GEORGIAN MATHEMATICAL JOURNAL: Vol. 3, No. 4, 1996, 343-362

ON THE MODE-CHANGE PROBLEM FOR RANDOM MEASURES

TS. HAHUBIA AND R. MNATSAKANOV

Abstract. The classical change-point problem in modern terms, i.e., the mode-change problem, is stated for sufficiently general set-indexed random processes, namely for random measures. A method is shown for solving this problem both in the general form and for the intensity of compound Poisson random measures. The results obtained are novel for the change-point problem, too.

1. Introduction

1. The goal of this paper is to state the classical change-point problem in modern terms, i.e., to formulate the mode-change problem for sufficiently general set-indexed random processes by which we mean random signed measures with realizations almost surely (a.s.) belonging to D(A) which is the space of set functions “outer continuous with inner limits” and where A, a domain of set functions, is a family of Borel subsets of a d-dimensional compact (d ≥ 1) under some entropy condition. An attempt is made to indicate one way of solving this problem in the general form and in a particular case, namely, for the intensity of compound Poisson random measures. The obtained results are novel for the classical change-point problem, too.

When studying various phenomena of nature and social life, one may come across a situation in which a certain observed process is flowing in time or evolving in space, or both flowing in time and evolving in space. In that case one is faced with the problem of understanding the variability character of this process in the sense whether it evolves in one and the same mode or there exists some moment (or several moments or some domain) after which a different mode of behavior sets in, that is to say, after which “the mode changes.” For example, when studying chronologically arranged archaeological data collected from a specific locality, one is interested in

343 finding out whether any abrupt change in the culture occurred there, and if it did, then after which moment. When studying a literary text, one is concerned with determining the identity of the author of the text or detecting “inserts” of other authors. The problem of determining the moment and place of the change in the seismic conditions in a certain area is an example of identifying the change in the mode both in time and in space.

1991 Mathematics Subject Classification. 62G05, 62G10, 60G10.

Key words and phrases. Lévy random measure, Lévy bridge, Lévy broken bridge, compound Poisson random measure, change-set.

1072-947X/96/0700-0343$09.50/0 \begin{matrix}c \\ ◯ \end{matrix} 1996 Plenum Publishing Corporation

The statistical “change-set” problem or, as otherwise termed, the “modechange” problem serves as good model for describing and solving the abovementioned problems. The mode-change problem consists of two parts:

(i) testing the hypotheses on the existence of a change-set and (ii) constructing a “sufficiently good,” say, consistent estimator for the

change-set if the latter exists.

The exact formulation of this problem for random signed measures is given in Section 3 and for the intensity of compound Poisson random measures in Section 4.

Set-indexed empirical processes, partial-sum processes, Brownian processes, Lévy (infinitely divisible) processes, compound Poisson processes, compound point processes, etc., — all these are examples of random signed measures.

Here one of the ways for solving the mode-change problem is indicated using the results of the fundamental papers by Dudley [1–4], Adler et al. [5, 6], and Bass and Pyke [7–9] devoted to the investigation of the set-indexed random processes. These results are stated as preliminaries in the greater part of Section 2.

2. Preliminaries

Let B be the Borel σ-field formed by subsets of some compact Γ ⊂ ℝ^{d}, say, Γ = I^{d}, where I^{d} = [0, 1]^{d}, d ≥ 1, and let A be a subfamily of B, A ⊂ B, satisfying the following conditions:

(a) A is a collection of closed subsets of Γ;

(b) A is closed with respect to the metric

d_{L}(A, B) = |A△B|, A, B ∈ A,

where | · | denotes the Lebesgue measure, A△B = (A ∩ \overline{B}) ∪ (\overline{A} ∩ B) is the symmetric difference, and \overline{A} = Γ\A;

(c) for each δ > 0 there is a finite subset A_{δ} (δ-net) of A such that whenever A ∈ A there exists B ∈ A_{δ} with A ⊂ B^{0} ⊂ B ⊂ A^{δ}, where B^{0} is the interior of B with respect to the relative topology on Γ and A^{δ} is the open δ-neighborhood of A;

(d) there are constants K > 0 and r ∈ (0, 1) such that for a sufficiently small δ > 0 we have H(δ) ≤ Kδ^{−r}, where H(δ) = ln(♯A_{δ}^{∗}) is the entropy and ♯A_{δ}^{∗} is the cardinality of the smallest δ-net A_{δ}^{∗} of A.

The family of closed convex sets in I^{d}, d > 1, for which r = (d − 1)/2 with d = 2, is an example of A covered by the above conditions (see [2]).

The conditions (a)–(c) imply that A is a compact separable and complete and, in particular, totally bounded under the metric d_{L}. The condition (d) will be discussed below (see Remark 2.5).

Definition 2.1. A function X : A → ℝ is said to be outer continuous with inner limits at A ∈ A if

(i) A ⊂ A_{n} ∈ A, d_{L}(A, A_{n}) → 0 implies X(A_{n}) → X(A) as n → ∞; (ii) A_{n} ∈ A, A_{n} ⊂A^{0}, d_{L}(A, A_{n})→0 implies the existence of lim_{n→∞} X(A_{n}).

Let D(A)={X:A → ℝ such that X is outer continuous with inner limits at each A ∈ A}.

Remark 2.1. The space D(A) is the intersection of the space D(A) from [8] and the space of cadlag functions from [6].

By analogy with [8] and [9] the metric on D(A) is defined by

d_{D}(X, Y) = d_{L}(G(X), G(Y)),

where G(X) is a graph of the function X, i.e., the closure of the set {(A, X(A)), A ∈ A} with respect to the metric

ρ((A_{1}, r_{1}), (A_{2}, r_{2})) = d_{L}(A_{1}, A_{2}) + |r_{1} − r_{2}|

on A × ℝ. The metric d_{D} is a pseudometric on D(A). We shall identify the functions X and Y for d_{D}(X, Y) = 0. (Note, for example, that for a ∈ [−1, 1] the functions X_{a}(t) = sin(\frac{1}{t}), t ∈ (0, 1], X_{a}(0) = a, are all identified by d_{D}).

Remark 2.2. Since the metric d_{L} is weaker than the Hausdorff metric d_{H} considered in [8, 9], the above metric d_{D} is also weaker than its analog in the same papers.

