ_{n2}^{+}}, again, using standard lower-tail approximations for the gamma distribution, v_{n2}^{+} ∼ {c_{n}(1 − 2θ_{n2}^{+})}^{α}/Γ(α + 1) which has exact rate (log log n)^{−α}. Since (log log n)^{α/2} log log log n/ log n → 0 for any α > 0, the first statement of condition F holds. Furthermore, the right hand side of

(17) becomes O(1)(log n)(log log n)^{1−α/2}, which tends to infinity for any α. The left-hand side becomes a constant times

sup_{θ∈Θ_{n3}^{+}} \frac{∫_{b_{n}}^{c_{n}} dF_{2θ}}{1 − c_{n}(1 − 2θ)} = O(1)

since sup_{θ∈Θ_{n3}} c_{n}(1 − 2θ) ≤ c_{n}(1 − 2θ_{n2}^{+}) → 0. Hence condition F is satisfied.

7 Proofs

7.1 Proof of Theorem 2.1

Lemma 7.1. If h_{n} and ℓ_{n} satisfy ℓ_{n} → ∞, h_{n} → 0 and h_{n}ℓ_{n} → ∞ and equation 4 in condition C holds, then defining R_{n}^{∗}(s) via

P \{sup(X_{n}(t), s ≤ t ≤ s + h_{n}) > ℓ_{n}\} = \frac{h_{n} exp\{−ℓ_{n}^{2}/2\}}{2π} [1 + R_{n}^{∗}(s)] ,

as n → ∞ we have

sup_{s,s+h_{n}∈T_{n1}} |R_{n}^{∗}(s)| → 0 .  (24)

Proof. Define the correlation functions

r_{n}^{sup}(h) = sup_{t,t+h∈T_{n1}} r_{n}(t, t + h) and r_{n}^{inf}(h) = inf_{t,t+h∈T_{n1}} r_{n}(t, t + h) .

Then by 4 we have, as h → 0,

r_{n}^{sup}(h) = 1 − h^{2}/2 + o(h^{2}) ,

and the same holds for r_{n}^{inf}(h). If we now define two sequences of stationary, standardised Gaussian processes { X_{n}^{sup}(t) | t ∈ T_{n1} } and { X_{n}^{inf}(t) | t ∈ T_{n1} } with respective correlation functions

EX_{n}^{sup}(t)X_{n}^{sup}(t + h) ≡ r_{n}^{sup}(h) and EX_{n}^{inf}(t)X_{n}^{inf}(t + h) ≡ r_{n}^{inf}(h)

then Theorem 2.1 of Hüsler (1990) and stationarity mean that (24) holds but with X_{n} replaced by both X_{n}^{sup} and X_{n}^{inf} in the definition of R_{n}^{∗}(s). But by two applications of Slepian’s Lemma (see for instance Leadbetter et al. (1983), Theorem 7.4.2), the maximum of X_{n} is stochastically between the maxima of X_{n}^{sup} and X_{n}^{inf}. This proves (24).

Lemma 7.2. Suppose that for a sequence of intervals { T_{n1} | n ∈ ℕ }, the standardised Gaussian processes { X_{n}(t) | t ∈ T_{n1} }, satisfy condition C. Then, for any level ℓ_{n} → ∞, if J_{n} = λ(T_{n1}) exp{−ℓ_{n}^{2}/2} /(2π) satisfies lim sup J_{n} < ∞ then

P { sup_{0≤t≤t_{n1}} X_{n}(t) ≤ ℓ_{n}} − exp{−J_{n}} → 0 .  (25)

Proof. The method of proof is almost identical to the proof of Theorem 4.1 in Hüsler (1990), taking α = 2 throughout. The only notable difference is that our Lemma 7.1 takes the place of Hüsler’s Theorem 2.2. Otherwise, the method of proof is identical; our equation (4) in condition C takes the place of Hüsler’s conditions (5) and (6), equation (5) takes the place of Hüsler’s condition (10), and since the boundary is constant Hüsler’s smoothness conditions (8) and (9) are trivially satisfied.

