On the low intensity bootstrap for triangular arrays of ... - Springer Link

Test (2009) 18: 283–301 DOI 10.1007/s11749-007-0077-3 O R I G I N A L PA P E R

On the low intensity bootstrap for triangular arrays of independent identically distributed random variables Eustasio del Barrio · Arnold Janssen · Carlos Matrán

Received: 11 April 2007 / Accepted: 3 September 2007 / Published online: 21 September 2007 © Sociedad de Estadística e Investigación Operativa 2007

Abstract In this work, we give a complete picture of the behavior of the low intensity bootstrap of linear statistics. Our setup is given by triangular arrays of independent identically distributed random variables and different normalizations related to the rates of bootstrap intensities. We show that the behavior of this low intensity bootstrap coincides with that of partial sums of a number of summands equal to the bootstrap resampling size. Agreement on the limit laws for different (small) bootstrap sizes is thus shown to be closely related to domains of attraction of α-stable laws. As a byproduct, we obtain local distributional properties of Lévy processes. Keywords Bootstrap · Resampling · Low intensity · Infinitely divisible laws · Stable laws · Domains of attraction · Lévy processes Mathematics Subject Classification (2000) Primary 62F40 · Secondary 60G51 1 Introduction After almost three decades since the introduction of bootstrap, it seems that there is a wide agreement in the statistical community concerning the remedies for situations where the usual bootstrap fails. Subsampling introduced by Politis and Romano (1994) (see also Politis et al. 1999) or low intensity bootstrap (Swanepoel 1986; The authors have been partially supported by the Spanish Ministerio de Educación y Ciencia and FEDER, grant MTM2005-08519-C02-01,02 and the Consejería de Educación y Cultura de la Junta de Castilla y León, grant PAPIJCL VA102A06. E. del Barrio () · C. Matrán Departamento de Estadística e Investigación Operativa, Universidad de Valladolid, Valladolid, Spain e-mail: [email protected] A. Janssen Mathematisches Institut, Heinrich-Heine Universität, Düsseldorf, Germany

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Athreya 1987) generally constitute valid alternatives to the usual bootstrap (see the discussion in Bickel et al. 1997). Recent works analyzing in greater detail the success of the low intensity bootstrap address even the choice of appropriate intensity of resampling when the usual bootstrap fails (see, e.g., Bickel and Sakov 2005). These remedies, however, should not be carelessly assumed to work universally; the consideration of a triangular array setup gives significant information about the subtleties that can occur. In the case of the low intensity bootstrap, which is the topic of interest in this paper, these facts have been studied for linear statistics in several papers, including Arcones and Giné (1989, 1991), Athreya (1987), Deheuvels et al. (1993), and later Cuesta-Albertos and Matrán (1998). This last paper treats the problem in the setup of arrays of row-wise independent identically distributed (i.i.d.) random variables, giving a general result on the behavior of the bootstrap of normalized sums which covers the so far available results on sequences of i.i.d. random variables. This triangular array setup was already considered in Mammen (1992a, 1992b) for the ordinary bootstrap and has also been considered recently in Janssen and Pauls (2003) and Janssen (2005) even for other resampling schemes. In the same framework of triangular arrays, necessary conditions for the bootstrap of the mean were obtained in del Barrio et al. (1999), while del Barrio et al. (2002) analyzed the stability of the behavior when the resampling order varies, warning about the consequences that small resampling orders can produce. See the survey paper by Csörgö and Rosalsky (2003) for additional references. The present paper continues the work of del Barrio et al. (2007) about low resampling statistics given by partial sums of triangular arrays. They show that general low resampling schemes (including the low intensity bootstrap) act as a filter for the normal part of the limiting distribution of the partial sums and its Poisson part is suppressed. In this paper, we focus on the low intensity bootstrap in which an additional normalization may be required. We provide a complete description of the performance of this low intensity bootstrap, characterizing the possible limits and normalizations in terms of those of the original array. For those sceptic of the relevance of considering triangular arrays, it should be enough to recall that the Poisson law of rare events can be only explained in this setup (see also the Introduction in Cuesta-Albertos and Matrán 1998). However, the following preliminary example should be an additional aid to recognize the different behaviors that can arise in this setting and the nature of the problems of interest. Example 1.1 Let {Xn,j , j = 1, . . . , n2 , n ∈ N} be an array of row-wise i.i.d. Bernoulli random variables of size k(n) = n2 with EXn,j = pn = n1 . By the Central Limit Theorem (CLT) n2

j =1 Xn,j

√

n−1

−n

→w N (0, 1);

by the Poisson law of rare events n  j =1

Xn,j →w Pois(1),

On the low intensity bootstrap for triangular arrays of independent

285

and, if m(n) n → 0, there is no possible normalization that produces a nondegenerate m(n) limit law of j =1 Xn,j . What can be expected by bootstrapping the row Xn,1 , . . . , Xn,n2 with a resampling → 0? size, m(n), such that m(n) n2 In what sense could it be said that the bootstrap with small resampling size works? We will answer these questions in the course of this the paper and reconsider this example later (see Example 2.2 and the final comments in Remark 4.1). It should be noted that (Cuesta-Albertos and Matrán 1998) stresses the difficulties in reconciling the behavior in the bootstrap of the normal and Poisson asymptotic parts of the sums of triangular arrays of i.i.d. random variables. However, CuestaAlbertos and Matrán (1998) only provides sufficient conditions to guarantee a bootstrap limit law related in some way to that of the original array. As will be shown in this paper, that result does not provide a good enough description of the behavior of the low intensity bootstrap. In fact, we will prove that this behavior coincides with that of partial sums with a number of summands equal to the bootstrap resampling size. We begin with a more formal description of the problem and the scope of this work. Let Xn,1 , . . . , Xn,k(n) , n = 1, 2, . . . , be a triangular array of row-wise independent identically distributed random variables. Throughout this paper, we are concerned with a bootstrap sample drawn by the m(n) out of k(n) bootstrap, ∗ ∗ , . . . , Xn,m(n) Xn,1