For the space D(A) the σ-field F = G^{−1}(B_{G}) is induced by the graph function G(·) and the Borel σ-field B_{G} generated by the family G of closed subsets of A × ℝ.

A weak convergence of probabilities P_{n} on (D(A), F) to the probability P is denoted by P_{n} ⟶^{w} P and understood in the usual sense: if ∫ fdP_{n} ⟶ ∫ fdP for all functionals f continuous and bounded on D(A) with respect to the metric d_{D}. If {X_{n}}_{n≥0} is a sequence of set-indexed random processes whose sample functions are from the space D(A) a.s., then X_{n} is said to converge weakly to X_{0} (denoted by X_{n} ⟶^{w} X_{0}) if the distribution laws of X_{n} converge weakly to those of X_{0}.

From [8] we have the following proposition:

Proposition 2.1. X_{n} ⟶^{w} X_{0} if:

(i) P_{A_{1},...,A_{k}}^{X_{n}} that are finite-dimensional distributions of X_{n} converge to those of X_{0} for all finite k and A_{1}, …, A_{k} ∈ A^{∗}, where A^{∗} is a dense subset of A with respect to d_{L};

(ii) the sequence {X_{n}}_{n≥1} is tight, that is to say, for all ε > 0 there exists a compact subset F^{ε} of D(A) such that

inf_{n} P {X_{n} ∈ F^{ε}} ≥ 1 − ε.

Definition 2.2. The mapping X : B × Ω → ℝ is a random measure (r.m.) if:

(i) X(A, ·) is a random variable (r.v.) for each A ∈ B;

(ii) X(·, ω) is a finitely additive measure on B a.s.

Let the restriction of the r.m. X to A ⊂ B be also denoted by X. Define var (X(A), A ∈ A) as total variation of X with respect to (Γ, B).

On account of Theorem 8.2 from [9] we have the following criterion for a sequence of r.m.’s to be tight:

Proposition 2.2. A sequence of r.m.’s {X_{n}(A), A ∈ A}_{n≥1} is tight if for any ε > 0 there exist positive numbers M_{ε}, δ_{ε}, an integer-valued function N(δ), and a positive-valued function h(δ) such that the conditions

sup_{n} P \{X_{n} ∈ \overline{F} _{k}^{ε}\} < ε/3, k = 1, 2, 3,

are satisfied for the following three subsets of D(A):

F_{1}^{ε} = {X : var (X(A), A ∈ A) ≤ M_{ε}},
F_{2}^{ε} = {X : every element of G(X) is within a distance ≤ δ_{ε}

from some (A_{i}, X(A_{i})), i = 1, …, N(δ_{ε}), with respect to ρ},

F_{3}^{ε} = {X : sup_{A_{i}⊂B⊂A_{i}^{h(δε)}} |X(B) − X(A_{i})| < δ_{ε}, i = 1, …, N(δ_{ε})}

(A_{1}, …, A_{N(δ_{ε})} ∈ A is some δ_{ε}-net of A possibly depending on X).

Definition 2.3. A r.m. X is called a Lévy r.m. if:

(i) it has independent increments, i.e., r.v.’s X(A_{1}), …, X(A_{k}) are independent for pairwise disjoint sets A_{1}…. , A_{k} ∈ B;

(ii) it is stochastically continuous, i.e., for any sequence A_{n} ∈ B and all ε > 0 we have P {|X(A_{n}) − X(A_{0})| > ε} → 0 whenever d_{L}(A_{n}, A_{0}) → 0 for an arbitrary set A_{0} ∈ B.

From [5–7] we have the following two propositions:

Proposition 2.3. If a Lévy r.m. actually exists, then the logarithm of its characteristic functions can be expressed for each A ∈ B by

ln E{e^{iuX(A)}} = Ψ_{A}^{(1)}(u) + Ψ_{A}^{(2)}(u),  (2.1)

where

Ψ_{A}^{(1)}(u) = iuM(A) − (1/2)u^{2}V (A),  (2.2)
Ψ_{A}^{(2)}(u) = ∫_{|x|<1}(e^{iux} − 1 − iux)ν(A, dx) +
+ ∫_{|x|≥1}(e^{iux} − 1)ν(A, dx)  (2.3)

and M is a real-valued function on B continuous in the metric d_{L}, V is a finite nonnegative measure on Γ, absolutely continuous with respect to the Lebesgue measure; for all A ∈ B, ν(A, ·) is a positive measure on ℝ with ν(Γ, {0}) = 0 and satisfying

∫ (x^{2} ∧ 1)ν(Γ, dx) < ∞;

for every Borel set B ⊂ ℝ with a positive distance to {0}, ν(·, B) is a nonnegative finite Borel measure on Γ, absolutely continuous with respect to the Lebesgue measure.

Proposition 2.4. There exists a set-indexed process with independent increments and logarithm of the characteristic function of form (2.1). Moreover, this process has a version which is a Lévy r.m. on B.

Remark 2.3. From (2.1)–(2.3) it is clear that every Lévy r.m. X can be expressed as the sum of the independent Gaussian r.m. X_{G} and the non-Gaussian Lévy r.m. X\overline{G}, i.e., X = X_{G} + X\overline{G}.

Definition 2.4. The Lévy r.m. is P -homogeneous with real numbers μ, σ > 0 and the Lévy measure ν(dx) and called the Lévy (P, μ, σ, ν)-r.m. if for each A ∈ B the logarithm of its characteristic function (2.1) has the form

ln E{e^{iuX(A)}} = P (A){iuμ − σ^{2}u^{2}/2 +
+ ∫_{|x|<1}(e^{iux} − 1 − iux)ν(dx) + ∫_{|x|≥1}(e^{iux} − 1)ν(dx)},  (2.4)

where P is some probability measure on B, absolutely continuous with respect to the Lebesgue measure.

According to Remark 2.3 the Lévy (P, μ, σ)-r.m. is the Gaussian r.m. if ν ≡ 0 and it is the Brownian P -r.m. if, additionally, μ = 0, σ ≠ 0. If ν ≢ 0 and μ = σ = 0, then it is the non-Gaussian (P, ν)-r.m.

