Hölderian invariance principles and some applications for testing epidemic changes

Alfredas Račkauskas^{1} and Charles Suquet^{2}

1 Vilnius University and Institute of Mathematics and Informatics, Department of Mathematics, Vilnius University, Naugarduko 24, Lt-2006 Vilnius, Lithuania

  Alfredas.Rackauskas@maf.vu.lt

2 Université des Sciences et Technologies de Lille, Laboratoire de Mathématiques appliquées F.R.E. CNRS 2222. Bât. M2, U.F.R. de Mathématiques, F-59655

  Villeneuve d’Ascq Cedex, France. Charles.Suquet@univ-lille1.fr

1 Introduction

The well known invariance principle of probability theory introduced by Donsker and Prokhorov received and continue to receive many compliments from both theoreticians and practitioners in very broad area of researches. Econometricians use it to tackle the “unit root” problems (see e.g. Stock (1994) and cited literature therein) to analyze structural changes (see e.g. Csörgő and Horváth (1997) and cited literature therein) or “bubbles” phenomenon (see e.g. Kirman and Teyssière (2002) and cited literature therein) etc. It is impossible to name all problems even in econometrics only, the solutions of which were obtained with help of invariance principle. In this review we present some of recent results (mainly obtained by the authors of this review) on invariance principles with respect to Hölder topologies. We emphasize a new construction of polygonal line process and applications to the structural change problems.

In more modern mathematical form, invariance principle can be formulated as follows. A sequence of stochastic processes {ξ_{n}(t), t ∈ T }_{n∈ℕ} induces a sequence of measures {μ_{n}, n ∈ ℕ} on some suitable topological function space, say, E. One proves then that the measures converge weakly to a measure μ corresponding to a limiting process, say {ξ(t), t ∈ T }. Weak convergence of measures, in the case where E is a separable metric space, gives

f(ξ_{n}) ⟶^{D} f(ξ),  (1)

for any f a real-valued function continuous μ-almost everywhere on E, where ⟶^{D} denotes convergence in distribution. Usually ξ_{n}, n ∈ ℕ, are polygonal line processes constructed from partial sums of random variables. More modern theory deals also with polygonal line processes based on data with values in a completely arbitrary, perhaps infinite-dimensional sample space.

This general approach is important for statisticians either working with data such as “functions” or “pictures” (seismographs, electrocardiograms, highdimensional biomedical data, high-frequency finance etc.) or dealing with “infinite-dimensional parameters” such as distribution function, density, characteristic function etc. For statistical inference it is usually necessary to rely on the asymptotic convergence result (1) for a certain class F of functions f. The class F aims to highlight appropriate geometry of paths of processes under consideration which on its turn, in the case of partial sums polygonal line process, reflects certain properties of the data under consideration. Examples of paths spaces customary include L_{2}, Skorohod’s D[0, 1] and C[0, 1] the space of continuous functions. Each of these spaces corresponds to a particular geometry of paths of processes. So, L_{2} spaces describe mean square properties (variance, mean square errors, energy etc.) the space D[0, 1] is responsible for extreme values and C[0, 1] deals with continuity (increments, modulus of continuity etc.) of processes.

This review bestow attention to Hölder spaces which particularly well highlight variation properties (p-variation, α-slopes) of processes. Variation of polygonal line process reflects various epidemic type changes in a structure of data (epidemic or bubble type changes in a mean, in distribution or other perhaps infinite-dimensional parameters). That’s precisely why we advocate to use Hölder continuous functionals of polygonal line processes as test statistics of structural stability in data under various epidemic alternatives. Another motivations to bestow Hölder spaces is many-sided properties of Wiener process. Paths of Wiener process are not only square integrable, bounded or continuous. They have also certain Hölder continuity. This fact to our opinion is not yet exploited in full extent.

The paper is organized as follows. In Section 2 we introduce Hölder spaces of continuous functions. Sections 3 and 4 are devoted to invariance principles with respect to Hölder spaces. Firstly we consider classical polygonal line process constructed from partial sums of random variables (elements with values in a Banach space). Necessary and sufficient conditions for their convergence in a Hölder space are discussed. Next we advocate another construction of polygonal line process. Differently from the classical case, this new construction puts abscissas of vertices at random points. It is shown that so constructed polygonal line process in many aspects behaves better than the classical one.

Sections 5 and 6 present in a rather broad perspective some applications of Hölderian invariance principles to the problem of testing stability in the sample under epidemic alternatives.

2 Some facts about Hölder spaces

The introduction of Hölder spaces answers the purpose to quantify the global smoothness of functions by controlling their modulus of uniform continuity. Let us precise this by an informal description of the most familiar case. For fixed 0 < α < 1, H^{α} is the set of functions x : [0, 1] → ℝ such that |x(t) − x(s)| ≤ K|t − s|^{α} for some constant K depending only on x and α. The best constant K in this uniform estimate defines a semi-norm on the vector space H^{α}. By adding |x(0)| to this semi-norm we obtain a norm ∥x∥_{α} which makes H^{α} a non separable Banach space. Clearly if 0 < α < β < 1, H^{β} is topologically embedded in H^{α} and all these Hölder spaces are topologically embedded in the classical Banach space C of continuous functions [0, 1] → ℝ.

To remedy the non separability drawback of H^{α}, one introduces its subspace H^{α,o} of functions x such that |x(t) − x(s)| = o(|t − s|^{α}) uniformly. This subspace is closed (hence also a Banach space for the same norm ∥x∥_{α}) and separable.

One interesting feature of the spaces H^{α,o} is the existence of a basis of triangular functions, see Ciesielski (1960). It is convenient to write this basis as a triangular array of functions, indexed by the dyadic numbers. Let us denote by D_{j} the set of dyadic numbers in [0, 1] of level j, i.e.

D_{0} = {0, 1}, D_{j} = {(2l − 1)2^{−j}; 1 ≤ l ≤ 2^{j−1}}, j ≥ 1.

Write for r ∈ D_{j}, j ≥ 0,

r^{−} := r − 2^{−j}, r^{+} := r + 2^{−j}.