from Xn,1 , . . . , Xn,k(n) ,

(1.1)

for small resampling orders. It is well known that this is a new array of random variables which, conditionally given {Xn,j , j = 1, . . . , k(n), n ∈ N}, are row-wise 1 k(n) i.i.d. with conditional distribution equal to the empirical distribution k(n) i=1 δXn,i , where δx denotes Dirac’s measure on x. Thus, writing P ∗ for the conditional probability given {Xn,j , j = 1, . . . , k(n), n ∈ N}, we have that ∗ = Xn,i ] = P ∗ [Xn,j

1 , k(n)

j = 1, . . . , m(n); i = 1, . . . , k(n),

if Xn,1 , . . . , Xn,kn are all different, with the obvious changes in the presence of ties. We assume that the resampling size, m(n), satisfies m(n) →0 k(n)

and

min{m(n), k(n)} → ∞

as n → ∞.

We use the expressions low intensity bootstrap or bootstrap with small resampling size as a short form for this setup. In Sect. 2, we show that the unconditional behavior of the m(n) bootstrap is the same as that of the partial sum with m(n) summands. This is obtained through a special construction in Theorem 2.1. With this equivalence in mind, we devote Sect. 3 to explore the conditions under which rescaled sums of a small fraction of an infinitesimal array attracted to some infinitely divisible law converge to some (nondegenerate) limit law. This will be later used to characterize when the low intensity bootstrap of

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the mean works, but we obtain also an easy application on the local behavior of Lévy processes. In Sect. 4, we analyze the conditional behavior in the light of the unconditional one and show that it is the same. This will be proved in Theorem 4.2. On the other hand, we study the conditions that ensure the stability of the possible bootstrap limit laws. Turning to notation, by E ∗ we will often denote the conditional expectation given {Xn,j , j = 1, . . . , k(n), n ∈ N}. By L(Z) we mean the distribution of a r.v. Z, and L∗ (Z) is the conditional distribution of a r.v. Z given {Xn,j , j = 1, . . . , k(n), n ∈ N}. We will often invoke the general Central Limit Theorem, for which we will use the approach given in Chapter 2 in Araujo and Giné (1980, Theorem 2.4.7, p. 61). Some related notation includes the characterization of the infinitely divisible laws in the form N (0, σ 2 ) ∗ cτ Pois μ. Here μ is a Lévy measure, and τ > 0 is a constant involved in a certain truncation procedure such that μ{−τ, τ } = 0. Moreover, given a random variable, X, and δ > 0, we write Xτ as a short form for XI{|X|≤τ } , where IA is the indicator of a set A. In particular, we use the notation Xn,j,δ := Xn,j I{|Xn,j |≤δ} , 1 X¯ n := kn

k(n)  j =1

Xn,j ,

∗ ∗ ∗ |≤δ} , Xn,j,δ := Xn,j I{|Xn,j

1 X¯ n,δ := k(n)

k(n) 

j = 1, . . . , k(n);

Xn,j,δ .

j =1

2 Asymptotic equivalence with a sequence of partial sums The resampling statistic  m(n)  k(n)   k(n) 1/2  ∗ m(n) 1/2  Tn∗ = Xn,i − Xn,i m(n) k(n) i=1

(2.1)

i=1

is a special form of a more general weighted bootstrap statistic, see Example 2.1 in del Barrio et al. (2007) and further results therein. It is shown there (Theorem 2.2) that Tn∗ vanishes in probability whenever the original partial sums Tn =

k(n) 

Xn,i →w ψ

(2.2)

i=1

have a limit law without normal component. Specific examples are given by stable variables with stability index α < 2, see Example 2.1 below. We will see that, under certain conditions, a proper normalization an Tn∗ of (2.1) with an → ∞ leads to a nondegenerate unconditional limit law. The following example highlights our methodology. Example 2.1 Let (Yi )i∈N be an i.i.d. sequence in the strict domain of attraction of an α-stable law ξ (α < 2), i.e., n 1  Yi →w ξ (2.3) cn i=1

for some normalizing sequence cn > 0.

On the low intensity bootstrap for triangular arrays of independent

If we take k(n) = n and put Xn,i := unconditional convergence m(n) 

1 cn Yi ,

287

then, as is well known, we have the

∗ Xn,i → 0 in probability.

(2.4)

i=1 1 However, if we set ξn,i = cm(n) Yi (that is, ξn,i = bn Xn,i for bn = m(n) i=1 ξn,i →w ξ . Theorem 2.1 below implies that m(n) 

∗ ξn,i = bn d

i=1

m(n) 

∗ Xn,i →ξ

cn cm(n) ),

then

(2.5)

i=1 1

∗ 2 in distribution, and taking an = ( m(n) n ) bn , we also have an Tn → ξ in distribution 1

2 for 1 < α < 2 (in this case, we have an ( m(n) n ) → 0, see Lemma 2.1 in del Barrio and ∗ Matrán 2000), and an Tn reproduces the limit law (2.3).