Lemma 7.3. Suppose that condition C and the first statement of condition B hold for some sequence of intervals { T_{n1} | n ∈ ℕ }. Then for any other sequence of intervals { T_{n0} | n ∈ ℕ } satisfying T_{n0} ⊂ T_{n1} for each n, defining ℓ_{n0} = inf { b_{n}(t) | t ∈ T_{n0} } and J_{n0} = λ(T_{n0}) exp{−ℓ_{n0}^{2}/2} /(2π), we have

P {X_{n}(t) ≤ b_{n}(t), t ∈ T_{n0}} − exp{−J_{n0}} → 0 .  (26)

Proof. Write m_{n0} = sup { b_{n}(t) | t ∈ T_{n0} }. Then using the notation introduced in the statement of condition B, we have ℓ_{n1} ≤ ℓ_{n0} ≤ m_{n0} ≤ m_{n1}, and since m_{n1}/ℓ_{n1} → 0 we have both

ℓ_{n0}/ℓ_{n1} → 1  (27)

and

m_{n0}/ℓ_{n0} → 1 .  (28)

Since λ(T_{n0}) ≤ λ(T_{n1}), (27) gives

lim sup J_{n0} ≤ lim sup λ(T_{n1}) exp{−ℓ_{n0}^{2}/2}
= lim sup λ(T_{n1}) exp{−ℓ_{n1}^{2}/2} [1 + o(1)] < ∞  (29)

from condition B, and since

P { sup_{t∈T_{n1}} X_{n}(t) ≤ ℓ_{n0}} ≤ P {X_{n}(t) ≤ b_{n}(t), t ∈ T_{n0}}
≤ P { sup_{t∈T_{n1}} X_{n}(t) ≤ m_{n0}} ,

(28) means that (25) holds with J_{n} replaced by J_{n0} and ℓ_{n} replaced by both ℓ_{n0} and m_{n0}, by condition C, (29) and Lemma 7.2. This proves (26).

Lemma 7.4. Suppose condition A and the sequence of intervals { T_{n1} | n ∈ ℕ } satisfies the second statement of condition B. Then

P {X_{n}(t) ≤ b_{n}(t), t ∈ T_{n} \ T_{n1}} → 1 .

Proof. If the second statement of condition B holds then for each j = 2, 3, …, k,

P {X_{n}(t) ≤ b_{n}(t), t ∈ T_{nj}} ≥ P {X_{n}(t) ≤ B_{nj}(t), t ∈ T_{nj}} .

We intend to show that for each j the probability on the right-hand side tends to 1.

For each n, define ℓ_{nj} = inf { B_{nj}(t) | t ∈ T_{nj} } and let {I_{njk}} form a partition of T_{nj} of intervals of width h_{nj} = (2π)^{1/2}/ℓ_{nj}. Condition A and equation 13.3.2 of Cramér & Leadbetter (1967) give us that for any u > 0,

EN_{u,n,j,k} = h_{nj} exp{−u^{2}/2} /(2π) ,  (30)

where N_{u,n,j,k} is the number of upcrossings of the level u by the process { X_{n}(t) | t ∈ I_{njk} }. Assume without loss of generality that inf I_{nj} ∈ I_{njk}. Since the processes are a.s. continous, we have

P { max_{t∈I_{njk}} X_{n}(t) > u} ≤ EN_{u,n,j,k} + P (X_{n}(inf I_{nj}) > u) ,  (31)

(see for instance Piterbarg (1996), Lemma F.1, page 28). Write now ℓ_{njk} = inf_{t∈I_{njk}} B_{nj}(t).

P { sup_{t∈T_{nj}}[X_{n}(t) − B_{nj}(t)] > 0} ≤ ∑_{k} P { sup_{t∈I_{njk}}[X_{n}(t) − B_{nj}(t)] > 0}  (32)
≤ ∑_{k} P { sup_{t∈I_{njk}} X_{n}(t) > ℓ_{njk}}
≤ ∑_{k} exp{−ℓ_{njk}^{2}/2} [h_{nj}(2π)^{−1} + ℓ_{njk}^{−1}(2π)^{−1/2}]
≤ π^{−1}h_{nj} ∑_{k} exp{−ℓ_{njk}^{2}/2} ,

