Limit theorems for U-statistics of Bernoulli data
Davide Giraudo
TL;DR
This paper establishes limit theorems such as the law of large numbers, iterated logarithm, and central limit theorem for U-statistics derived from stationary Bernoulli data modeled as functionals of i.i.d. sequences, under dependence conditions.
Contribution
It introduces new limit theorems for U-statistics of dependent Bernoulli data, extending classical results to dependent sequences with a novel approach.
Findings
Proves strong law of large numbers for these U-statistics.
Establishes a bounded law of the iterated logarithm.
Derives a central limit theorem under dependence conditions.
Abstract
In this paper, we consider U-statistics whose data is a strictly stationary sequence which can be expressed as a functional of an i.i.d. one. We establish a strong law of large numbers, a bounded law of the iterated logarithms and a central limit theorem under a dependence condition. The main ingredients for the proof are an approximation by U-statistics whose data is a functional of i.i.d. random variables and an analogue of the Hoeffding's decomposition for U-statistics of this type.
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Taxonomy
TopicsProbability and Risk Models · Random Matrices and Applications · Financial Risk and Volatility Modeling
Limit theorems for U-statistics of Bernoulli data
Davide Giraudo
Ruhr-Universität Bochum, Germany
Abstract.
In this paper, we consider U-statistics whose data is a strictly stationary sequence which can be expressed as a functional of an i.i.d. one. We establish a strong law of large numbers, a bounded law of the iterated logarithms and a central limit theorem under a dependence condition. The main ingredients for the proof are an approximation by U-statistics whose data is a functional of i.i.d. random variables and an analogue of the Hoeffding’s decomposition for U-statistics of this type.
Key words and phrases:
Law of large numbers; law of the iterated logarithms; central limit theorem; U-statistics; functionals of i.i.d.
1. Introduction and main results
1.1. Context
Let be a strictly stationary sequence, in the sense that the vectors and have the same distribution for all and . The U-statistic of kernel and data is defined as
[TABLE]
The study of the asymptotic behavior of the sequence properly normalized is a question of interest in probability theory and the applications. We will be interested in the following three limit theorems.
- (1)
Law of large numbers: let ; the following convergence holds
[TABLE] 2. (2)
Bounded law of the iterated logarithms: the random variable
[TABLE]
is almost surely finite, where is defined by and . 3. (3)
Central limit theorem: there exists a such that
[TABLE]
where is a standard normal random variable.
Usually, the conditions for guaranting this kind of limit theorems are on the dependence of the sequence and also on the kernel , for example by requiring some integrability conditions on . We will first review a few results for the case where the data is i.i.d. and the kernel is symmetric.
- (1)
If and , then (1.2) holds [14]. 2. (2)
If , then the random variable defined by (1.3) is almost surely finite. Moreover, for all , according to Theorem 2.5 in [5], the following inequality holds:
[TABLE] 3. (3)
The convergence (1.4) has been established in [16].
The extension of these results to the case of stationary dependent data is a challenging problem. The law of large numbers has been established under a -mixing assumption in [3, 12]. In the case , the independence assumption can be replaced by a -dependence assumption, that is, for all -uple of distinct integers , the collection of random variables is independent. If the sequence is identically distributed, then the law of large numbers hold (Theorem 1 in [4]). The question of an equivalent of the ergodic theorem when the data is any strictly stationary sequence was considered in [1]. The question of the law of the iterated logarithms was also adressed in [11] where the data is allowed to be -mixing or a functional of a -mixing sequence. A central limit theorem has been established under a -mixing assumption in [2] and under an -mixing condition in [10]. The -dependent case has been addressed in [18]. The case of associated random variables has also been investigated in [13]. Note that there are also results dealing with the asymptotic behavior of -statistics of a semimartingale [20] and of Poisson processes [21].
In this paper, we will be interested in establishing the law of large number, the law of the iterated logarithms and the central limit theorem when the data can be expressed as a functional of an i.i.d. sequence. In a similar context on the data but for weighted U-statistics, the central limit theorem was investigated in [17].
Let us precise the context. Let be an independent identically distributed sequence with values in , . Given measurable functions and , we are interested in the asymptotic behavior of the U-statistic of order two defined by
[TABLE]
that is, letting , is a U-statistic of kernel and the data is the strictly stationary sequence . More precisely, we are interested in conditions involving the kernel and the sequence which guarantee the previously mentioned limit theorems.