For r ∈ D_{j}, j ≥ 1, the triangular Faber-Schauder functions Λ_{r} are continuous, piecewise affine with support [r^{−}, r^{+}] and taking the value 1 at r:

Λ_{r}(t) = \{ \begin{matrix}2^{j}(t − r^{−}) & if t ∈ (r^{−}, r]; \\ 2^{j}(r^{+} − t) & if t ∈ (r, r^{+}]; \\ 0 & else. \end{matrix}

When j = 0, we just take the restriction to [0, 1] in the above formula, so

Λ_{0}(t) = 1 − t, Λ_{1}(t) = t, t ∈ [0, 1].

The sequence {Λ_{r}; r ∈ D_{j}, j ≥ 0} is a Schauder basis of C. Each x ∈ C has a unique expansion

x = ∑_{j=0}^{∞} ∑_{r∈D_{j}} λ_{r}(x)Λ_{r},  (2)

with uniform convergence on [0, 1]. The Schauder scalar coefficients λ_{r}(x) are given by

λ_{r}(x) = x(r) − \frac{x(r^{+}) + x(r^{−})}{2}, r ∈ D_{j}, j ≥ 1,  (3)

and in the special case j = 0 by

λ_{0}(x) = x(0), λ_{1}(x) = x(1).  (4)

The partial sum ∑_{j=0}^{n} in the series (2) gives the linear interpolation of x by a polygonal line between the dyadic points of level at most n.

[figure: Λ_{r}(t); 1; 0; t; r^{−}; r; r^{+}; 2^{−j}]

Fig. 1. The Faber-Schauder triangular function Λ_{r}

Ciesielski (1960) proved that {Λ_{r}; r ∈ D_{j}, j ≥ 0} is also a Schauder basis of each space H^{α,o} (hence the convergence (2) holds in the H^{α} topology when x ∈ H^{α,o}) and that the norm ∥x∥_{α} is equivalent to the following sequence norm :

∥x∥_{α}^{seq} := sup_{j≥0} 2^{jα} max_{r∈D_{j}} |λ_{r}(x)|.

This equivalence of norms provides a very convenient discretization procedure to deal with Hölder spaces and is extended in Račkauskas and Suquet (2001b) to the more general setting of Hölder spaces of Banach space valued functions x, with a modulus of continuity controlled by some weight function ρ.

Let (𝔹, ∥ ∥) be a separable Banach space. We write C_{𝔹} for the Banach space of continuous functions x : [0, 1] → 𝔹 endowed with the supremum norm ∥x∥_{∞} := sup{∥x(t)∥; t ∈ [0, 1]}. Let ρ be a real valued non decreasing function on [0, 1], null and right continuous at 0. Put

ω_{ρ}(x, δ) := sup_{\begin{matrix}s,t∈[0,1], \\ 0<t−s<δ \end{matrix}} \frac{∥x(t) − x(s)∥}{ρ(t − s)}.

Denote by H_{𝔹}^{ρ} the set of continuous functions x : [0, 1] → 𝔹 such that ω_{ρ}(x, 1) < ∞. The set H_{𝔹}^{ρ} is a Banach space when endowed with the norm

∥x∥_{ρ} := ∥x(0)∥ + ω_{ρ}(x, 1).

Define

H_{𝔹}^{ρ,o} = {x ∈ C_{𝔹} : lim_{δ→0} ω_{ρ}(x, δ) = 0}.

Then H_{𝔹}^{ρ,o} is a closed separable subspace of H_{𝔹}^{ρ}. We shall abbreviate C_{ℝ}, H_{ℝ}^{ρ} and H_{ℝ}^{ρ,o} in C, H^{ρ} and H^{ρ,o} correspondingly. Our main examples of Hölder spaces use as weight function ρ = ρ_{α,β}, 0 < α < 1, β ∈ ℝ defined by:

ρ_{α,β}(h) := h^{α} ln^{β}(c/h), 0 < h ≤ 1,

for a suitable constant c. For ρ = ρ_{α,β}, we shall write H_{𝔹}^{α,β} and H_{𝔹}^{α,β,o} for H_{𝔹}^{ρ} and H_{𝔹}^{ρ,o} respectively and we abbreviate H_{𝔹}^{α,0,o} in H_{𝔹}^{α,o}. As above, the subscript 𝔹 will be omitted when 𝔹 = ℝ.

In what follows, we assume that the modulus of smoothness ρ satisfies the following technical conditions where c_{1}, c_{2} and c_{3} are positive constants:

ρ(0) = 0, ρ(δ) > 0, 0 < δ ≤ 1;  (5)
ρ is non decreasing on [0, 1];  (6)
ρ(2δ) ≤ c_{1}ρ(δ), 0 ≤ δ ≤ 1/2;  (7)
∫_{0}^{δ} \frac{ρ(u)}{u} du ≤ c_{2}ρ(δ), 0 < δ ≤ 1;  (8)
δ ∫_{δ}^{1} \frac{ρ(u)}{u^{2}} du ≤ c_{3}ρ(δ), 0 < δ ≤ 1.  (9)

For instance, elementary computations show that the functions ρ_{α,β} satisfy Conditions (5) to (9), for a suitable choice of the constant c, namely c ≥ exp(β/α) if β > 0 and c > exp(−β/(1 − α)) if β < 0. For any ρ satisfying (5) to (9), we have the equivalence of norms :

∥x∥_{ρ} ∼ ∥x∥_{ρ}^{seq} := sup_{j≥0} \frac{1}{ρ(2^{−j})} max_{r∈D_{j}} ∥λ_{r}(x)∥ ,

where the 𝔹-valued coefficients λ_{r}(x) are still defined by (3) and (4).

The space E = H_{𝔹}^{ρ,o} may be used as the topological framework for functional central limit theorems. Among various continuous functionals f to which we can then apply the convergence (1), let us mention the norms f_{1}(x) = ∥x∥_{ρ} and f_{2}(x) = ∥x∥_{ρ}^{seq}, which are closely connected to the test statistics UI(n, ρ) and DI(n, ρ) proposed below for the detection of epidemic changes. Other examples of Hölder continuous functional and operators like p-variation, fractional derivatives are given in Hamadouche (2000).

To deal with convergences in distribution of stochastic processes considered as random elements in H_{𝔹}^{ρ,o}, a key tool is the following tightness criterion established in Račkauskas and Suquet (2003b).