Definition 2.5. For α ∈ (0, 2) the non-Gaussian (P, ν)-r.m. is called the Lévy (P, α)-stable r.m. if ν is the stable Lévy measure of the exponent α having the form

ν(dx) = \begin{cases}c_{1}x^{−(1+α)}dx, & x > 0, \\ c_{2}|x|^{−(1+α)}dx, & x < 0, \end{cases}  (2.5)

where c_{1}, c_{2} are positive finite constants.

If c_{1} = c_{2} = c, then we say that the Lévy (P, α)-stable r.m. X is symmetric with logarithm of characteristic function having the form

ln E{e^{iuX(A)}} = −cP (A)|u|^{α}, α ∈ (0, 2).  (2.6)

If α ∈ (1, 2), then EX(A) exists, and for the symmetric Lévy (P, α)- stable r.m. EX(A) ≡ 0. Note that the converse statement is also valid, that is to say, if EX(A) ≡ 0, then the Lévy (P, α)-stable r.m. is symmetric.

If the Lévy measure ν is concentrated at a single point x_{0}, i.e., if

ν(dx) = λdδ(x − x_{0}),  (2.7)

where λ > 0 and δ(·) is the Dirac measure, then the non-Gaussian (P, ν)- r.m. is the Poisson P -r.m. π with the intensity J(A) = Eπ(A) = λP (A).

Let {x_{n}}_{n≥1} be a sequence of independent identically distributed (i.i.d.) r.v.’s not depending on the sequence of Poisson P -r.m.’s {π_{n}(A), A ∈ A}, n ≥ 1, with λ = n for each n = 1, 2, . . .. Then

K_{n}(A) = ∑_{i≤π_{n}(A)} x_{i}  (2.8)

is called the compound Poisson P -r.m. Its intensity is equal to

J_{n}(A) = EK_{n}(A) = naP (A),  (2.9)

if Ex_{i} = a. Note that the compound Poisson P -r.m. can also be written in the form

K_{n}(A) = ∑_{i≤π_{n}(Γ)} x_{i}I_{A}(ξ_{i}),  (2.10)

where ξ_{i} are d-dimensional r.v.’s not depending on x_{i} and π_{n}(Γ) and having the distribution P and I_{A}(ξ_{i}) = I{ξ_{i} ∈ A}, where I{·} is the indicator function.

Under certain conditions compound Poisson r.m.’s can be approximated by compound point r.m.’s, namely, it is easy to prove

Proposition 2.5. If the conditions (a)−(d) are applied to the subfamily A and r.v.’s x_{i} lie in the domain of normal attraction of the stable law of the index α ∈ (1, 2), then the three normalized r.m.’s

(π_{n}(Γ))^{−1/α} ∑_{i≤π_{n}(Γ)}(x_{i} − a)I_{A}(ξ_{i}),

n^{−1/α} ∑_{i≤π_{n}(Γ)}(x_{i} − a)I_{A}(ξ_{i}), n^{−1/α} ∑_{i≤n}(x_{i} − a)I_{A}(ξ_{i}), A ∈ A,

converge weakly to the same limit as n → ∞.

When the domain of the Lévy (P, μ, σ, ν)-r.m. X is the whole Borel σalgebra, then its sample functions are not all bounded as a rule. A question arises here how its domain A ⊂ B should be restricted for its sample functions to be bounded and continuous or to be at least elements of the space D(A) a.s. The following propositions from [1, 2, 7–9] give the answer to this question.

Proposition 2.6. Let the entropy H of the subfamily A ⊂ B satisfy the condition

∫_{0}^{1} (H(x)/x)^{1/2}dx < ∞.  (2.11)

Then the sample functions of the Lévy (P, μ, σ, ν)-r.m. X(A), A ∈ A, are continuous with respect to the metric d_{L} a.s. if and only if it is the Gaussian (P, μ, σ)-r.m.

Therefore the sample functions of the non-Gaussian (P, ν)-r.m. are discontinuous a.s.

Remark 2.4. If condition (2.11) is not satisfied for the entropy H of A, then according to [4] sample functions of the Gaussian (P, μ, σ)-r.m. X(A), A ∈ A, may not even be bounded.

Proposition 2.7. If the Lévy measure ν is such that

∫ (|x| ∧ 1)ν(dx) < ∞,  (2.12)

then the sample functions of the non-Gaussian (P, ν)-r.m. belong to the space D(B) a.s. (i.e., no entropy condition is required).

Thus according to (2.5) and (2.7) the sample functions of the Lévy (P, α)- stable r.m. X(A) with α < 1 and of the Poisson P -r.m. π(A), A ∈ B, are elements of the space D(B) a.s.

Proposition 2.8. Let the entropy H of the subfamily A ⊂ B satisfy the condition

∫_{0}^{1} (H(x)/x)^{1−1/α}dx < ∞,  (2.13)

where α ∈ (1, 2); then the sample functions of the Lévy (P, α)-stable r.m. X(A), A ∈ A, belong to the space D(A) a.s.

Proposition 2.9. Let there be positive constants K, r such that for the entropy H of the subfamily A ⊂ B the condition

H(x) ≤ Kx^{−r}  (2.14)

is satisfied for sufficiently small x > 0. Then the sample functions of the non-Gaussian (P, ν)-r.m. X(A), A ∈ A, belong to the space D(A) a.s.

Remark 2.5. If in condition (2.14) we additionally require that r < 1, then conditions (2.11) and (2.13), too, will be fulfilled for the entropy H. Thus according to Remarks 2.3 and 2.4 the condition (d) above ensures the belonging of the sample functions of the Lévy (P, μ, σ, ν)-r.m. X(A), A ∈ A, to the space D(A) a.s.

Let us assume that in the sequel the subfamily A ⊂ B will always be subjected to the conditions (a)–(d) above. We shall consider the Lévy (P, μ, σ, ν)-r.m. L(A), A ∈ A, with EL ≡ 0. By analogy with Dudley [3] we introduce

Definition 2.6. The r.m.

L(A) = L(A) − P (A)L(Γ), A ∈ A,  (2.15)

is called the Lévy (P, μ, σ, ν)-bridge.