In all the paper, the kernel is supposed to be symmetric in the sense that for all .
The paper is organized as follows. In Subsection 1.2, we will introduce a measure of dependence of a U-statistic whose data is a functional of an i.i.d. sequence. In Subsection 1.3, we formulate an analogue of the Hoeffding decomposition for such U-statistics. Subsections 1.4, 1.5 and 1.6 are devoted respectively to the statements of the law of large numbers, the bounded law of the iterated logarithms and the central limit theorems for U-statistics of Bernoulli data. In Subsection 1.7, we give examples of kernels for which the measure of dependence can be estimated only with the help of the dependence of the data. Section 2 is devoted to the proofs of the previously mentioned results.
1.2. Measure of dependence
The extension of the results of the i.i.d. case requires a measure of dependence. Let be an i.i.d. sequence, a measurable function and . In order to deal with the dependence which comes into play in , we need the following notations. Denote and define the random vectors
[TABLE]
The random variable can be writen as a function of and by stationarity and Lemma A.6, the involved function does not depend on . Therefore, we write
[TABLE]
We then define for and the coefficient of dependence
[TABLE]
and for , .
In particular, finiteness of allows to write
[TABLE]
where the convergence takes place in and almost surely. The interest of the decomposition (1.10) is that it reduces the treatmeant of the original U-statistic to that of U-statistics whose data is a strictly stationnary sequence which is a functional of independent identically distributed random variables. Nevertheless, this task requires some work in order to be reduced to U-statistics of independent data.
1.3. A generalized Hoeffding’s decomposition
A usefull tool to establish limit theorems for U-statistics with i.i.d. data is the Hoeffdings’s decomposition [16]. Let be a symmetric measurable function and be an i.i.d. sequence. We write decompose in the following way:
[TABLE]
where ,
[TABLE]
[TABLE]
In this way, the following equality holds
[TABLE]
The part involving can be treated by martingale techniques, since the sequence is a martingale differences sequence with respect to the filtration and the terms can be treated thanks to a reverse martingale differences property.
We would like to extend this to the setting mentioned in 1.1, that is, , where is an i.i.d. sequence. We introduce the following notation
[TABLE]
where is an i.i.d. sequence of random variables with values in and is a measurable function.
One naturally expects the decomposition to be more complicated than in the independent case. Let us point out the major differences and common points. Like in the independent case, the decomposition of the centered U-statistic involves a stationary sequence and a degenerated U-statistic. But in the context of Bernoulli data, one get a series involving stationary sequences and a series of degenerated U-statistics. An other difference is that we also have remainder terms which are not directly associated to the involved stationary sequences or degenerated U-statistic. The origin of these terms will be explained during the proof.
Proposition 1.1**.**
Let be an i.i.d. sequence of random variables, an independent copy of and be a measurable function. Let and . Let be a function such that
[TABLE]
Let be a symmetric measurable function. Assume that the following convergence holds almost surely for all :
[TABLE]
Then the following equality holds:
[TABLE]
where for each and all , the U-statistic has independent data and is degenerated, and the remainder terms are defined as
[TABLE]
[TABLE]
[TABLE]
[TABLE]
[TABLE]
[TABLE]
[TABLE]
with
[TABLE]
1.4. Law of large numbers
We first present a result on the Marcinkievicz law of large numbers for U-statistics of i.i.d. data. In order to extend it to the context of functional of i.i.d., we need a control on a maximal function.
Proposition 1.2** (Marcinkievicz-Zygmund law of large numbers).**
Let be an i.i.d. sequence with values in and let be a measurable function. For , define
[TABLE]
Assume that is finite and that for all , . Then the following statements hold:
- (1)
the sequence converges to zero almost surely ; 2. (2)
for any positive ,
[TABLE]
where is bigger than and depends only on .
Corollary 1.3**.**
Let be an i.i.d. sequence with values in and let be a measurable function. For , define
[TABLE]
Assume that is finite and . Then the following statements hold:
- (1)
the sequence converges to zero almost surely ; 2. (2)
for any positive ,
[TABLE]
where is bigger than and depends only on .