Theorem 1. Suppose the Banach space 𝔹 is separable. Then the sequence (ξ_{n})_{n≥1} of random elements in H_{𝔹}^{ρ,o} is tight if and only if it satisfies the two following conditions.

(i) For each dyadic t ∈ [0, 1], the sequence of 𝔹-valued random variables (ξ_{n}(t))_{n≥1} is tight on 𝔹.

(ii)For each positive ε,

lim_{J→∞} sup_{n≥1} P \{sup_{j>J} \frac{1}{ρ(2^{−j})} max_{r∈D_{j}} ∥λ_{r}(ξ_{n})∥ > ε\} = 0.

3 Hölderian invariance principles

Let X_{1}, …, X_{n}, … be i.i.d. random elements in the separable Banach space 𝔹. Set S_{0} = 0,

S_{k} = X_{1} + ⋯ + X_{k}, for k = 1, 2, …

and consider the partial sums processes

ξ_{n}(t) = S_{[nt]} + (nt − [nt])X_{[nt]+1}, t ∈ [0, 1].

In the familiar case where 𝔹 is the real line ℝ, classical Donsker-Prohorov invariance principle states, that if E X_{1} = 0 and E X_{1}^{2} = σ^{2} < ∞, then

n^{−1/2}σ^{−1}ξ_{n} ⟶^{D} W,  (10)

in C[0, 1], where (W (t), t ∈ ℝ) is a standard Wiener process. The finiteness of the second moment of X_{1} is clearly necessary here, since (10) yields that n^{−1/2}ξ_{n}(1) satisfies the central limit theorem.

Lamperti (1962) was the first who considered the convergence (10) with respect to some Hölderian topology. He proved that if 0 < α < 1/2 and E |X_{1}|^{p} < ∞, where p > p(α) := 1/(1/2 − α), then (10) takes place in H^{α,o}. This result was derived again by Kerkyacharian and Roynette (1991) by another method based on Ciesielski (1960) analysis of Hölder spaces by triangular functions. Further generalizations were given by Erickson (1981) (partial sums processes indexed by [0, 1]^{d}), Hamadouche (2000) (weakly dependent sequence (X_{n})), Račkauskas and Suquet (2002) (Banach space valued X_{i}’s and Hölder spaces built on the modulus ρ(h) = h^{α} ln^{β}(1/h)). The following result is proved in Račkauskas and Suquet (2002a).

Theorem 2. Let 0 < α < 1/2 and p(α) = 1/(1/2 − α). Then

n^{−1/2}σ^{−1}ξ_{n} ⟶_{n→∞}^{D} W in the space H^{α,o}

if and only if E X_{1} = 0 and

lim_{t→∞} t^{p(α)}P{|X_{1}| ≥ t} = 0.  (11)

Condition (11) yields the existence of moments E |X_{1}|^{p} for any 0 ≤ p < p(α). If α approaches 1/2 then p(α) → ∞. Hence, stronger invariance principle requires higher moments.

To describe more general results, some preparation is needed. Let 𝔹′ be the topological dual of 𝔹. For a random element X in 𝔹 such that for every f ∈ 𝔹′, E f(X) = 0 and E f^{2}(X) < ∞, the covariance operator Q = Q(X) is the linear bounded operator from 𝔹′ to 𝔹 defined by Qf = E f(X)X, f ∈ 𝔹′. A random element X ∈ 𝔹 (or covariance operator Q) is said to be pregaussian if there exists a mean zero Gaussian random element Y ∈ 𝔹 with the same covariance operator as X, i.e. for all f, g ∈ 𝔹′, E f(X)g(X) = E f(Y)g(Y). Since the distribution of a centered Gaussian random element is defined by its covariance structure, we denote by Y_{Q} a zero mean Gaussian random element with covariance operator Q.

For any pregaussian covariance Q there exists a 𝔹-valued Brownian motion W_{Q} with parameter Q, a centered Gaussian process indexed by [0, 1] with independent increments such that W_{Q}(t) − W_{Q}(s) has the same distribution as |t − s|^{1/2}Y_{Q}.

We say that X_{1} satisfies the central limit theorem in 𝔹, which we denote by X_{1} ∈ CLT(𝔹), if n^{−1/2}S_{n} converges in distribution in 𝔹. This implies that E X_{1} = 0 and X_{1} is pregaussian. It is well known (e.g. Ledoux and Talagrand, 1991) that the central limit theorem for X_{1} cannot be characterized in general in terms of integrability of X_{1} and involves the geometry of the Banach space 𝔹.

We say that X_{1} satisfies the functional central limit theorem in 𝔹, which we denote by X_{1} ∈ FCLT(𝔹), if n^{−1/2}ξ_{n} converges in distribution in C_{𝔹}. Kuelbs (1973) extended the classical Donsker-Prohorov invariance principle to the case of 𝔹-valued partial sums by proving that n^{−1/2}ξ_{n} converges in distribution in C_{𝔹} to some Brownian motion W if and only if X_{1} ∈ CLT(𝔹) (in short X_{1} ∈ CLT(𝔹) if and only if X_{1} ∈ FCLT(𝔹)). Of course in Kuelbs theorem, the parameter Q of W is the covariance operator of X_{1}.

The convergence in distribution of n^{−1/2}ξ_{n} in H_{𝔹}^{ρ,o}, which we denote by X_{1} ∈ FCLT(𝔹, ρ), is clearly stronger than X_{1} ∈ FCLT(𝔹).

An obvious preliminary requirement for the FCLT in H_{𝔹}^{ρ,o} is that the 𝔹valued Brownian motion has a version in H_{𝔹}^{ρ,o}. From this point of view, the critical ρ is ρ_{c}(h) = \sqrt{h ln(1/h)} due to Lévy’s Theorem on the modulus of uniform continuity of the Brownian motion. So our interest will be restricted to functions ρ generating a weaker Hölder topology than ρ_{c}. More precisely, we consider the following class R of functions ρ.

Definition 1. We denote by R the class of functions ρ satisfying

i) for some 0 < α ≤ 1/2, and some function L which is normalized slowly varying at infinity,

ρ(h) = h^{α}L(1/h), 0 < h ≤ 1,  (12)

ii) θ(t) = t^{1/2}ρ(1/t) is C^{1} on [1, ∞), iii) there is a β > 1/2 and some a > 0, such that θ(t) ln^{−β}(t) is non decreasing

on [a, ∞).