using (30), (31), the usual upper bound for the standard normal upper tail and the fact that each ℓ_{njk}^{−1} ≤ ℓ_{nj}^{−1} = (2π)^{−1/2}h_{nj}. The last expression is π^{−1} times the upper Riemann sum based on intervals of width h_{nj} which approximates the integral ∫_{T_{nj}} exp{−B_{nj}^{2}(t)} dt → 0 by condition B. Since the integrand is continuous and monotone over T_{nj}, the upper and lower Riemann sums approximating this integral only differ by at most two terms, corresponding to the endpoints of each interval, since the upper bound of the integrand over one interval is the lower bound for an adjacent interval. Thus the most the upper and lower Riemann sums could differ by is 2h_{nj} exp{−ℓ_{nj}^{2}/2} = O (ℓ_{nj}^{−1} exp{−ℓ_{nj}^{2}/2}) → 0 since ℓ_{nj} → ∞ by condition B. Since any lower Riemann sum is at most the integral it approximates, this completes the proof.

Theorem 2.1 is proved by using Lemma 7.3 with T_{n0} = T_{n1} and Lemma 7.4.

7.2 Proof of Theorem 4.1

Note firstly that since 0 ∈ T_{n1} for all n,

1 ≥ sup_{t∈T_{n1}} v_{n}[θ_{n}(t)] ≥ v_{n}(0) = 1 − 2/n → 1 ,

and so writing l_{n1}(x) = inf_{t∈T_{n1}} b_{n,x}(t) we have

J_{n}(x) = λ(T_{n1}) exp{−ℓ_{n1}^{2}(x)} /(2π) ∼ λ(T_{n1}) exp{−u_{n}^{2}(x)/2} /(2π) → e^{−x} .

This, together with the fact that

{a_{n} [ sup_{θ∈Θ_{n}} Z̃_{n}(θ) − a_{n}] + log 2π ≤ x} = {X_{n}(t) ≤ b_{n,x}(t), t ∈ T_{n}} ,

shows that the statement (16), with Z_{n} replaced by Z̃_{n}, follows from Theorem 2.1.

Next note that for any sequence of intervals { T_{n0} | n ∈ ℕ } satisfying T_{n0} ⊂ T_{n1}, with a_{n0} = [2 log λ(T_{n0})]^{1/2} Lemma 7.3 implies that,

P {a_{n0} [ sup_{t∈T_{n0}} Z̃_{n}(θ_{n}(t)) − a_{n0}] + log 2π ≤ x} → exp{−e^{−x}} ,

and so sup_{t∈T_{n0}} Z̃_{n}(θ_{n}(t)) = O_{p} (a_{n0}). In particular, taking T_{n0} ≡ T_{n1}, we have by Theorem 2.1 that

sup_{θ∈Θ_{n}} Z̃_{n}(θ) = sup_{t∈T_{n1}} Z̃_{n}(θ_{n}(t)) = O_{p} (a_{n}) = O_{p} (log λ(T_{n1}))^{1/2} ,

whereas if we define T_{n0} = { t ∈ T_{n1} | |t| ≤ exp{[log λ(T_{n1})]^{1/2}} }, we have

sup_{t∈T_{n0}} Z̃_{n}(θ_{n}(t)) = O_{p} (a_{n0}) = O_{p} (log(T_{n0}))^{1/2} = O_{p} (log λ(T_{n1}))^{1/4} .

So with probability tending to 1, sup_{θ∈Θ_{n}} Z̃_{n}(θ) = sup_{t∈T_{n}\T_{n0}} Z̃_{n}(θ_{n}(t)).

Note that we can write F_{n}g_{θ} = ρ_{n}(0, θ) and ρ_{n}(0, θ_{n}(t)) = r_{n}(0, t). Furthermore, sup_{t∈T \T_{n0}} r_{n}(0, t) = o (log λ(T_{n0}))^{−1} by equation (5) in condition C. So

sup_{t∈T_{n}\T_{n0}} Z̃_{n}(θ_{n}(t)) = sup_{t∈T_{n}\T_{n0}} [Z_{n}(θ_{n}(t)) + O_{p} (1) r_{n}(0, t)]
= sup_{t∈T_{n}\T_{n0}} Z_{n}(θ_{n}(t)) + o_{p} (log λ(T_{n1}))^{−1/2} .