From now on assume that the m(n)-portion of the partial sum m(n) 

Xn,i

is shift tight,

(2.6)

i=1

m(n) (i.e., for some sequence of constants (hn )n , the sequence i=1 Xn,i − hn is tight). Observe that the Xn,i correspond to the normalized variables ξn,i in our Example 2.1. ∗ , . . . , X∗ Theorem 2.1 Suppose that (2.6) holds, and let Xn,1 n,m(n) be a m(n) out of

k(n) bootstrap sample with m(n) → ∞ and

m(n) k(n)

→ 0 as n → ∞. We can define, on

a possibly enlarged probability space, i.i.d. random variables, X˜ n,1 , . . . , X˜ n,m(n) with L(X˜ n,i ) = L(Xn,1 ), i = 1, . . . , m(n), such that m(n)  i=1

∗ Xn,i

−

m(n) 

X˜ n,i → 0 in probability.

(2.7)

i=1

Proof We consider w.l.o.g. that the multinomial weights defining the resampling variables are based on a triangular array of weight functions on another probability space Wn,i : (Ω  , A , P  ) → R, 1 ≤ i ≤ k(n). In this way, via projections, all random vari∗ ) may be considered on the ables under consideration (in particular, the Xn,i , Xn,j    same probability space (Ω × Ω , A ⊗ A , P ⊗ P ). We will define the new triangular array on this product space with Ω rich enough to allow consideration of i.i.d. random k(n)+m(n)−1 variables {Xn,i }i=1 for every n. The new triangular array X˜ n,i , 1 ≤ i ≤ m(n), will be defined by induction. For each i, we find an index j (i) with ∗ Xn,i = Xn,j (i) .

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Set X˜ n,1 = Xn,j (1) . For i > 1, define again X˜ n,i = Xn,j (i) whenever the index j (i) did not appear earlier, i.e., j (i) = j (k) for all k < i. Otherwise, we choose X˜ n,i to m(n) be Xn,l , l the first unused index with l > k(n). Observe that {X˜ n,i }i=1 are i.i.d. with d X˜ n,i = Xn,1 . Let now   m(n)  (2.8) N m(n) := ∪i=1 j (i) be the number of different variables in our bootstrap sample. Lemma 2.1 Suppose that

m(n) k(n)

→ c ∈ [0, 1).

(m(n)) ) → τc = (a) Then we have E( N m(n)

(b) In the case c = 0, we have

N (m(n)) m(n)

1−exp(−c) c

with τ0 = 1.

→ 1 in probability.

Proof (a) Let k(n) be fixed. Consider N (j ) for j < k(n). We have N (1) = 1. The conditional expectation of N (j + 1) given N (j ) = r equals   r2 k(n) − r k(n) − 1 E N(j + 1)|N (j ) = r = + (r + 1) = r + 1. k(n) k(n) k(n) The expectation is then    k(n) − 1  E N (j ) + 1. E N (j + 1) = k(n) If we set ρ =

k(n)−1 k(n) ,

the recursion yields the solution j −1

   i 1 − ρj . ρ = E N (j ) = 1−ρ i=0

In the case c > 0, we obviously have 

  E(N (m(n))) 1 − exp(−c) k(n) 1 m(n) → = 1− 1− . m(n) m(n) k(n) c On the other hand, it is easy to see by the extended mean value theorem that j →

E(N (j )) j

is decreasing.

Thus, τc ≤ τ0 ≤ 1 implies τ0 = 1 as c ↓ 0. Assertion (b) follows from part (a), since 0≤1− is convergent in L1 .

N (m(n)) →0 m(n) 

On the low intensity bootstrap for triangular arrays of independent

289

Final part of the proof of Theorem 2.1: The special construction of the X˜ n,i implies that m(n) 

∗ Xn,i −

m(n) 

i=1

X˜ n,i = A˜ n − B˜ n

i=1

with d A˜ n = An :=

∗ m(n) 

i=1

∗ Xn,i

m(n)∗ hn − m(n)

d and B˜ n = Bn :=

∗ m(n) 

i=1

Xn,i −

m(n)∗ hn , m(n)

m(n)∗

:= m(n) − N(m(n)), and hn is the centering considered in the shift where tightness property. Hence, it suffices to show that An → 0 and Bn → 0 in probability. We first prove that Bn → 0 in probability. Since the counting variable N (m(n)) (m(n)) is independent of the data and N m(n) → 1 by Lemma 2.1, this follows from the shift tightness assumption (2.6) and from the following elementary fact, which can be proved, e.g., with a straightforward modification of Proposition 9.10 in Breiman (1968). Fact. Let Yn,i , i = 1, . . . , kn , n ≥ 1, with kn → ∞ be a triangular array of random variables which are i.i.d. for each n ≥ 1. Whenever kn 

Yn,i

i=1

is tight, then for any sequence of integers ln such that 1 ≤ ln ≤ kn and kn / ln → ∞, ln 

Yn,i →Pr 0.

i=1

We now consider An . Roughly speaking, we can see it as an m(n) − N (m(n)) bootstrap of (2.6) with new sample size m(n) and low resampling intensity. To be m(n) precise, we assume, without loss of generality, that i=1 Xn,i − hn is convergent and also that m(n)∗ m(n) − N (m(n)) = →0 (2.9) m(n) m(n) almost surely (we could consider convergent subsequences if necessary). The same proof as in Proposition 12 in Cuesta-Albertos and Matrán (1998) gives the conditional convergence ∗ m(n) 

i=1

∗ Xn,i −

m(n)∗ hn → 0, m(n)

given the resampling sample size m(n)∗ , which implies Bn →Pr 0 and (2.7).



Statement (2.5) of Example 2.1 is a consequence of Theorem 2.1 if we are going to bootstrap the scheme (ξn,i )i≤k(n) .