The following result is proved in Račkauskas and Suquet (2002a).

Theorem 3. Let ρ ∈ R. Then X_{1} ∈ FCLT(𝔹, ρ) if and only if X_{1} ∈ CLT(𝔹) and for every A > 0,

lim_{t→∞} tP{∥X_{1}∥ ≥ At^{1/2}ρ(1/t)} = 0.  (13)

If ρ ∈ R with α < 1/2 in (12) then it suffices to check condition (13) for A = 1 only. Of course the special case 𝔹 = ℝ and ρ(h) = h^{α} gives back Theorem 2.

In the case where ρ(h) = h^{1/2} ln^{β}(c/h) with β > 1/2, Condition (13) is equivalent to

E exp(d ∥X_{1}∥^{1/β}) < ∞, for each d > 0.

Let us note, that for the spaces 𝔹 = L_{p}(0, 1), 2 ≤ p < ∞, as well as for each finite dimensional space, Condition (13) yields X_{1} ∈ CLT(𝔹). On the other hand it is well known that for some Banach spaces existence of moments of any order does not guarantee central limit theorem. It is also worth noticing that like in Kuelbs FCLT, all the influence of the geometry of the Banach space 𝔹 is absorbed by the condition X_{1} ∈ CLT(𝔹).

It would be useful to extend the Hölderian FCLT to the case of dependent X_{i}’s. This was done by Hamadouche (2000) in the special case where 𝔹 = ℝ and under weak dependence (association and α-mixing). The result presented in Račkauskas and Suquet (2002a) provides a general approach for 𝔹-valued X_{i}’s and any dependence structure, subject to obtaining a good estimate of the partial sums. Laukaitis and Račkauskas (2002) obtained Hölderian FCLT for a polygonal line process based on residual partial sums of a stationary Hilbert space valued autoregression (ARH(1)) and applied it to the problem of testing stability of ARH(1) model under different types of alternatives.

4 Adaptive self-normalized partial sums process

Various partial sums processes can be built from the sums S_{n} = X_{1} + ⋯ + X_{n} of independent identically distributed mean zero random variables. In Račkauskas and Suquet (2001a) the so called adaptive self-normalized partial sums processes are defined. Self-normalized means that the classical normalization by \sqrt{n} is replaced by

V_{n} = (X_{1}^{2} + ⋯ + X_{n}^{2})^{1/2}.

Adaptive means that the vertices of the corresponding random polygonal line have their abscissas at the random points V_{k}^{2}/V_{n}^{2} (0 ≤ k ≤ n) instead of the deterministic equispaced points k/n. By this construction the slope of each line adapts itself to the value of the corresponding random variable.

By ζ_{n} (respect. ξ_{n}) we denote the random polygonal partial sums process defined on [0, 1] by linear interpolation between the vertices (V_{k}^{2}/V_{n}^{2}, S_{k}), k = 0, 1, …, n (respect. (k/n, S_{k}), k = 0, 1, …, n), where

S_{k} = X_{1} + ⋯ + X_{k}, V_{k}^{2} = X_{1}^{2} + ⋯ + X_{k}^{2}.

For the special case k = 0, we put S_{0} = 0, V_{0} = 0. By convention the random functions V_{n}^{−1}ξ_{n} and V_{n}^{−1}ζ_{n} are defined to be the null function on the event {V_{n} = 0}. Figure 2 displays the polygonal lines n^{−1/2}ξ_{n} and V_{n}^{−1}ζ_{n} built on a simulated sample of size n = 1000 of the symmetric distribution given by P(|X_{1}| > t) = 0.5 t^{−2.3}1_{[1,∞)}(t).

>From this picture one can observe how adaptive partition of the time interval improves slopes of polygonal line process.

Fig. 2. Partial sums processes n^{−1/2}ξ_{n} and V_{n}^{−1}ζ_{n}

Recall that “X_{1} belongs to the domain of attraction of the normal distribution” ( denoted by X_{1} ∈ DAN) means that there exists a sequence b_{n} ↑ ∞ such that

b_{n}^{−1}S_{n} ⟶^{D} N(0, 1).

The following result is proved in Račkauskas and Suquet (2001a).

Theorem 4. Assume that ρ satisfies Conditions (5) to (9) and

lim_{j→∞} \frac{2^{j}ρ^{2}(2^{−j})}{j} = ∞.  (14)

If X_{1} is symmetric then

V_{n}^{−1}ζ_{n} ⟶^{D} W, in H^{ρ,o}

if and only if X_{1} ∈ DAN.

When tested with ρ(h) = h^{1/2} ln^{β}(c/h), (14) reduces to j^{2β−1} → ∞. Due to the inclusions of Hölder spaces, this shows that Theorem 4 gives the best result possible in the scale of the separable Hölder spaces H^{α,β,o}. Moreover, no high order moments are needed except the condition X_{1} ∈ DAN which due to well known O’Briens result is equivalent to

V_{n}^{−1} max_{1≤k≤n} |X_{k}| ⟶^{P} 0.

It seems worth noticing here, that without adaptive construction of the polygonal process, the existence of moments of order bigger than 2 is necessary for Hölder weak convergence. Indeed, if the process V_{n}^{−1}ξ_{n} converges weakly to W in H^{α,o} for some α > 0, then its maximal slope n^{−1/2}V_{n}^{−1} max_{1≤k≤n} |X_{k}| converges to zero in probability. This on its turn yields V_{n}^{−1} max_{1≤k≤n} |X_{k}| → 0 almost surely, and according to Maller and Resnick (1984), E X_{1}^{2} < ∞. Hence n^{−1}V_{n}^{2} converges almost surely to E X_{1}^{2} by the strong law of large numbers. Therefore n^{−1/2}ξ_{n} converges weakly to W in H^{α,o} and by Theorem 2 the moment restriction (11) is necessary.

Naturally it is very desirable to remove the symmetry assumption in Theorem 4. Although the problem remains open, we can propose the following partial result in this direction (for more on this problem see Račkauskas and Suquet (2001a)).

Theorem 5. If for some ε > 0, E |X_{1}|^{2+ε} < ∞, then for any β > 1/2, V_{n}^{−1}ζ_{n} converges weakly to W in the space H^{1/2,β,o}.