If L is the Gaussian (P, 0, σ)-r.m., then L is the Brownian (P, σ,)-bridge and denoted by L_{G}. If L is the non-Gaussian (P, ν)-r.m., then due to the fact that its sample functions are discontinuous a.s. the r.m. (2.15) is called the Lévy broken (P, ν)-bridge and denoted by L_{b}. Thus, generally speaking, L = L_{G} + L_{b}, where L_{G} and L_{b} are independent of each other.

It is clear that L is a stochastically continuous r.m. but without independent increments. Note that L(∅) = L(Γ) = 0.

One can easily prove the following two propositions:

Proposition 2.10. Let {L_{n}}_{n≥1} be the sequence of Lévy (P, μ, σ, ν)-

r.m.’s with P_{n} ⟶^{w} P_{0}, μ_{n} ⟶ μ_{0}, σ_{n} ⟶ σ_{0}, ν_{n} ⟶^{w} ν_{0} as n → ∞, and

let L_{0} be the Lévy (P_{0}, μ_{0}, σ_{0}, ν_{0})-r.m.; then L_{n} ⟶^{w} L_{0}.

Remark 2.6. When Γ = [0, 1], A = {[0, t], 0 ≤ t ≤ 1}, and P_{n} is the Lebesgue measure, a similar proposition was proved by Skorokhod [10].

Let {X_{n}}_{n≥1} be the sequence of r.m.’s with EX_{n} ≡ 0, and {Δ_{n}(A) − EΔ_{n}(A), A ∈ A}_{n≥1}, be the sequence of Lévy (P_{n}, μ_{n}, σ_{n}, ν_{n})-r.m.’s with

P_{n} ⟶^{w} P_{0}, μ_{n} ⟶ μ_{0}, σ_{n} ⟶ σ_{0}, ν_{n} ⟶^{w} ν_{0}, and nonnegative EΔ_{n} be of

order n^{β} as n → ∞ (β > 0) and EΔ_{n}(·)/EΔ_{n}(Γ) ⟶^{w} Q(·), where Q is some probability measure on Γ. These two sequences are independent of each other. Consider the sequence of r.m.’s {Y_{n}}_{n≥1} where Y_{n} = X_{n} + ε_{n}Δ_{n} and ε_{n} ↘ 0 as n → ∞.

Proposition 2.11. If X_{n} ⟶^{w} L, where L is the Lévy (P_{0}, μ, σ, ν)-r.m. with EL ≡ 0 and ε_{n} is asymptotically equivalent to {EΔ_{n}(Γ)}^{−1}, then

X_{n}(·) − P_{n}(·)X_{n}(Γ) ⟶^{w} L(·)

and

Y_{n}(·) − P_{n}(·)Y_{n}(Γ) ⟶^{w} L(·) + M(·),

where L is the Lévy (P_{0}, μ, σ, ν)-bridge and M = Q − P_{0}.

3. The Mode-Change Problem in a General Form

Let {X_{n}(A), A ∈ A}_{n≥1} and {Δ_{n}(A), A ∈ A}_{n≥1}, be two independent sequences of r.m.’s with sample functions from the space D(A) a.s. Consider two probability spaces (D(A), F, P_{kn}), k = 0, 1, where P_{0n} and P_{1n} are induced by X_{n} and X_{n} + ε_{n}Δ_{n}, respectively, with ε_{n} ↘ 0 as n → ∞.

The mode-change problem can be stated in a general form as follows: On account of the realizations of some observed r.m.’s X̃_{n}(A), A ∈ A, we have:

(i) to test the null hypotheses

H_{0n} : X̃_{n}(A) =^{d} X_{n}(A)

against the alternatives

H_{1n} : there exists some change-set C ∈ A, such that X̃_{n}(A) =^{d} X_{n}(A) + ε_{n}Δ_{n}(A ∩ C)

(=^{d} denotes the equality of the distribution laws); (ii) to construct a consistent estimator Ĉ_{n} for the change-set C. The consistency of the estimator is understood in the sense that d_{L}(Ĉ_{n},C)

→ 0 in probability as n → ∞. The mode-change problem reduces to the classical change-point one when Γ = [0, 1] and A = {[0, t], 0 ≤ t ≤ 1}, C = (t_{0}, 1], where t_{0} is the unknown change-point.

Remark 3.1. One can formulate the mode-change problem with C not necessarily from A, but in that case it is not quite clear how the consistency of the estimator should be determined for the change-set. In such a situation one should evidently take into consideration each time the specific character of the class A. In the classical change-point problem this question is set aside, since here the structure of the class A does not affect the understanding of the consistency of the estimator for the change-point.

Remark 3.2. The mode-change problem reduces to that of testing the hypotheses when the change-set C = Γ and to that of goodness of fit when C = ∅.

From our point of view, to solve the mode-change problem we have to study the asymptotic behavior of an object such that it will be possible not only to distinguish between the hypotheses H_{0n} and H_{1n} but also to construct a consistent estimator for the change-set C.

In the case of the existence of E_{0}X_{n} = λ_{n} ≢ 0 and E_{1}Δ_{n} = δ_{n} ≢ 0, where E_{k} denotes the mathematical expectation when the hypotheses H_{kn}, k = 0, 1, are valid, we think that such an object is the observed r.m. X̃_{n} which is normalized in an appropriate manner and is centered by the conditional expectation, i.e., we have

T_{n}(A) = n^{−γ}{X̃_{n}(A) − E_{0}[X̃_{n}(A)|X̃_{n}(Γ)]}, A ∈ A, γ > 0. (3.1)

Then for our purpose it appears sufficient to center by the linear regression a_{n}(A)X̃_{n}(Γ) where the set-function a_{n}(A) is determined from the condition

E_{0}[a_{n}(A)X̃_{n}(Γ)] = E_{0}[E_{0}[X̃_{n}(A)|X̃_{n}(Γ)]].

Therefore

a_{n}(A) = λ_{n}(A)/λ_{n}(Γ),

and instead of the r.m. T_{n} from (3.1) we can consider

Z_{n}(A) = n^{−γ}{X̃_{n}(A) − (λ_{n}(A)/λ_{n}(Γ))X̃_{n}(Γ)}, A ∈ A. (3.2)

Obviously, E_{0}Z_{n} ≡ 0 and E_{1}Z_{n}(A) = n^{−γ}ε_{n}δ_{n}(C)M_{n}(A), where M_{n}(A) = δ_{n}(A ∩ C)/δ_{n}(C) − λ_{n}(A)/λ_{n}(Γ).