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Remark 2.1 The construction of Theorem 2.1 works for some other resampling schemes. In particular, in subsampling, there is no need for enlarging the probability space Ω, we have N (m(n)) = m(n), and Bn is null, so we trivially have (2.7). In this subsampling case, the construction here is simply a reconsideration of the fact, already observed in Politis and Romano (1994), that the subsample is a sample of size m(n) of the underlying model. X

The next application for the low resampling bootstrap covers schemes { bn,i }i≤k(n) n which may be in the domain of strict attraction of stable laws like in Example 2.1. This result provides strong motivation for relating the behavior of the low intensity bootstrap to that of small subarrays as in the general Theorem 4.3 below. Theorem 2.2 (About unconditional convergence of low intensity bootstrap) Suppose that (1.1) holds and also that m(n) 

Xn,i − an →w ξ.

(2.10)

∗ Xn,i − an →w ξ.

(2.11)

i=1

Then m(n)  i=1 1 dn

If, in addition, we assume that with dn → ∞ such that

k(n) i=1

Xn,i is tight, where dn is a sequence of reals

dn m(n) → 0, k(n)

(2.12)

then we have the convergence 

m(n) k(n)

1/2

Tn∗ − an →w ξ.

(2.13)

Proof Consider the copies X˜ n,i of Xn,i constructed in Theorem 2.1. Our assumptions imply that 

m(n) k(n)

1/2

Tn∗

−

m(n)  i=1

m(n)  Xn,i → 0 X˜ n,i = oP (1) − k(n) k(n)

(2.14)

i=1

in probability as n → ∞. The rest of the theorem is obvious.



Example 2.2 Consider again the triangular array of Bernoulli random variables of Example 1.1.

On the low intensity bootstrap for triangular arrays of independent m(n) n2

m(n) n

→ ∞. Then the Central Lindeberg–Feller Limit m(n) Theorem applies to the partial sum i=1 Xn,i , and we have

(a) Suppose that

→ 0 and

291

m(n) i=1



(b) Assume that

m(n) n

∗ − Xn,i

m(n) n

m(n) 1 n (1 − n )

→w N (0, 1).

→ λ > 0. Then m(n) 

∗ Xn,i →w Pois(λ).

i=1

m(n) (c) In the case m(n) n → 0, we have P ( i=1 Xn,i > 0) → 0. Then, for any normalm(n) ∗ → 0 in probability. ization, we have d1n i=1 Xn,i

3 Limits of scaled partial sums of small subarrays The asymptotic behavior (and stability) of the bootstrap with small resampling size, m(n), of linear statistics computed over row-wise i.i.d. triangular arrays {Xn,j : j = 1, . . . , k(n)} has been shown to be equivalent to that of the corresponding rescaled version computed from the subarray {Xn,j : j = 1, . . . , m(n)}. Here we explore the conditions under which rescaled sums of a small fraction of an infinitesimal array attracted to some infinitely divisible law are convergent to some (nondegenerate) limit. To be precise, we assume that k(n) 

Xn,j − an →w γ .

(3.1)

j =1

Here γ must be infinitely divisible and admits therefore the representation γ = N(0, σ 2 ) ∗ cτ Pois μ for some Lévy measure μ on R, where cτ Pois μ is the probability law having the characteristic function     itx φ(t) = exp e − 1 − itxI |x| ≤ τ dμ(x) . R−{0}

Also, following the notation in p. 82 of Araujo and Giné (1980), the stable laws will be denoted as δc ∗ c Pois μ(c1 , c2 ; α). As shown in the introductory example, it is possible that a rescaled subarray {Xn,j /rn : j = 1, . . . , m(n)} with m(n)  ∞ and m(n) k(n) → 0 has partial sums converging in law to some infinitely divisible law, namely, m(n)   1  Xn,j − bn →w N 0, β 2 ∗ cτ Pois ν. rn j =1

(3.2)

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The scaling sequence, rn (which must obviously satisfy rn → 0), can usually be chosen as a smooth function of the relative size of the subarray, that is, rn = f ( m(n) k(n) ) for some continuous function f : (0, ∞) → (0, ∞). We show here that if (3.2) holds for every subarray of small order, then the limit in (3.2) must be α-stable of order α ∈ (0, 2]. If α ∈ (0, 2), then μ must have regularly varying tails of order −α, while, for α = 2, either σ 2 > 0 (and then rescaled small subarrays simply filter out the δ nonnormal part) or the truncated moment −δ x 2 dμ(x) is a slowly varying function at the origin. The proof of this result relies on a technical Lemma concerning infinitely divisible laws. We will use the following notation. For a given infinitely divisible law, γ = N(0, σ 2 ) ∗ cτ Pois μ, we will write γ ∗c) for the law N (0, cσ 2 ) ∗ cτ Pois(cμ) (this notan)

tion is consistent with the property γ ∗n) = γ ∗ · · · ∗γ for n ∈ N). For any probability measure on R, γ , and r > 0, γr will stand for the scaled measure γr (A) = γ (rA) (that is, if X has the distribution law γ , then that of X/r is γr ). With this notation, we have the following analogue of the classical characterization of the domains of attraction in the Central Limit problem: Lemma 3.1 Assume that γ = N (0, σ 2 ) ∗ cτ Pois μ is an infinitely divisible law such that γf∗c) (c) →w ρ

(3.3)

for some measurable function f : (0, ∞) → (0, ∞) as c → 0. Then ρ is α-stable for some α ∈ (0, 2]. Furthermore, (i) (3.3) holds with a Gaussian limit iff √ in this case, we can take f (t) = t, and then ρ = N (0, σ 2 ) or

δ σ 2 = 0 and x 2 dμ(x) is slowly varying at 0; in this case, f must be σ 2 > 0;

−δ

√ regularly varying at 0 with exponent 1/2 and such that f (t) = o( t ). (ii) (3.3) holds with a non Gaussian α-stable limit iff σ 2 = 0, μ[−δ, δ]c varies regularly at 0 with exponent − α lim