Finally, note that sup_{t∈T_{n0}} Z_{n}(θ_{n}(t)) = sup_{t∈T_{n0}} Z̃_{n}(θ)+O_{p} (1) = O_{p} (log λ(T_{n1}))^{1/4}. Combining these gives that with probability tending to 1, sup_{θ∈Θ_{n}} Z_{n}(θ) = sup_{θ∈Θ_{n}} Z̃_{n}(θ) + o_{p} (a_{n})^{−1}, and so

a_{n} [ sup_{θ∈Θ_{n}} Z_{n}(θ) − a_{n}] = a_{n} [ sup_{θ∈Θ_{n}} Z̃_{n}(θ) − a_{n}] + o_{p} (1)

and so (16) holds with Z̃_{n} replaced by Z_{n}.

7.3 Proof of Theorem 5.1

Firstly note that by condition F each Θ_{n1}, and thus each T_{n1}, contains 0.

Conditions D and E imply C

Differentiation yields that

r_{n}^{(0,2)}(t, s) = ρ_{n}^{(1,1)}θ_{n}(s), θ_{n}(s))^{−1}ρ_{n}^{(0,2)}θ_{n}(t), θ_{n}(s)) −  (33)
ρ_{n}^{(1,1)}θ_{n}(s), θ_{n}(s))^{−2}ρ_{n}^{(0,1)}θ_{n}(s), θ_{n}(s))ρ_{n}^{(1,2)}θ_{n}(s), θ_{n}(s))

Let E_{nj}(θ) = ∫_{A_{n}} x^{j} dF_{θ}(x)/ ∫_{A_{n}} dF_{θ}, denote the j-th conditional moment of X ∼ F_{θ} given X ∈ A_{n}. Then ρ_{n}^{(0,1)}(θ, η) = ρ_{n}(θ, η) [E_{n1}(θ + η) − E_{n1}(2η)]. Note also that the derivatives with respect to θ satisfy E′_{nj}(θ) = E_{n(j+1)}(θ) − E_{nj}(θ)E_{n1}(θ) for all j ≥ 1. Repeated differentiation yields that ρ_{n}^{(i,j)}(θ, η) equals ρ(θ, η) times a linear combination of terms of the form ∏_{k=1}^{i+j} E_{na_{k}}^{b_{k}}(θ_{k}), where ∑_{k} a_{k}b_{k} ≤ (i + j) and each θ_{k} equals either θ + η, 2θ or 2η.

So the first term on the right hand side of (33) is ρ(θ_{n}(t), θ_{n}(s)) times a ratio of such linear combinations, with i + j = 2; the numerator is evaluated at values θ_{n}(s) + θ_{n}(t), 2θ_{n}(t) or 2θ_{n}(s) but the denominator is only evaluated at 2θ_{n}(s). The second term is simply a ratio of such linear combinations with i + j = 4 evaluated only at 2θ_{n}(s).

Suppose condition D holds for some { Θ_{n1} | n ∈ ℕ }. Then r_{n}^{(0,2)}(t, s) = r_{0}^{(0,2)}(t, s)[1 + o(1)], with o(1) uniform over t, s ∈ T_{n1} = { τ_{n}(θ) | θ ∈ Θ_{n1} }. A fortiori, we have also r_{n}(t, s) = r_{0}(t, s)[1 + o(1)] with the same uniformity properties. Thus for any h_{n} → 0 we have

sup_{t,t+h_{n}∈T_{n1}} |r_{n}^{(0,2)}(t, t + h_{n})/r_{0}^{(0,2)}(t, t + h_{n}) − 1|
≤ sup_{t,s∈T_{n1}} |r_{n}^{(0,2)}(t, t + s)/r_{0}^{(0,2)}(t, t + s) − 1| → 0 ,

and the first statement of condition E implies that for any h_{n} → 0,

r_{n}^{(0,2)}(t, t + h_{n}) = r_{0}^{(0,2)}(t, t + h_{n})[1 + o(1)] = −1 + o(1)