If δ_{n} and λ_{n} are nonnegative or nonpositive measures on A and for the normalized measures P_{n}(A) = λ_{n}(A)/λ_{n}(Γ) and Q_{n}(A ∩ C) = δ_{n}(A ∩ C)/δ_{n}(C) the condition

Q_{n}(A ∩ C) > P_{n}(A ∩ C)

is fulfilled for all A ∈ A for which A∩C ≠ ∅, then the set-function M_{n}(A) = Q_{n}(A ∩ C) − P_{n}(A) has the unique maximum which is attained only on the change-set C. Indeed, for all A ≠ C we have

M_{n}(C) − M_{n}(A) =
= Q_{n}(C\(A ∩ C)) − P_{n}(C\(A ∩ C)) + P_{n}(A ∩ \overline{C}) > 0.  (3.3)

By the same argument it is clear that M_{n} attains its minimum only on the complement to the change-set C and M_{n}(C) = −M_{n}(\overline{C}). Therefore the form of E_{1}Z_{n}(A) is the key to constructing a consistent estimator for the change-set C.

Remark 3.3. From the above reasoning it is clear why the observed r.m. X̃_{n} should not be centered by E_{0}X̃_{n}. Though for the r.m.

Y_{n} = n^{−γ}(X̃_{n} − E_{0}X̃_{n})  (3.4)

we have E_{0}Y_{n} ≡ 0 and E_{1}Y_{n}(A) = n^{−γ}ε_{n}δ_{n}(C)Q_{n}(A ∩ C), the change-set C is not the unique set of attaining the maximum for Q_{n}(A ∩ C). Any set A ∈ A containing C is also a set of attaining the maximum.

Let E_{0}X_{n} = λ_{n} and E_{1}Δ_{n} = δ_{n} exist, λ_{n} and δ_{n} being nonnegative or nonpositive measures on A, both being of order n^{β} (β > 0) as n → ∞. The normalized measures P_{n} ⟶^{w} P, Q_{n} ⟶^{w} Q where the probability measures P and Q are absolutely continuous with respect to the Lebesgue measure and

Q(A ∩ C) > P (A ∩ C)  (3.5)

for all A ∈ A for which A ∩ C ≠ ∅.

Further assume that the sequence of r.m.’s {n^{−γ}(X_{n} − λ_{n})}_{n≥1} (0 < γ < β) satisfies the conditions of Proposition 2.2 and for all finite k ≥ 1 and A_{1}, …, A_{k} ∈ A^{∗} (A^{∗} is a dense subset of A) their finite-dimensional distributions P_{n}^{A_{1},...,A_{k}} converge to those of some Lévy (P, μ, σ, ν)-r.m. L with EL ≡ 0. The r.m.’s n^{−γ}(Δ_{n} − δ_{n}) are the Lévy (P_{n}, μ_{n}, σ_{n}, ν_{n})-r.m.’s

with P_{n} ⟶^{w} P, μ_{n} → μ_{0}, σ_{n} → σ_{0}, ν_{n} ⟶^{w} ν_{0}, as n → ∞, where σ_{0} > 0, μ_{0}

are real numbers and ν_{0} is some Lévy measure.

Under the above assumptions one way of solving the mode-change problem is via the following two theorems:

Theorem 3.1. If n^{−γ}ε_{n}|δ_{n}(C)| → 1 as n → ∞, then under the hypotheses H_{0n}:

Y_{n} ⟶^{w} L, Z_{n} ⟶^{w} L

and under the alternatives H_{1n}:

Y_{n} ⟶^{w} L + m, Z_{n} ⟶^{w} L + M in the case δ_{n} > 0

and

Y_{n} ⟶^{w} L − m, Z_{n} ⟶^{w} L − M in the case δ_{n} < 0,

where L is the Lévy (P, μ, σ, ν)-bridge and

m(A) = Q(A ∩ C), M(A) = Q(A ∩ C) − P (A), A ∈ A.

Theorem 3.2. If n^{β−γ}ε_{n} → ∞ as n → ∞, then the estimators

Ĉ_{n} = \begin{cases}arg max_{A∈A} Z_{n}(A) & if δ_{n} > 0, \\ arg min_{A∈A} Z_{n}(A) & if δ_{n} < 0 \end{cases}

and

C′_{n} = arg max_{A∈A} |Z_{n}(A)| (if the sign of δ_{n} is unknown)

are consistent estimators for the change-set C.

Example. Consider the classical change-point problem:

Let x_{1}, …, x_{n} be independent observations on the r.v. ξ distributed with the density f̃(x). On account of these observations we have:

(i) to test the null hypothesis

H_{0} : f̃(x) = f(x)

against the alternative:

H_{1} : there exists a change-point 1 < r_{0} < n or t_{0} = r_{0}/n such that x_{1}, …, x_{r_{0}} are observations on the r.v. distributed with the density f̃(x) = f(x) while x_{r_{0}+1}, …, x_{n} are observations on the r.v. distributed with the density f̃(x) = f_{n}(x) = f(x) + ε_{n}h_{n}(x) where ε_{n} ↓ 0 and h_{n} → h_{0} in some sense as n → ∞;

(ii) to construct a consistent estimator for the change-point if the latter exists.

This problem can be reduced to the mode-change problem for random measures provided that we take into account the following reasoning:

The probability density of the sum ξ + ε_{n}Δ_{n} of independent r.v’.s can be written in the form f + ε_{n}Ag_{n}, where

Ag(x) = − ∫ [y ∫_{0}^{1} f′(x − ε_{n}yt)dt]g(y)dy,

f and g_{n} are the probability densities of ξ and Δ_{n}, respectively.

When g runs through a closed set G of probability densities with finite first moments, the functions Ag will run through the closed family H^{∗} ⊂ H, where H is a family of functions h with ∫ h(x)dx = 0 and finite ∫ xh(x)dx. It is obvious that h_{n} ∈ H.