δ→0

and

μ(−∞, −δ) = p ∈ [0, 1]. μ[−δ, δ]c

In this case, ρ = cτ Pois μ(c1 , c2 ; α) with c1 /(c1 + c2 ) = p, and f must be regularly varying at 0 with exponent 1/α. Proof By standard arguments about infinitely divisible laws (e.g., Proposition 2.1 in del Barrio et al. 2002) (3.3) implies that ρ is infinitely divisible, namely, ρ = N(0, β 2 ) ∗ cτ Pois ν, and this holds iff

On the low intensity bootstrap for triangular arrays of independent

293

(a)   cμ δf (c), ∞ → ν(δ, ∞) (b)

for every δ such that ν{δ} = 0 (and similarly for the lower tail) and  

δf (c)

δ c 2 2 2 σ + x dμ(x) → β + x 2 dν(x) f (c)2 −δf (c) −δ

as c → 0

for every δ such that ν{−δ, δ} = 0. Now, if ν = 0, then regular variation theory (see, e.g., Proposition 0.4(ii) in Resnick 1987) gives that μ[−δ, δ]c varies regularly at 0 with some exponent −α, ν[−δ, δ]c = Cδ −α (hence, necessarily, α ∈ (0, 2), since ν is a Lévy measure), and ν(−∞, −δ) = c1 δ −α , ν(δ, ∞) = c2 δ −α with ci ≥ 0, c1 + c2 = C > 0 (note that we cannot have different orders of regular variation on the left and right tails unless c1 = 0 or c2 = 0). We must also have limδ→0 μ(−∞,−δ) μ[−δ,δ]c = p ∈ [0, 1] in this case and, therefore, ρ = c Pois μ(c1 , c2 ; α). We also have that f must be regularly varying at 0 with exponent 1/α. Thus, c/f (c)2 → ∞ as c → 0. Then (b) implies that, in this case, necessarily, σ 2 = 0. Assume 0. Then ρ is Gaussian, ρ = N (0, β 2 ). If σ 2 > 0, we set √ now that ν = √ √ f (t) = t and write cμ(δ c, ∞) = cμ(δ c, 1] + cμ(1, ∞). The second term in the right-hand side of this last equation vanishes. For the first term, we have t 2 I(t,1) (x) ≤ x 2 for t, x ∈ (0, 1). Hence, by dominated convergence, √ 1 cμ(δ c, 1] = 2 δ

0

1

δ 2 cI(δ √c,1) (x) dμ(x) → 0.

Consequently, we have (a) and (b) with ν = 0 and β 2 = σ 2 . Finally, let us consider the case where both σ 2 = 0 and ν = 0. Then (b) reduces to c f (c)2

δf (c) −δf (c)

x 2 dμ(x) → β 2

for every δ > 0. Obviously, we must have c/f (c)2 → ∞ as c → 0 (or β 2 = 0) and,  t 2using again Proposition 0.4.1 of Resnick (1987), we obtain that U (t) := at 0 and f (t) regularly varying at 0 with −t x dμ(x), t > 0 must be slowly varying √ exponent 1/2 and such that f (t) = o( t ). This completes the proof.  Remark 3.1 Now let us relate the result just proved to the limiting behavior of Lévy processes at 0. Assume that {X(t)}t≥0 is a Lévy process with L(X(1)) = N(0, σ 2 ) ∗ cτ Pois μ. Then X(t) → 0 in probability as t → 0, and we might wonder about possible scaling functions f > 0 such that X(t) →w ρ f (t)

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as t → 0 for some nondegenerate law ρ. Lemma 3.1 then gives that the only possible limits are α-stable laws. We always have   X(t) √ →w N 0, σ 2 t and, if σ 2 = 0 (a pure jump Lévy process), then wemust use a rescaling of smaller t order. It must be regularly varying of order 1/2 if −t x 2 dμ(x) is slowly varying. Otherwise, μ must have regularly varying tails, and the rescaling function must be regularly varying of order 1/α. However easy, we are not aware of this result in the literature about Lévy processes. For related work, we refer to Doney and Maller (2002). There is also a clear connection with the results in Wschebor (1995). We now formulate the main consequence for row-wise i.i.d. triangular arrays. Theorem 3.1 Assume that {Xn,j : j = 1, . . . , k(n); n ∈ N} is a row-wise i.i.d. triangular array satisfying (3.1) with γ = N (0, σ 2 ) ∗ cτ Pois μ. If there exists a nondegenerate law, ρ, and a continuous function f : (0, ∞) → (0, ∞) satisfying that, for every sequence of integers {m(n)}n such that m(n)  ∞ and m(n)/k(n) → 0, m(n) 1  Xn,j − bn →w ρ rn

(3.4)

j =1

with rn = f (m(n)/k(n)) and suitable centering constants bn , then ρ is α-stable for some α ∈ (0, 2]. Furthermore, (3.4) holds with ρ Gaussian iff: √ (i) σ 2 > 0, and we can take rn = m(n)/k(n), else δ (ii) −δ x 2 dμ(x) is slowly varying at 0. In this case, f must be regularly varying at √ 0 with exponent 1/2 and such that f (t) = o( t ). Finally, (3.4) holds with ρ non-Gaussian α-stable iff μ[−δ, δ]c lim

δ→0

varies regularly at 0 with exponent −α

and

μ(−∞, −δ) = p ∈ [0, 1]. μ[−δ, δ]c

In this case, f must be regularly varying at 0 with exponent 1/α. Proof The general CLT for triangular arrays (see, e.g., Theorem 2.4.7 in Araujo and Giné 1980) and (3.1) imply that if mn is a sequence of integers satisfying mn → ∞ and mn /k(n) → c ∈ (0, 1), then  m n  1 ∗c) Xn,j − mn EXn,1,τf (cn ) →w γf (c) , (3.5) f (cn ) j =1

where cn = mn /k(n). Arguing as in the proof of Theorem 2.7 in del Barrio et al. (2002), we consider a sequence {el }l  0 and denote by d a distance metrizing the