with o(1) uniform over t such that t and t+h_{n} are in T_{n1} defined above. Finally, since r_{n}^{(0,1)}(t, t) ≡ 0, a Taylor series expansion with mean-value remainder gives that r_{n}(t, t + h_{n}) = 1 − r_{n}^{(0,2)}(t, t + h_{n}^{∗}), for h_{n}^{∗} between 0 and h_{n}. Equation (34) then implies that equation (4) in condition C holds. In the same way, with appropriately uniform o(1), r_{n}(t, s) = r_{0}(t, s)[1 + o(1)] and so the second statement of condition E then implies that equation (5) holds.

Conditions D and F imply B

Two consequences of D are that

v_{n}(θ) = 1 + o(1)  (34)

and

γ_{n}(θ) = K′′(2θ)[1 + o(1)] ,  (35)

with o(1) uniform over θ ∈ Θ_{n1}. The first of these implies that for each x, the boundary functions defined in Theorem 4.1, with T_{n1} = { τ_{n}(θ) | θ ∈ Θ_{n1} } satisfy the first statement of condition B. Note also that (35) implies that

λ(T_{n1}) = τ_{n}(θ_{n1}^{+}) − τ_{n}(θ_{n1}^{−}) = ∫_{θ_{n1}^{−}}^{θ_{n1}^{+}} γ_{n}^{1/2}(θ) dθ = ∫_{θ_{n1}^{−}}^{θ_{n1}^{+}} K′′(2θ)^{1/2} dθ[1 + o(1)]
= [τ_{0}(θ_{n1}^{+}) − τ_{0}(θ_{n1}^{−})] [1 + o(1)] .  (36)

We prove the second statement of condition B in two parts. The minimum boundary height tends to infinity by construction, so we only need to verify the second statement. We partition T_{n} = { τ_{n}(θ) | θ ∈ Θ_{n} } into T_{n1}, T_{n2}^{+} = T_{n} ∩ { τ_{n}(θ) | θ_{n1}^{+} < θ < θ_{n2}^{+} }, T_{n3}^{+} = T_{n} ∩ { τ_{n}(θ) | θ ∈ Θ_{n}, θ ≥ θ_{n2}^{+} }, T_{n2}^{−} = T_{n} ∩ { τ_{n}(θ) | θ_{n2}^{−} < θ < θ_{n1}^{−} }, and T_{n3}^{−} = T_{n} ∩ { τ_{n}(θ) | θ ∈ Θ_{n}, θ ≤ θ_{n2}^{−} }.

The argument for T_{nj}^{+} also applies to T_{nj}^{−}, for j = 2, 3. We assume for simplicity that T_{n2}^{+} and T_{n3}^{+} are both non-empty. Write v_{n2}^{+} = inf_{θ∈Θ_{n2}^{+}} v_{n}(θ). We henceforth drop the + superscript. We use the inequality VarX ≥ EVar(X|A) ≥ Var(X|A)P (A), which when A = {X ∈ A_{n}} and unconditionally X ∼ F_{2θ} reduces to

γ_{n}(θ) ≤ K′′(2θ)/v_{n}(θ) .  (37)

Using this gives

λ(T_{n2}) = τ_{n}(θ_{n2}) − τ_{n}(θ_{n1}) = ∫_{θ_{n1}}^{θ_{n2}} γ_{n}^{1/2}(θ) dθ ≤ v_{n2}^{−1/2} ∫_{θ_{n1}}^{θ_{n2}} K′′(2θ)^{1/2} dθ
= v_{n2}^{−1/2} [τ_{0}(θ_{n2}) − τ_{0}(θ_{n1})] .

On T_{n2}, b_{n}(·) is bounded below by the constant (thus montone and continuous) function B_{n2} ≡ u_{n}(x). Since exp{−u_{n}^{2}(x)/2} = λ(T_{n1})^{−1}, the previous display, together with (36) give that ∫_{T_{n2}} exp{−B_{n2}^{2}(t)/2} dt → 0.

For θ > θ_{n}^{+}, the mean of F_{2θ} given by K′(2θ) is strictly greater than q_{n}^{+}.