Some extensions of this result for the non i.i.d. case are given in Račkauskas and Suquet (2003a).

5 Hölder norm test statistics

An important question in the large area of change point problems involves testing the null hypothesis of no parameter change in a sample versus the alternative that parameter changes do take place at an unknown time. For a survey we refer to the books by Brodsky and Darkhovsky (1993, 2000) or Csörgő and Horváth (1997). A widely accepted methodology to construct tests is to start with a functional limit theorems for processes build from partial sums S(k), k = 0, …, n. Functionals of paths of these processes can be used as tests statistics for null hypothesis under certain alternatives. In this way various test statistics are proposed in the literature. To name a few, the KP SS statistic by Kwiatkowski et al. (1992) and the V/S statistic by Giraitis et al. (2003) are based on the L_{2} space continuous functionals of partial sums process whereas Hurst (1951) R/S exploits D[0, 1] continuous functionals.

It is clear that a larger choice of admissible functionals on paths of polygonal line process corresponds to a bigger class of possible alternatives. Therefore it is important to have functional limit theorems established in a sufficiently rich functional framework. Working in the stronger topological framework of a space of Hölder functions allows to consider functionals based on increments (Ŝ(k) − Ŝ(j), 0 ≤ k < j ≤ n) as well as on their ϑ-slopes (|Ŝ(k) − Ŝ(j)|/ϑ(k − j), 0 ≤ k < j ≤ n), where ϑ is certain function.

Using this methodology in Račkauskas and Suquet (2003c) we propose some new statistics for testing change in the mean under the so called epidemic alternative. More precisely, given a sample X_{1}, X_{2}, …, X_{n} of random variables, we want to test the standard null hypothesis of constant mean

(H_{0}): X_{1}, …, X_{n} all have the same mean denoted by μ_{0},

against the epidemic alternative

(H_{A}): there are integers 1 < k^{∗} < m^{∗} < n and a constant μ_{1} ≠ μ_{0} such that E X_{i} = μ_{0} + (μ_{1} − μ_{0})1_{{k^{∗}<i≤m^{∗}}}, i = 1, 2, …, n.

Let us note, that other type of epidemics can be considered as well. If either k^{∗} = 1 or m^{∗} = n, then we have one abrupt change in the mean. This is a quite well studied case in the literature. If eiher k^{∗}/n ≥ c_{0} > 0 or m^{∗}/n ≤ c_{1} < 1 then the problem can be considered with existing tests based on weighted partial sums processes. The situation when one naturally comes to Hölder topology concerns the cases, where either l^{∗}/n → 0 or l^{∗}/n → 1. Here l^{∗} := m^{∗} − k^{∗} denotes the length of the epidemic.

To motivate the definition of some test statistics, let us assume just for a moment that the changes times k^{∗} and m^{∗} are known. Suppose moreover under (H_{0}) that the (X_{i}, 1 ≤ i ≤ n) are i.i.d. with finite variance σ^{2}. Under (H_{A}), suppose that the (X_{i}, i ∈ 𝕀_{n}) and the (X_{i}, i ∈ 𝕀_{n}^{c}) are separately i.i.d. with the same finite variance σ^{2}, where

𝕀_{n} := {i ∈ {1, …, n}; k^{∗} < i ≤ m^{∗}}, 𝕀_{n}^{c} := {1, …, n} \ 𝕀_{n}.

Then we simply have a two sample problem with known variances. It is then natural to accept (H_{0}) for small values of the statistics |Q| and to reject it for large ones, where

Q := \frac{S(𝕀_{n}) − l^{∗}S(n)/n}{\sqrt{l^{∗}}} − \frac{S(𝕀_{n}^{c}) − (n − l^{∗})S(n)/n}{\sqrt{n − l^{∗}}}.

After some algebra, Q may be recast as

Q = \begin{matrix}√ & \overline{n} \\ √ & \overline{l^{∗}(n − l^{∗})} \end{matrix}(S(𝕀_{n}) − \frac{l^{∗}}{n} S(n))[\sqrt{1 − l^{∗}/n} + \sqrt{l^{∗}/n}]

As the last factor into square brackets ranges between 1 and \sqrt{2}, we may drop it and introducing the notation

t_{k} = t_{n,k} := \frac{k}{n}, 0 ≤ k ≤ n,

replace |Q| by the statistics

R = n^{−1/2} \frac{|S(m^{∗}) − S(k^{∗}) − S(n)(t_{m^{∗}} − t_{k^{∗}})|}{[(t_{m^{∗}} − t_{k^{∗}})(1 − (t_{m^{∗}} − t_{k^{∗}}))]^{1/2}}.

Now in the more realistic situation where k^{∗} and m^{∗} are unknown it is reasonable to replace R by taking the maximum over all possible indexes for k^{∗} and m^{∗}. This leads to consider

UI(n, 1/2) := max_{1≤i<j≤n} \frac{|S(j) − S(i) − S(n)(t_{j} − t_{i})|}{[(t_{j} − t_{i})(1 − (t_{j} − t_{i}))]^{1/2}}.

It is worth to note that the same statistics arises from likelihood arguments in the special case where the observations X_{i} are Gaussian, see Yao (1993). The asymptotic distribution of UI(n, 1/2) is unknown, due to difficulties caused by the denominator (for historical remarks see Csörgő and Horváth p.183).