For any given h_{n} let us consider

H_{n}^{∗} = {h : h ∈ H^{∗}, ∫ xh(x)dx = ∫ xh_{n}(x)dx},

and find the function h_{n}^{∗} such that ϱ(h_{n}, h_{n}^{∗}) = inf_{h∈H_{n}^{∗}} ϱ(h_{n}, h), where ϱ is the metric in whose sense the convergence h_{n} → h_{0} is considered.

Instead of f_{n} = f + ε_{n}h_{n} we can consider f_{n}^{∗} = f + ε_{n}h_{n}^{∗}, where h_{n}^{∗} → h_{0}^{∗} in the metric ϱ and f_{n}^{∗} is the convolution of f with some g_{n} ∈ G, which is a quasisolution for the following Fredholm integral equation of the first kind:

Ag = h_{n}.  (3.6)

However, from the asymptoptic viewpoint there is no difference whether we consider f_{n} or f_{n}^{∗} as an alternative.

In this example Γ = [0, 1], A = [0, t], 0 ≤ t ≤ 1, and D(A) is in fact the Skorokhod space D_{[0,1]}. (In D(A) the metric d_{D} is stronger than the Skorokhod metric.) Using the observations x_{1}, …, x_{n} let us construct the random measures, i.e., the processes

X̃_{n}(t) = X̃_{n}([0, t]) = ∑_{i≤nt} x_{i}, t ∈ [0, 1].

On account of these processes we have:

(i) to test the null hypotheses

H_{0n} : X̃_{n}([0, t]) =^{d} X_{n}([0, t])

against the alternatives

H_{1n} : there exists some change-set (t_{0}, 1] such that

X̃_{n}([0, t]) =^{d} X_{n}([0, t]) + ε_{n}Δ′_{n}([0, t] ∩ (t_{0}, 1]),

where

X_{n}(t) = X_{n}([0, t]) = ∑_{i≤nt} ξ_{i}, Δ′_{n}([0, t] ∩ (t_{0}, 1]) = ∑_{nt_{0}<i≤n} Δ_{n}^{(i)},

where ξ_{1}, …, ξ_{n}, Δ_{n}^{(1)}, …, Δ_{n}^{(n)} is the sequence of independent r.v’.s with ξ_{1}, …, ξ_{n} being distributed identically with the density f and Δ_{n}^{(1)}, ..., Δ_{n}^{(n)} being distributed identically with the density g_{n}, which is a quasisolution of equation (3.6);

(ii) to construct a consistent estimator for the change-set provided that the latter exists.

If ∫ xf(x)dx = a ≠ 0, ∫ xh_{n}(x)dx = b_{n}, n = 0, 1, 2, …, and b_{n} → b_{0} ≠ 0 as n → ∞, then

λ_{n}([0, t]) = nat,
δ_{n}([0, t] ∩ C) = nb_{n}(t − t_{0})I{t > t_{0}}, t ∈ [0, 1],

and

δ_{n}(C) = nb_{n}(1 − t_{0}).

Therefore

P_{n}([0, t]) = P ([0, t]) = t, P ([0, t] ∩ C) = I{t > t_{0}}(t − t_{0}),
Q_{n}([0, t] ∩ C) = Q([0, t] ∩ C) = I{t > t_{0}}(t − t_{0})/(1 − t_{0}).

Since t_{0} ∈ (0, 1), condition (3.5) is fulfilled automatically. All other required conditions are also fulfilled here.

If the variances of r.v.’s x_{1} and Δ_{n}^{(1)} are finite, then γ = 1/2, and processes (3.4) and (3.2) take, respectively, the form

Y_{n}^{(1)}(t) = n^{−1/2} ∑_{i≤nt}(x_{i} − a),
Z_{n}^{(1)}(t) = n^{−1/2}( ∑_{i≤nt} x_{i} − t ∑_{i≤n} x_{i}) = n^{−1/2} ∑_{l≤nt}(x_{i} − \overline{x}_{n}),

where \overline{x}_{n} = n^{−1} ∑_{i≤n} x_{i}.

If ε_{n} = O(n^{−1/2}), then by Theorem 3.1, under the hypotheses H_{0n},

Y_{n}^{(1)} ⟶^{w} L_{G}, Z_{n}^{(1)} ⟶^{w} L_{G}

and, under the alternatives H_{1n},

Y_{n}^{(1)} ⟶^{w} L_{G} + b_{0}m, Z_{n}^{(1)} ⟶^{w} L_{G} + b_{0}M,

where L_{G} is the Wiener process, L_{G} is the Brownian bridge, and

m(t) = (t_{0} − t)I{t > t_{0}},
M(t) = I{t > t_{0}}(t − t_{0})/(1 − t_{0}) − t, t ∈ [0, 1].

The function M has the unique minimum only at the change-point t_{0}, while the maximum of the function m is attained all over the interval [0, t_{0}]. (Note that M(t_{0}) = M([0, t_{0}].) The set [0, t_{0}] is a complement to the change-set (t_{0}, 1], and on which the function M has its minimum.)

If n^{1/2}ε_{n} → ∞, then by Theorem 3.2 the estimators

t̂_{n} = \begin{cases}arg max_{t∈[0,1]} Z_{n}^{(1)}(t) & if b_{0} < 0, \\ arg min_{t∈[0,1]} Z_{n}^{(1)}(t) & if b_{0} > 0 \end{cases}

and

t′_{n} = arg max_{t∈[0,1]} |Z_{n}^{(1)}(t)|

are the consistent estimators for the change-point t_{0}.

Remark 3.4. The statistics based on Y_{n}^{(1)} and Z_{n}^{(1)} are discussed in [11].

If the variances of r.v.’s x_{1} and Δ_{n}^{(1)} are infinite and these r.v.’s lie in the domain of normal attraction of the stable law of an index α ∈ (1, 2), then γ = 1/α and processes (3.4) and (3.2) take, respectively, the form

Y_{n}^{(2)}(t) = n^{−1/α} ∑_{i≤nt}(x_{i} − a),
Z_{n}^{(2)}(t) = n^{−1/α} ∑_{i≤nt}(x_{i} − \overline{x}_{n}), t ∈ [0, 1].