However, the derivative

v′_{n}(θ) = ∫_{A_{n}} 2[x − K′(2θ)] dF_{2θ}
= 2v_{n}(θ) {[∫_{A_{n}} dF_{2θ}]^{−1} ∫_{A_{n}} x dF_{2θ} − K′(2θ)}
≤ 2v_{n}(θ)[q_{n}^{+} − K′(2θ)] ,  (38)

since the first term inside the curly brackets is the conditional mean of X given X ∈ A_{n}, which is at most q_{n}^{+}. This is strictly negative for any θ > θ_{n}^{+}. Thus

v′_{n}(θ) < 0 for θ ≥ θ_{n2} .  (39)

Consequently, the continuous function t ↦ b_{n}(t) is strictly increasing over T_{n3}. Hence

∫_{T_{n3}} exp{−b_{n}^{2}(t)/2} dt ≤ ∫_{θ_{n2}}^{θ_{n}(∞)} γ_{n}^{1/2}(θ) exp{−u_{n}^{2}(x)v_{n}^{−1}(θ)/2} dθ
≤ ∫_{θ_{n2}}^{θ_{n}(∞)} v_{n}^{−1/2}(θ)K′′(2θ) exp{−u_{n}^{2}(x)v_{n}^{−1}(θ)/2} dθ

using (37). Inequality (39) implies that the function θ ↦ 1/v_{n}(θ) is strictly increasing. Make a change of variable in the last integral to y = 1/v_{n}(θ). If we write y ↦ w_{n}(y) as the inverse of this transformation, then dθ = w′_{n}(y)dy, where w′_{n}(y) = −v_{n}^{2}(w_{n}(y))/v′_{n}(w_{n}(y)). This integral is at most

∫_{1/v_{n}(θ_{n2})}^{∞} K′′(2w_{n}(y))^{1/2}y^{1/2}e^{−u_{n}^{2}(x)y/2}w′_{n}(y) dy
≤ 2u_{n}^{−3}(x) ∫_{u_{n}^{2}(x)/[2v_{n}(θ_{n2})]}^{∞} z^{1/2}e^{−z} dz sup_{y≥1/v_{n}(θ_{n2})} K′′(2w_{n}(y))w′_{n}(y) (40)

An integration by parts approach (see Barndorff-Nielsen & Cox (1989) or the appendix of Stewart (2002)) gives that as c → ∞, ∫_{c}^{∞} z^{1/2}e^{−z} dz = O(c^{1/2}e^{−c}).

Implementing this enables us to rewrite the integral in (40) as

O(1)u_{n}(x)v_{n2}^{−1/2} exp{−u_{n}^{2}(x)v_{n2}^{−1}/2} .

Also the supremum in (40) can be written as

sup_{θ≥θ_{n2}} K′′(2θ)^{1/2}v_{n}^{2}(θ)/[−v′_{n}(θ)] ≤ sup_{θ≥θ_{n2}} K′′(2θ)^{1/2}v_{n}(θ)/ {2[K′(2θ) − q_{n}^{+}]}

Combining permits us to rewrite (40) as

O(1)u_{n}^{−2}(x)v_{n2}^{−1/2} exp{−u_{n}^{2}(x)v_{n2}^{−1}/2} sup_{θ≥θ_{n2}} K′′(2θ)^{1/2}v_{n}(θ)/ {2[K′(2θ) − q_{n}^{+}]} (41)

Since 1 ≥ v_{n2} → 0 by condition F, 0 ≤ v_{n2}^{−1} − 1 → ∞ and so

exp{−u_{n}^{2}(x)v_{n2}^{−1}/2} = exp{−u_{n}^{2}(x)/2} exp{−u_{n}^{2}(x)[v_{n2}^{−1} − 1]/2} = o(λ(τ_{0}(Θ_{n1})))^{−1} ,

since exp{−u_{n}^{2}(x)/2} = O(λ(T_{n1}))^{−1} = O(L_{n1})^{−1}, where L_{n1} = λ(τ_{0}(Θ_{n1})).

Finally, the supremum in (41) is O(v_{n2}^{1/2}L_{n1} log L_{n1}) by condition F. This com-

pletes the proof.

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