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weak convergence of probability measures on R. Then, for a fixed l ∈ N, if we take nl large enough and mnl := el k(nl ), then 

   mn l 1 1 ∗el ) Xn,j − mnl EXn,1,τf (cnl ) , γf (el ) ≤ . d f (cnl ) l

(3.6)

j =1

We can take w.l.o.g. nn < nl+1 . Let now {m(n)} be a sequence of integers such that m(n)  ∞, m(n)/k(n) → 0, and m(nl ) = mnl . Then (3.4) implies that 

m(n)   1  Xn,j − m(n)EXn,1,τ rn , ρ → 0. d rn

(3.7)

j =1

Combining (3.6) and (3.7) and noting the arbitrariness of the sequence {el }l , we conclude that γf∗c)  (c) →w ρ. The result follows from Lemma 3.1.

4 When does the low intensity bootstrap work for partial sums of i.i.d. random variables? Our goal in this section is to characterize when does the limiting distribution of a scaled bootstrapped linear statistic mimic that of the original version or, at least, does not depend on the convergence rate of m(n) k(n) → 0. Necessary conditions for the bootstrap of the mean of a triangular array of rowwise i.i.d. random variables were obtained in del Barrio et al. (1999). We quote, for the sake of completeness, the part of the main result there, Theorem 1.1, which deals with the low intensity case. Here we correct a missprint in that statement in del Barrio et al. (1999), namely, the scaling constants should be assumed to satisfy rn → 0. Theorem 4.1 Assume that m(n)  ∞ and m(n)/k(n) → 0. If there exist a probability measure ρ, constants {rn }∞ n=1 , rn → 0, and random variables An , with An measurable on the σ -field σ (Xn,j : j = 1, . . . , k(n)), n ∈ N, satisfying  L

∗

m(n)  1  ∗ w Xn,i − An → ρ in probability, rn i=1

then: (i) ρ is infinitely divisible, thus, ρ = N (a, α 2 ) ∗ cτ Pois ν. (ii) There exist constants {bn }∞ n=1 such that   1 w Xn,i − bn → N a, α 2 ∗ cτ Pois ν. rn m(n) i=1

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The constants bn must satisfy     m(n)πn Xn,1 − πn = a, lim bn − − m(n)E n→∞ rn rn τ where πn is a median for Xn,j . Theorem 4.1 combined with Sects. 2 and 3 suggest how to complete the picture of the behavior of the low intensity bootstrap of linear statistics in the general setup of triangular arrays of row-wise i.i.d. random variables. The construction in Theorem 2.1 and the independence of the original sample and the resampling scheme (the associated multinomial weights) lead us to think that, similarly as in the unconditional result in Theorem 2.2, the conditional bootstrap limit laws also are asymptotically the same as those obtained from partial sums of the original sample with the same number of summands. This is stated in the following theorem, which completely characterizes the behavior of the bootstrap sample mean for small orders of resampling intensity. We note that in this theorem there is no assumption on the behavior of the original k(n) sum j =1 Xn,j . The remarkable fact is that the behavior of the bootstrap of the sum with small resample sizes always mimics the corresponding partial sum with the same order of summands. Theorem 4.2 Let {Xn,j , j = 1, . . . , k(n), n ∈ N} be an infinitesimal triangular array ∗ , . . . , X∗ of row-wise i.i.d. random variables, and let Xn,1 n,m(n) be a bootstrap sample drawn, by the m(n) out of k(n) bootstrap, from Xn,1 , . . . , Xn,k(n) , where and min{m(n), k(n)} → ∞ as n → ∞. If m(n) 

  Xn,i − an →w ρ = N 0, σ 2 ∗ cτ Pois ν.

m(n) k(n)

→0

(4.1)

i=1

Then L

∗

m(n) 

 ∗ Xn,i

− m(n)X¯ n,τ

→w ρ in probability.

(4.2)

i=1

Note that there is no loss of generality in the assumption on ρ, because it must be infinitely divisible and the sequence of constants can be chosen to get a centered normal part. Also note that, according to Theorem 4.1,the condition in the theorem also is necessary. Proof Our proof is based on the well-known arguments related to the Central Limit Theorem (see also Cuesta-Albertos and Matrán 1998 in relation with the present framework); thus, they will be only sketched. ∗ , j = 1, . . . , m(n), n ∈ N} in First note that the infinitesimality of the array {Xn,j ∗ ∗ probability means that P (|Xn,j | > ) → 0 in probability for every  > 0. This is

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∗ | > )) → 0 for every  > 0, or, equivalently, to obviously equivalent to E(P ∗ (|Xn,j



 k(n)   1  E I(|Xn,j |>) = P |Xn,1 | >  → 0, k(n) j =1

i.e., to the infinitesimality of the array {Xn,j , j = 1, . . . , k(n), n ∈ N}, which is guaranteed by hypothesis. By resorting to the découpage de Lévy in the version given in Proposition 2 of Cuesta-Albertos and Matrán (1998) and following the same scheme as in Proposition 7 of Cuesta-Albertos and Matrán (1998), it is straightforward to reduce the problem to the consideration of the pure cases (a) ρ = N (0, σ 2 ) and (b) ρ = cτ Pois ν. ∗ )=X ¯ n,τ rn and the CLT as stated in Corollary 4.8 Taking into account that E ∗ (Xn,1 of Araujo and Giné (1980) (see also Proposition 3 of Cuesta-Albertos and Matrán 1998), to prove (4.2) under (a) it suffices to show that ∗ ∗ (a.i) m(n)E ∗ (Xn,1,τ − E ∗ Xn,1,τ )2 →p σ 2 and ∗ ∗ (a.ii) m(n)P (|Xn,1 | > ) →p 0.