Here if ε_{n} = O(n^{−(1−1/α)}), then according to Theorem 3.1, under the hypotheses H_{0n},

Y_{n}^{(2)} ⟶^{w} L_{α}, Z_{n}^{(2)} ⟶^{w} L_{b}

and, under the alternatives H_{1n},

Y_{n}^{(2)} ⟶^{w} L_{α} + b_{0}m, Z_{n}^{(2)} ⟶^{w} L_{b} + b_{0}M,

where L_{α} is the symmetric Lévy (P, α)-stable random process with P being the Lebesgue measure on [0, 1], L_{b} is the broken bridge corresponding to L_{α}, and the functions m and M are as above.

If n^{1−1/α}ε_{n} → ∞, then by Theorem 3.2 the consistent estimators have the same form as t̂_{n} and t′_{n}.

Remark 3.5. These last results are novel for the change-point problem, too. Proof of Theorem 3.1. Under the hypotheses H_{0n}

Y_{n}(A) =^{d} n^{−γ}{X_{n}(A) − λ_{n}(A)},
Z_{n}(A) =^{d} n^{−γ}{(X_{n}(A) − λ_{n}(A)) − P_{n}(A)(X_{n}(Γ) − λ_{n}(Γ))}

and under the alternatives H_{1n}

Y_{n}(A) =^{d} n^{−γ}{X_{n}(A) − λ_{n}(A)} + ε_{n}n^{−γ}δ_{n}(C)Q_{n}(A ∩ C),
Z_{n}(A) =^{d} n^{−γ}{(X_{n}(A) − λ_{n}(A)) − P_{n}(A)(X_{n}(Γ) − λ_{n}(Γ))} +

+ ε_{n}n^{−γ}{(Δ_{n}(A ∩ C) − δ_{n}(A ∩ C)) − P_{n}(A)(Δ_{n}(C) − δ_{n}(C))} +

+ ε_{n}n^{−γ}δ_{n}(C)(Q_{n}(A ∩ C) − P_{n}(A)).

Both under the hypotheses H_{0n} and under the alternatives H_{1n} the weak convergence of Y_{n} and Z_{n} follows from Propositions 2.1, 2.2, 2.10, 2.11, and by Remark 2.5 we have

ε_{n}n^{−γ}{(Δ_{n}(A ∩ C) − δ_{n}(A ∩ C)) − P_{n}(A)(Δ_{n}(C) − δ_{n}(C))} → 0

in probability uniformly on A.

Proof of Theorem 3.2. Assume that δ_{n} > 0. If n^{β−γ}ε_{n} → ∞, then because of the fact that the Lévy bridge L is bounded a.s. we have

Z_{n}/ε_{n}n^{γ}δ_{n}(C) ⟶^{w} M,

M is a continuous set-function on A and due to condition (3.5) it is proved by analogy with (3.3) that M has the unique maximum at the change-set C. Therefore by the fact that then arg max_{A∈A}(Z_{n}(A)/ε_{n}n^{−γ}δ_{n}(C)) is a continuous functional with respect to the metric d_{D} in the space D(A) it turns out that d_{L}(Ĉ_{n}, C) → 0 in probability. For δ_{n} < 0 the proof is similar. Since M has the unique minimum only on the complement to the changeset C and M(C) = −M(\overline{C}) with \overline{C} ∉ A, there follows the consistency of the estimator C′_{n}.

4. The Mode-Change Problem for the Intensity of Compound Poisson Random Measures

Due to (2.8) and (2.9) the mode-change problem in terms of the mean values for compound Poisson P -r.m.’s can be stated as follows:

On the basis of realizations of the observed compound Poisson P_{n}-r.m.’s K̃_{n}(A), A ∈ A, we have

(i) to test the null hypotheses H′_{0n} : P_{n}(A) = P_{0}(A), A ∈ A, against the alternatives H′_{1n} : there exists some change-set C ∈ A such that

P_{n}(A) = P_{0}(A) + ε_{n}δ_{1n}(A ∩ C) + o(ε_{n})δ_{2n}(A ∩ C),

where ε_{n} ↘ 0 as n → ∞ and {δ_{kn}}_{m≥1}, k = 1, 2 are the sequences of measures on A, which are absolutely continuous with respect to the Lebesgue measure. The measures δ_{1n} are nonnegative or nonpositive and have a weak limit δ_{1}, while δ_{2n} are bounded;

(ii) to construct a consistent estimator Ĉ_{n} for the change-set C.

Remark 4.1. Keeping in mind the representation of compound Poisson P_{n}-r.m.’s in form (2.10), the foregoing statement of the mode-change problem concerns only the change occurring in the distribution of r.v.’s ξ_{i}.

Let the r.v.’s x_{i} in the representation of the observed compound Poisson P_{n}-r.m.’s in form (2.8) or (2.10) be in the domain of normal attraction of the stable law of an index α ∈ (1, 2) with Ex_{i} = a ≠ 0. Consider the objects

Y′_{n} = (π_{n}(Γ))^{−1/α}(K̃_{n} − E_{0}K̃_{n})  (4.1)

and

Z′_{n}(A) = (π_{n}(Γ))^{−1/α}{K̃_{n}(A) − P_{0}(A)K̃_{n}(Γ)}, A ∈ A.  (4.2)

According to (2.9) we have

E_{0}K̃_{n}(A) = J_{0n}(A) = naP_{0}(A),
E_{1}K̃_{n}(A) = J_{1n}(A) = na[P_{0}(A) + ε_{n}δ_{1}(A ∩ C) + o(ε_{n})δ_{2n}(A ∩ C)].

Hence γ = 1/α and β = 1 (compare with the general formulation in Section 3). Since we are interested in the asymptotic behavior of r.m.’s (4.1) and (4.2), by Proposition 2.5 it is sufficient to investigate the asymptotic behavior of the r.m.’s

Y′′_{n}(A) = n^{−1/α}[ ∑_{i≤n} x_{i}I_{A}(ξ_{i}) − naP_{0}(A)]  (4.3)

and

Z′′_{n}(A) = n^{−1/α}[ ∑_{i≤n} x_{i}I_{A}(ξ_{i}) − P_{0}(A) ∑_{i≤n} x_{i}].  (4.4)

These equalities can be rewritten as

Y′′_{n} = L_{n} + n^{1/2−1/α}aV_{n}  (4.5)

and

Z′′_{n}(A) = Y′′_{n}(A) − P_{0}(A)Y′′_{n}(Γ),  (4.6)

where

L_{n}(A) = n^{−1/α} ∑_{i≤n}(x_{i} − a)I_{A}(ξ_{i}),
V_{n}(A) = n^{−1/2} ∑_{i≤n} [I_{A}(ξ_{i}) − P_{0}(A)].