In our setting, the X_{i}’s are not supposed to be Gaussian. Moreover it seems fair to pay something in terms of normalization when passing from R to n^{−1/2}UI(n, 1/2). Intuitively the cost should depend on the moment assumptions made about the X_{i}’s. To discuss this, consider the polygonal partial sums process ξ_{n}. Then UI(n, 1/2) appears as the discretization through the grid (t_{k}, 0 ≤ k ≤ n) of the functional T_{1/2}(ξ_{n}) where

T_{1/2}(x) := sup_{0<s<t<1} \frac{|x(t) − x(s) − (x(1) − x(0))(t − s)|}{[(t − s)(1 − (t − s))]^{1/2}}.  (15)

This functional is continuous in the Hölder space H^{1/2,o}. Obviously finite dimensional distributions of n^{−1/2}σ^{−1}ξ_{n} converge to those of a standard Brownian motion W. However Lévy’s theorem on the modulus of uniform continuity of W implies that H^{1/2,o} has too strong a topology to support a version of W. So n^{−1/2}ξ_{n} cannot converge in distribution to σW in the space H^{1/2,o}. This forbid us to obtain limiting distribution for T_{1/2}(n^{−1/2}ξ_{n}) by invariance principle in H^{1/2,o} via continuous mapping. As discussed in the previous section, Hölderian invariance principles do exist for, roughly speaking, all the scale of Hölder spaces H^{ρ,o}, provided that the weight function ρ satisfies lim_{h↓0} ρ(h)(h ln |h|)^{−1/2} = ∞. This leads us to replace T_{1/2}(n^{−1/2}ξ_{n}) by T_{α}(n^{−1/2}ξ_{n}) obtained substituting the denominator in (15) by (t − s)^{α}(1 − (t − s))^{α}. Going back to the discretization we finally suggest the class UI (uniform increments) of statistics which includes particularly

UI(n, α) := max_{1≤i<j≤n} \frac{|S(j) − S(i) − S(n)(j/n − i/n)|}{[(j/n − i/n)(1 − (j − i)/n)]^{α}}.

and similar ones UI(n, ρ) built with a general weight ρ(h) instead of h^{α}. Together with UI we consider the class of DI (dyadic increments) statistics, which includes particularly

DI(n, α) = max_{1≤j≤log n} 2^{jα} max_{r∈D_{j}} |S(nr) − \frac{1}{2}(S(nr + n2^{−j}) + S(nr − n2^{−j}))|,

where D_{j} is the set of dyadic numbers of the level j. We use here log to denote the logarithm with basis 2, while ln denotes the natural logarithm (log(2^{t}) = t = ln(e^{t})). DI(n, α) and UI(n, α) have similar asymptotic behaviors. Moreover, dyadic increments statistics are of particular interest since their limiting distributions are completely specified (see Theorem 11 below).

To simplify notation put

ϱ(h) := ρ(h(1 − h)), 0 ≤ h ≤ 1.

For ρ ∈ R, define (recalling that t_{k} = k/n, 0 ≤ k ≤ n),

UI(n, ρ) = max_{1≤i<j≤n} \frac{|S(j) − S(i) − S(n)(t_{j} − t_{i})|}{ϱ(t_{j} − t_{i})}

and

DI(n, ρ) = max_{1≤j≤log n} \frac{1}{ρ(2^{−j})} max_{r∈D_{j}}|S(nr) − \frac{1}{2}(S(nr^{+}) + S(nr^{−}))|.

Let W = (W (t), t ∈ [0, 1]) be a standard Wiener process and B = (B(t), t ∈ [0, 1]) the corresponding Brownian bridge B(t) = W (t) − tW (1), t ∈ [0, 1]. Consider for ρ in R, the following random variables

UI(ρ) := sup_{0<t−s<1} \frac{|B(t) − B(s)|}{ϱ(t − s)}  (16)

and

DI(ρ) = sup_{j≥1} \frac{1}{ρ(2^{−j})} max_{r∈D_{j}}|W (r) − \frac{1}{2}W (r^{+}) − \frac{1}{2}W (r^{−})| = ∥B∥_{ρ}^{seq}. (17)

These variables serve as limiting for uniform increment (UI) and dyadic increment (DI) statistics respectively. No analytical form seems to be known for the distribution function of UI(ρ), whereas the distribution of DI(ρ) is completely specified below.

To obtain limiting distribution for statistics UI(n, ρ) and DI(n, ρ) we consider a bit stronger null hypothesis, namely

(H′_{0}): X_{1}, …, X_{n} are i.i.d. random variables.

Theorem 6. Under (H′_{0}), assume that ρ ∈ R and for every A > 0,

lim_{t→∞} t P{|X_{1}| > At^{1/2}ρ(1/t)} = 0.

Then

σ^{−1}n^{−1/2}UI(n, ρ) ⟶_{n→∞}^{D} UI(ρ) and σ^{−1}n^{−1/2}DI(n, ρ) ⟶_{n→∞}^{D} DI(ρ),

where σ^{2} = Var X_{1} and UI(ρ), DI(ρ) are defined by (16) and (17).

Remark 1. In the case where the variance σ^{2} is unknown the results remain valid if σ^{2} is substituted by its standard estimator σ^{2}.

Remark 2. Under (H_{0}), the statistics n^{−1/2}UI(n, ρ) keeps the same value when each X_{i} is substituted by X′_{i} := X_{i} − E X_{i}. This property is only asymptotically true for n^{−1/2}DI(n, ρ). For practical use of DI(n, ρ), it is preferable to replace X_{i} by X′_{i} if μ_{0} is known or else by X_{i}−\overline{X} where \overline{X} = n^{−1}(X_{1}+⋯+X_{n}). This will avoid a bias term which may be of the order of |E X_{1}| ln^{−β} n in the worst cases.

To see a consistency of tests to reject null hypothesis versus epidemic alternative H_{A} for large values of n^{−1/2}UI(n, ρ), we naturally assume that the numbers of observations k^{∗}, m^{∗} − k^{∗}, n − m^{∗} before, during and after the epidemic go to infinity with n. Write l^{∗} := m^{∗} − k^{∗} for the length of the epidemic.

Theorem 7. Let ρ ∈ R. Assume under (H_{A}) that the X_{i}’s are independent and σ_{0}^{2} := sup_{k≥1} Var X_{k} is finite. If

lim_{n→∞} n^{1/2} \frac{h_{n}}{ρ(h_{n})} = ∞, where h_{n} := \frac{l^{∗}}{n} (1 − \frac{l^{∗}}{n} ),  (18)

then

n^{−1/2}UI(n, ρ) ⟶_{n→∞}^{P} ∞, and n^{−1/2}DI(n, ρ) ⟶_{n→∞}^{P} ∞.

When ρ(h) = h^{α}, (18) allows us to detect short epidemics such that l^{∗} = o(n) and l^{∗}n^{−δ} → ∞, where δ = (1 − 2α)(2 − 2α)^{−1}. Symmetrically one can detect long epidemics such that n − l^{∗} = o(n) and (n − l^{∗})n^{−δ} → ∞.