Similarly, under (b), it suffices to show that  ∗ ){|x| > δ} → ν|{|x| > δ} in probability for every δ > 0 such (b.i) m(n)L∗ (Xn,1 w that ν{−δ, δ} = 0 and ∗ ∗ (b.ii) lim sup E(m(n)E ∗ (Xn,1,τ − E ∗ (Xn,1,τ ))2 ) ≤ g(τ ), where g(τ ) → 0 as τ → 0. As in the proof of (relation (13) in) Theorem 13 in Cuesta-Albertos and Matrán (1998), to show (a.i) take into account that   k(n) − 1 ∗ ∗ E(Xn,1,τ − EXn,1,τ )2 → σ 2 − E ∗ Xn,1,τ )2 = m(n) E m(n)E ∗ (Xn,1,τ k(n) by (a) and the necessary conditions for the CLT. Moreover, from the expression for the variance of the sample variance and from the inequality Var Z 2 ≤ 16τ 2 Var Z which holds if |Z| ≤ τ a.s. (see Proposition 4 in Cuesta-Albertos and Matrán 1998 for a proof), we have   ∗ ∗ − E ∗ Xn,1,τ )2 Var m(n)E ∗ (Xn,1,τ   1 sn 2 2 Var(Xn,1,τ − EXn,1,τ ) + = m(n) k(n) k(n)2 ≤

m(n)2 m(n)2 64τ 2 Var(Xn,1,τ ) + sn k(n) k(n)2

(4.3)

for some bounded sequence (sn )n . But the facts that m(n) = o(k(n)) and m(n) Var Xn,1,τ → σ 2 (by (a)) imply that (4.3) converges to 0, and thus (a.i) is proved.

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(a.ii) trivially follows from   ∗    E m(n)P ∗ |Xn,1 | >  = m(n)P |Xn,1 | >  → 0 by (a). This completes the proof of (4.2) under (a). If we assume (b), the necessary conditions for the CLT give that   m(n)L(Xn,j ) |x| > δ →w ν  |x| > δ

(4.4)

for every δ > 0 such that ν{−δ, δ} = 0 and that g(δ) := lim sup m(n)E(Xn,1,δ − EXn,1,δ )2 → 0

as δ → 0.

(4.5)

Regarding (b.i), we have  2 ∗ E m(n)L∗ (Xn,1 )(δ, ∞) − m(n)L(Xn,j )(δ, ∞)  2 k(n) 1  2 = m(n) E I(Xn,j >δ) − P (Xn,j > δ) k(n) j =1

=

  m(n)2 P (Xn,j > δ) 1 − P (Xn,j > δ) → 0, k(n)

since m(n)P (Xn,j > δ) → ν(δ, ∞) by (4.4) and m(n) k(n) → 0. The proof for the set (−∞, δ) is identical. Finally, for (b.ii), we have   ∗ ∗ E m(n)E ∗ (Xn,1,δ − E ∗ Xn,1,δ )2 = m(n)

k(n) − 1 E(Xn,1,δ − EXn,1,δ )2 k(n)

≤ m(n)E(Xn,1,δ − EXn,1,δ )2 =: gn (δ) and g(δ) = lim sup gn (δ) → 0 as δ → 0, showing (b.ii).



Now we are ready to obtain the characterization of the right behavior of the low intensity bootstrap for the sum of arrays of i.i.d. random variables. Theorem 4.3 Assume that {Xn,j : j = 1, . . . , k(n); n ∈ N} is a row-wise i.i.d. infinitesimal triangular array satisfying k(n) 

Xn,j − an →w γ .

(4.6)

j =1

If there exists a nondegenerate law, ρ, and a continuous function f : (0, ∞) → (0, ∞) satisfying that, for every sequence of integers {m(n)}n such that m(n)  ∞

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and m(n)/k(n) → 0,  L

∗

 m(n) 1  ∗ Xn,j − Cn →w ρ rn

in probability

(4.7)

j =1

with rn = f (m(n)/k(n)) and suitable centering, Cn , random variables measurable on the σ -fields σ (Xn,j : j = 1, . . . , kn ), n ∈ N, then ρ is α-stable for some α ∈ (0, 2], and the description for ρ and f given in Theorem 3.1 holds. Conversely, under the hypothesis of Theorem 3.1, the bootstrap with low resampling intensity works:  L

∗

m(n) k(n) 1  ∗ m(n) 1  Xn,j − Xn,j,τ rn rn k(n) rn j =1

 →w ρ

in probability,

(4.8)

j =1

where rn = f ( m(n) k(n) ) is that given in Theorem 3.1, m(n) is any sequence of positive integers such that m(n)  ∞ and m(n)/k(n) → 0, and ρ is α-stable with α ∈ (0, 2]. Proof Convergence (4.7) implies, by Theorem 1.1 of del Barrio et al. (1999), that ρ is an infinitely divisible law, ρ = N (a, α 2 ) ∗ cτ Pois ν, and that, for a suitable sequence of constants (bn )n , we have m(n) 1  Xn,j − bn →w ρ. rn

(4.9)

j =1

If we assume that this is true for every sequence m(n) → ∞ with

m(n) k(n)