Since the subfamily A satisfies the conditions (a)–(d), according to [3], under the hypotheses H′_{0n}, we have V_{n} ⟶^{w} L_{G}, where L_{G} is the Brownian P_{0}- bridge. Because of the fact that L_{G} has bounded realizations a.s.

an^{1/2−1/α} sup_{A∈A} |V_{n}(A)| → 0 in probability.  (4.7)

If ε_{n}n^{1−1/α}|aδ_{1n}(C)| → 1, then under the alternatives H′_{1n}, n^{1/2−1/α}aV_{n} converges in probability to m′ or −m′ depending on whether aδ_{1n}(C) is greater or smaller than 0, where the set-function m′(A) = δ_{1}(A ∩ C)/δ_{1}(C).

Both under the hypotheses H′_{0n} and under the alternatives H′_{1n}

L_{n} ⟶^{w} L_{α},  (4.8)

where L_{α} is the Lévy (P_{0}, α)-stable symmetric r.m. This fact is proved using Proposition 2.1. The convergence of finite-dimensional distributions of L_{n} to those of L_{α} and the tightness of the sequence {L_{n}}_{n≥1} both under the hypotheses H′_{0n} and under the alternatives H′_{1n} is proved quite similarly to the proof of Proposition 5.1 in [8] and to the proof of the tightness of the sequence of random processes from Example 6.3 in [9], respectively. The only difference consists in that instead of the Lebesgue measure of sets A ∈ A we should consider the measure P_{0} under the hypotheses H′_{0n} and the measure P_{n}(A) = P_{0}(A) + ε_{n}δ_{1}(A ∩ C) + o(ε_{n})δ_{2n}(A ∩ C) under the alternatives H′_{1n}.

Thus by (4.5)–(4.8) under the hypotheses H′_{0n} we have

Y′′_{n} ⟶^{w} L_{α}, Z′′_{n} ⟶^{w} L_{b}

and therefore by Proposition 2.6

Y′_{n} ⟶^{w} L_{α}, Z′_{n} ⟶^{w} L_{b},

where L_{b} is the Lévy broken (P_{0}, α)-bridge.

Under the alternatives H′_{1n} we have

Y′′_{n} ⟶^{w} L_{α} + m′, Z′′_{n} ⟶^{w} L_{b} + M′ for aδ_{1} > 0

and

Y′′_{n} ⟶^{w} L_{α} − m′, Z′′_{n} ⟶^{w} L_{b} − M′ for aδ_{1} < 0,

where the set-function M′(A) = δ_{1}(A ∩ C)/δ_{1}(C) − P_{0}(A). Therefore by Proposition 2.5 the r.m.’s Y′_{n} and Z′_{n} will have the same asymptotic behavior as Y′′_{n} and Z′′_{n}, respectively.

If

δ_{1}(A ∩ C)/δ_{1}(C) > P_{0}(A)  (4.9)

for all A ∈ A with A ∩ C ≠ ∅, then, as shown above, the unique maximum of M′ is attained only on the change-set C.

Thus, in common with Theorems 3.1 and 3.2, the following two theorems are valid.

Theorem 4.1. If ε_{n}n^{1−1/α}|aδ_{1n}(C)| → 1 for α ∈ (1, 2) as n → ∞, then under the hypotheses H′_{0n}

Y′_{n} ⟶^{w} L_{α}, Z′_{n} ⟶^{w} L_{b}

and under the alternatives H′_{1n}

Y′_{n} ⟶^{w} L_{α} + m′, Z′_{n} ⟶^{w} L_{b} + M′ for aδ_{1} > 0

and

Y′_{n} ⟶^{w} L_{α} − m′, Z′_{n} ⟶^{w} L_{b} − M′ for aδ_{1} < 0.

Theorem 4.2. If n^{1−1/α}ε_{n} → ∞ for α ∈ (1, 2) as n → ∞, then by condition (4.9) the estimators

Ĉ_{n} = \begin{cases}arg max_{A∈A} Z′_{n}(A) & if aδ_{1} > 0, \\ arg min_{A∈A} Z′_{n}(A) & if aδ_{1} < 0 \end{cases}

and

C′_{n} = arg max_{A∈A} |Z′_{n}(A)| (if the sign of aδ_{1} is unknown)

are consistent estimators for the change-set C.

Remark 4.2. If P_{0} is unknown, we should use the processes

Ẑ_{n}(A) = π_{n}(Γ))^{−1/α}[ ∑_{i≤π_{n}(Γ)} x_{i}I_{A}(ξ) − P̂_{n}(A) ∑_{i≤π_{n}(Γ)} x_{i}],

where P̂_{n}(A) = π_{n}(A)/π_{n}(Γ). The processes Ẑ_{n} have the same asymptotic behavior as the processes Ẑ_{n}. Thus, the estimator

C′′_{n} = arg max_{A∈A} |Ẑ_{n}(A)|

will also be consistent.

Remark 4.3. If x_{i} are degenerate r.v.’s, in particular, x_{i} ≡ 1, then K̃_{n}(A) are Poisson P_{n}-r.m.’s π̃_{n}(A). The objects

n^{−1/2}[π̃_{n}(A) − nP_{0}(A)] and n^{−1/2}[π̃_{n}(A) − P_{0}(A)π̃_{n}(Γ)], A ∈ A,

weakly converge under the hypotheses H′_{0n} to the Brownian P_{0}-r.m. L_{G} and to the Brownian P_{0}-bridge L_{G}, respectively, and under the alternatives H′_{1n} to L_{G} ± m′ and L_{G} ± M′, respectively.

Acknowledgement

The authors express their gratitude to Academician Yu. V. Prokhorov for suggesting to them the topic of the change-point problem and to Professor E. Khmaladze for useful remarks.

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(Received 20.04.1994; revised version 12.07.1995)

Authors’ address: A. Razmadze Mathematical Institute Georgian Academy of Sciences 1, M. Alexidze St., Tbilisi 380093 Republic of Georgia