When ρ(h) = h^{1/2} ln^{β}(c/h) with β > 1/2, (18) is satisfied provided that h_{n} = n^{−1} ln^{γ} n, with γ > 2β. This leads to detection of short epidemics such that l^{∗} = o(n) and l^{∗} ln^{−γ} n → ∞ as well as of long ones verifying n−l^{∗} = o(n) and (n − l^{∗}) ln^{−γ} n → ∞.

One of the disadvantages of DI and UI statistics is the moment restriction for their convergence. The introduction of adaptive self-normalized analogues for these statistics resolves this problem.

For any index set A ⊂ {1, …, n}, define

V^{2}(A) := ∑_{i∈A} X_{i}^{2}.

Then V_{k}^{2} = V^{2}({1, …, k}), V_{0}^{2} = 0. To simplify notation we write v_{k} for the random points of [0, 1]

v_{k} := \frac{V_{k}^{2}}{V_{n}^{2}} , k = 0, 1, …, n.

For ρ ∈ R define

SUI(n, ρ) := max_{1≤i<j≤n} \frac{|S(j) − S(i) − S(n)(v_{j} − v_{i})|}{ϱ(v_{j} − v_{i})}.

Introduce for any t ∈ [0, 1],

τ_{n}(t) := max{i ≤ n; V_{i}^{2} ≤ tV_{n}^{2}}

and

J_{n} := log(V_{n}^{2}/X_{n:n}^{2}) where X_{n:n} := max_{1≤k≤n} |X_{k}|.

Now define

SDI(n, ρ) := max_{1≤j≤J_{n}} \frac{1}{ρ(2^{−j})} max_{r∈D_{j}}|S(τ_{n}(r)) − \frac{1}{2}[S(τ_{n}(r^{+})) + S(τ_{n}(r^{−}))]|.

Theorem 8. Assume that under (H′_{0}), μ_{0} = 0 and the random variables X_{1}, …, X_{n} are either symmetric and belong to the domain of attraction of normal law or E |X_{1}|^{2+ε} < ∞ for some ε > 0. Then for every ρ ∈ R,

V_{n}^{−1}SUI(n, ρ) ⟶_{n→∞}^{D} UI(ρ) and V_{n}^{−1}SDI(n, ρ) ⟶_{n→∞}^{D} DI(ρ).

We restricted the null hypothesis assuming that μ_{0} = 0. Practically the reduction to this case by centering requires the knowledge of μ_{0}. Although this seems a quite reasonable assumption, it should be pointed out that this knowledge was not required for the UI and DI statistics.

Theorem 9. Let ρ ∈ R. Under (H_{A}), assume that the X_{i}’s satisfy

S(𝕀_{n}) = l^{∗}μ_{1} + O_{p}(\sqrt{l^{∗}}), S(𝕀_{n}^{c}) = O_{p}(\sqrt{n − l^{∗}}),  (19)

and

\frac{V^{2}(𝕀_{n})}{l^{∗}} ⟶_{n→∞}^{P} b_{1}, \frac{V^{2}(𝕀_{n}^{c})}{n − l^{∗}} ⟶_{n→∞}^{P} b_{0},  (20)

for some finite constants b_{0} and b_{1}. Assume that l^{∗}/n converges to a limit c ∈ [0, 1] and when c = 0 or 1, assume moreover that

lim_{n→∞} n^{1/2} \frac{h_{n}}{ρ(h_{n})} = ∞, where h_{n} := \frac{l^{∗}}{n} (1 − \frac{l^{∗}}{n} ).  (21)

Then

V_{n}^{−1}SUI(n, ρ) ⟶_{n→∞}^{P} ∞.

Note that in Theorem 9, no assumption is made about the dependence structure of the X_{i}’s. It is easy to verify the general hypotheses (19) and (20) in various situations of weak dependence like mixing or association. Under independence of the X_{i}’s (without assuming identical distributions inside each block 𝕀_{n} and 𝕀_{n}^{c}), it is enough to have sup_{k≥1} E |X_{k}|^{2+ε} < ∞ and E V^{2}(𝕀_{n})/l^{∗} → b_{1}, E V^{2}(𝕀_{n}^{c})/(n − l^{∗}) → b_{0}. This follows easily from Lindebergh’s condition for the central limit theorem (giving (19)) and of Theorem 5 p.261 in Petrov (1975) giving the weak law of large numbers required for the X_{i}’s.

Theorem 10. Let ρ ∈ R and set X′_{i} := X_{i}−E (X_{i}). Under (H_{A}) with μ_{0} = 0, assume that the random variables X′_{i} are independent identically distributed and that E |X′_{1}|^{2+ε} is finite for some ε > 0. Then under (21),

V_{n}^{−1}SDI(n, ρ) ⟶_{n→∞}^{P} ∞.

The distribution function of DI(ρ) may be conveniently expressed in terms of the error function:

erf x = \frac{2}{\sqrt{π}} ∫_{0}^{x} exp(−s^{2}) ds.

Theorem 11. Let c = lim sup_{j→∞} j^{1/2}/θ(2^{j}), where θ(t) = t^{1/2}ρ(1/t).

i) If c = ∞ then DI(ρ) = ∞ almost surely. ii) If 0 ≤ c < ∞, then DI(ρ) is almost surely finite and its distribution func-

tion is given by

P(DI(ρ) ≤ x) = ∏_{j=1}^{∞}{erf(θ(2^{j})x)}^{2^{j−1}}, x > 0.

The distribution function of DI(ρ) is continuous with support [c\sqrt{ln 2}, ∞).

Figure 3 below displays a table of the distribution function of DI(ρ) with ρ(h) = h^{0.4}.

Theorems 6 to 11 are proved in Račkauskas and Suquet (2003a).