→ 0,

f ( m(n) k(n) ),

where rn = as in Theorem 3.1, then we are just under the assumptions of this theorem. Finally, by Theorem 4.2 the converse implication leading to (4.8) is obvious.  Remark 4.1 Now we can answer the question about when the bootstrap of the mean works for small resampling intensities. If one wants to ensure γ in (4.6) to be the same as ρ in (4.8) for every low resampling size, then γ = ρ should be normal (and

m(n) then rn = m(n) k(n) ) or α-stable with 0 < α < 2 (and then rn = f ( k(n) ), and f must be regularly varying at 0 with exponent 1/α). However, it must be stressed that (as was noted in Example 1.1) even the fact that γ is normal does not guarantee that ρ is normal for every low rate m(n). Only if every subarray {Xn,j , j = 1, . . . , m(n), n ∈ N} with m(n) k(n) → 0 also satisfies (3.4) with ρ = γ this slow resampling bootstrap will always work. Finally, regarding Example 1.1 again, the analysis in Example 2.2 and Theorem 4.2 shows the obvious (conditional) behavior that the small resampling bootstrap can produce. Recalling the assumptions in Example 2.2, for the conditional m(n) ∗ law L∗ ( i=1 Xn,i − m(n)X¯ n ), we will respectively obtain the weak convergence in probability to the N (0, 1) law under assumption (a), to the centered Poisson law of

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parameter λ in probability under (b), and the convergence to zero in probability (and the same under any re-scaling) under (c). Our last example nicely explains how a whole spectrum of different limit laws for low resampling intensity may occur. Example 4.1 Let (Yt )t≥0 be a Lévy process with EYt = 0 and Var Yt < ∞ for all t ≥ 0. Choose the size of a grid Δn > 0 with Δn → 0 as n → ∞, and let k(n) be the sample size with k(n)Δn → ∞ and m(n) such that m(n)Δn → 1. The triangular array is now defined by the increments of the Lévy process as follows: Xn,j := Yj Δn − Y(j −1)Δn , 1 ≤ j ≤ k(n). The Central Limit Theorem implies the weak convergence of the normalized partial sums k(n) Yk(n)Δn j =1 Xn,j  = →w Z, (4.10) VarYk(n)Δn VarYk(n)Δn where Z is standard normal. In contrast to this fact, the low resampling bootstrap with resampling sample size [m(n)t], t > 0, produces by Theorem 4.2 a whole spectrum of limit laws L

∗

[m(n)t] 

 ∗ Xn,j

→w Yt

in probability.

(4.11)

j =1

We have cYt =d Z for some constant c > 0 if and only if (Yt )t≥0 is a Brownian motion. Only in this case there exists a suitable renormalization of (4.11) such that the limit law of the partial sums (4.10) can be reproduced by the low resampling bootstrap.

References Araujo A, Giné E (1980). The central limit theorem for real and Banach valued random variables. Wiley, New York Arcones M, Giné E (1989) The bootstrap of the mean with arbitrary bootstrap sample size. Ann Inst Henri Poincaré Probab Stat 25:457–481 Arcones M, Giné E (1991) Additions and correction to “the bootstrap of the mean with arbitrary bootstrap sample size. Ann Inst Henri Poincaré Probab Stat 27:583–595 Athreya KB (1987) Bootstrap of the mean in the infinite variance case. Ann Stat 15:724–731 del Barrio E, Matrán C (2000) The weighted bootstrap mean for heavy-tailed distributions. J Theor Probab 13:547–569 del Barrio E, Cuesta-Albertos JA, Matrán C (1999) Necessary conditions for the bootstrap of the mean of a triangular array. Ann Inst Henri Poincaré 35:371–386 del Barrio E, Cuesta-Albertos JA, Matrán C (2002) Asymptotic stability of the bootstrap sample mean. Stoch Proc Appl 97:289–306 del Barrio E, Janssen A, Matrán C (2007). Resampling schemes with low resampling intensity and their applications in testing hypotheses (submitted)

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Bickel PJ, Sakov A (2005) On the choice of m in the m out of n bootstrap and its application to confidence bounds for extreme percentiles. Preprint Bickel PJ, Götze F, van Zwet WR (1997) Resampling fewer than n observations: gains, losses, and remedies for losses. Stat Sin 7:1–31 Breiman L (1968) Probability. Addison–Wesley, Reading Csörgö S, Rosalsky A (2003) A survey of limit laws for bootstrapped sums. Int J Math Math Sci 45:2835– 2861 Cuesta-Albertos JA, Matrán C (1998) The asymptotic distribution of the bootstrap sample mean of an infinitesimal array. Ann Inst Henri Poincaré Probab Stat 34:23–48 Deheuvels P, Mason DM, Shorack GR (1993) Some results on the influence of extremes on the bootstrap. Ann Inst Henri Poincaré 29:83–103 Doney RA, Maller RA (2002) Stability and attraction to normality for Lévy processes at zero and at infinity. J Theoret Probab 15:751–792 Janssen A (2005) Resampling student T -type statistics. Ann Inst Math Stat 57:507–529 Janssen A, Pauls T (2003) How do bootstrap and permutation tests work?. Ann Stat 31:768–806 Mammen E (1992a) When does bootstrap work? Asymptotic results and simulations. Lecture notes in statistics, vol 77. Springer, New York Mammen E (1992b) Bootstrap, wild bootstrap, and asymptotic normality. Probab Theory Relat Fields 93:439–455 Politis DN, Romano JP (1994) Large sample confidence regions based on subsamples under minimal assumptions. Ann Stat 22:2031–2050 Politis DN, Romano JP, Wolf M (1999) Subsampling. Springer, New York Resnick SI (1987) Extreme values, regular variation and point processes. Springer, New York Swanepoel JWH (1986) A note in proving that the (modified) bootstrap works. Commun Stat Theory Meth 15(11):3193–3203 Wschebor M (1995) Almost sure weak convergence of the increments of Lévy processes. Stoch Proc Appl 55:253–270