6 Testing changes of infinite dimensional parameters

As mentioned in the introduction, polygonal line processes based on sample taking values in an infinite dimensional sample space gives tools for statisticians working with functional data or using nonparametric approach to the testing problems. In this section we just provide one example showing how general approach to the simple problem of testing stability in the mean allows

\begin{matrix}x & F (x) & x & F (x) & x & F (x) & x & F (x) \\ 1.06 & 0.0000075 & 1.4 & 0.5110985 & 1.74 & 0.9424161 & 2.08 & 0.9940030 \\ 1.08 & 0.0000573 & 1.42 & 0.5615323 & 1.76 & 0.9495661 & 2.1 & 0.9947547 \\ 1.1 & 0.0003004 & 1.44 & 0.6084394 & 1.78 & 0.9558321 & 2.12 & 0.995413 \\ 1.12 & 0.0011653 & 1.46 & 0.6515890 & 1.8 & 0.9613221 & 2.14 & 0.9959895 \\ 1.14 & 0.0035455 & 1.48 & 0.6909226 & 1.82 & 0.9661314 & 2.16 & 0.9964943 \\ 1.16 & 0.0088640 & 1.5 & 0.7265088 & 1.84 & 0.9703438 & 2.18 & 0.9969363 \\ 1.18 & 0.0188969 & 1.52 & 0.7585053 & 1.86 & 0.9740332 & 2.2 & 0.9973232 \\ 1.2 & 0.0353857 & 1.54 & 0.7871269 & 1.88 & 0.9772645 & 2.22 & 0.9976618 \\ 1.22 & 0.0596012 & 1.56 & 0.8126221 & 1.9 & 0.9800944 & 2.24 & 0.9979582 \\ 1.24 & 0.0920395 & 1.58 & 0.8352537 & 1.92 & 0.9825729 & 2.26 & 0.9982175 \\ 1.26 & 0.1323403 & 1.6 & 0.8552866 & 1.94 & 0.9847436 & 2.28 & 0.9984444 \\ 1.28 & 0.1794090 & 1.62 & 0.8729782 & 1.96 & 0.9866448 & 2.3 & 0.9986428 \\ 1.3 & 0.2316604 & 1.64 & 0.8885729 & 1.98 & 0.9883099 & 2.32 & 0.9988163 \\ 1.32 & 0.2872905 & 1.66 & 0.9022985 & 2. & 0.9897684 & 2.34 & 0.9989680 \\ 1.34 & 0.3445122 & 1.68 & 0.9143643 & 2.02 & 0.9910458 & 2.36 & 0.9991006 \\ 1.36 & 0.4017209 & 1.7 & 0.9249609 & 2.04 & 0.9921647 & 2.38 & 0.9992165 \\ 1.38 & 0.4575890 & 1.72 & 0.9342602 & 2.06 & 0.9931446 & 2.4 & 0.9993176 \end{matrix}

Fig. 3. Distribution function F (x) = P(DI(h^{0.4}) ≤ x)

to obtain prototypical Cramér-von Mises test statistic for change in distribution problem. In Račkauskas and Suquet (2003d) more examples are available including the prototypical Kolmogorov-Smirnov test.

Let X_{1}, …, X_{n} be the real valued random variables with distribution functions F_{1}, …, F_{n} respectively. Consider the null hypothesis

(H_{0}): F_{1} = ⋯ = F_{n} = F

and the following epidemic alternative:

(H_{A}): there are integers 1 < k^{∗} < m^{∗} < n such that F_{1} = F_{2} = ⋯ = F_{k^{∗}} = F_{m^{∗}+1} = ⋯ = F_{n}, F_{k^{∗}+1} = ⋯ = F_{m^{∗}} and F_{k^{∗}} ≠ F_{k^{∗}+1}.

For simplicity let us consider only the case of continuous function F. Then we can restrict to uniform empirical process.

Let U_{1}, …, U_{n} be independent uniformly on [0, 1] distributed random variables. Define u_{0} = 0,

u_{v}(t) = ∑_{j≤v} (1_{{U_{j}≤t}} − t), 1 ≤ v ≤ n; t ∈ [0, 1].

For r ∈ D set

D^{2}u_{nr} = u_{nr} − \frac{1}{2}(u_{nr^{+}} + u_{nr^{−}})

and

T^{2}(n, r) = ∫_{0}^{1}|D^{2}u_{nr}(t)|^{2} dt.

The Cramér-von Mises type dyadic increments statistic is defined by

T^{2}(n, ρ) = max_{1≤j≤log n} \frac{1}{ρ^{2}(2^{−j})} max_{r∈D_{j}} T^{2}(n, r).

Its limiting distribution is completely defined by the limiting distribution of the classical Cramér-von Mises statistic, namely, by the distribution function

L_{2}(x) = P {∫_{0}^{1} B^{2}(t) dt ≤ x} , x ≥ 0,

where B(t), t ∈ [0, 1] is a standard Brownian bridge. The distribution function L_{2}(x) is well known and several its representations are available (see Shorack and Wellner, 1986).

Theorem 12. If ρ ∈ R then

lim_{n→∞} P(n^{−1}T^{2}(n, ρ) ≤ x) = L_{2,ρ}(x),

for each x > 0, where

L_{2,ρ}(x) = ∏_{j=1}^{∞}[L_{2}(2θ(2^{j})x)]^{2^{j−1}}.

The result is an application of functional central limit Theorem 3 to the random elements Y_{1}, …, Y_{n} with values in the space L_{2}(0, 1) defined by

X_{k}(t) = 1_{{U_{k}≤t}} − t, t ∈ [0, 1], k = 1, …, n.

If ∑_{j} jθ^{−2}(2^{j}) < ∞, for practical uses the following estimate of the tails of L_{2,ρ} can be helpful. Define for x > 0

L_{2,ρ}^{(J)}(x) = ∏_{j=1}^{J}[L_{2}(2θ(2^{j})x)]^{2^{j−1}}.

Proposition 1. Put for J = 0, 1, 2, …

c_{J} := inf_{γ>0} \{γ + 8 ∑_{j=J+1}^{∞} \frac{2^{j}}{θ^{2}(2^{j})} exp (\frac{−γθ^{2}(2^{j})}{4})\}.

i) If x > 0 then

1 − L_{2,ρ}(x) ≤ 4 exp \{ − \frac{3x}{4c_{0}} \}.

ii) For each x > 0 and J ≥ 1

(1 − 4 exp \{ − \frac{3x}{8c_{J}} \})L_{2,ρ}^{(J)}(x) ≤ L_{2,ρ}(x) ≤ L_{2,ρ}^{(J)}(x).

The null hypothesis (H_{0}) is rejected for big values of the statistic T^{2}(n, ρ) calculated with U_{i} = F (X_{i}), i = 1, …, n.

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