The previous Proposition gives a control in terms of the weak--semi-norm, defined as , . However, the triangular inequality may fail, which is not convenient in view of the decomposition obtained in Proposition 1.1. For this reason, we introduce the weak--norm, defined as
[TABLE]
Notice that for all fixed , there exists constants and such that for all random variable , . Moreover, the observation that for any sequence of non-negative random variables ,
[TABLE]
will be useful in the sequel.
We are now in position to present our first result, which gives a control on the maximal function of a U-statistic whose data is a functional of an i.i.d. sequence.
Theorem 1.4**.**
Let be an i.i.d. sequence, and be measurable functions. Let . Then
[TABLE]
The second result of this Subsection is a Marcinkievicz law of large numbers.
Theorem 1.5** (Marcinkievicz law of large numbers).**
Let be an i.i.d. sequence, and be measurable functions. Let . Suppose that
[TABLE]
Then the following almost sure convergence holds:
[TABLE]
A law of large numbers has been obtained in [8], Theorem 6. They considered the classical law of large numbers () but for functions of a -mixing process. When applied to a Lipschitz-continuous kernel and a function of an i.i.d. process, their condition requires
[TABLE]
When is a linear process with Bernoulli innovations, that is, and , this is equivalent, by Marcinkievicz-Zygmund inequality, to
[TABLE]
while our condition read and is less restrictive. However, in [8], the case of functional of -mixing process and of a not necessarily continuous kernel is treated there and not in the present paper.
1.5. Bounded law of the iterated logarithms
Let us present our next result concerning the bounded law of the iterated logarithms. Like in the independent case, one can control the moments of order of the maximal function of U-statistic with the normalisation .
Theorem 1.6**.**
Let be an i.i.d. sequence, and be measurable functions. Let . For all , the following inequality holds:
[TABLE]
where depends only on .
1.6. Central limit theorem
Let us present our result concerning the central limit theorem. We will make essentially three assumptions on the dependence of our U-statistic. The first and second one involve the coefficients and respectively and the third condition is imposed is summability of the family of covariances of a strictly stationary sequence which comes from the generalized Hoeffding decomposition of Proposition 1.1.
Theorem 1.7**.**
Let be a measurable function, be a measurable function and be an i.i.d. sequence. Let
[TABLE]
Suppose that
[TABLE]
and that
[TABLE]
where the random variable is defined by the following -convergence
[TABLE]
and , , where is an independent copy of .
Then the following convergence in distribution holds:
[TABLE]
where
[TABLE]
Let us give some comments about this results and how it relates with existing results.
Remark 1.8*.*
It is not clear to us whether the condition (1.40) guarantees (1.41). Nevertheless, conditions of the type
[TABLE]
guarantee (1.41) and are not too hard to check for the choices of kernel involved in Subsection 1.7.
Remark 1.9*.*
In the context of Theorem 1.7, it is possible the variance equals [math]. In this case, the normalization should change and a deeper analysis is required in order to identify the limiting process.
Remark 1.10*.*
It does not seem to be an easy task to compare the condition of Theorem 1.7 with those in [17] in full generality. One way to measure the dependence in the latter paper is given (in the case where the weights are all equal to one) by the coefficients
[TABLE]
A sufficient condition (which follows from a combination of Theorem 3 and Proposition 4) for the central limit theorem is
[TABLE]
In a particular case, this condition can be compared with ours. Let and be a causal linear process:
[TABLE]
where is i.i.d. centered and square integrable and is square summable, conditon (1.47) reads
[TABLE]
(see Theorem 9). With the same kernel and definition of , our condition is more restrictive, since the convergence of implies that of .
Nevertheless, our result deal with not necessarily causal process.
Remark 1.11*.*
A central limit theorem has been obtained in [8], Theorem 7, for functions of a -mixing process. When applied to a Lipschitz-continuous kernel and a function of an i.i.d. process, their condition requires
[TABLE]
When is a linear process with Bernoulli innovations, that is, , this is equivalent to
[TABLE]
while our condition read and is less restrictive. However, in [8], the case of functional of -mixing process and of a not necessarily continuous kernel is treated there and not in the present paper.
Remark 1.12*.*
In [6], an approach based on empirical processes has been done in order to derive a central limit theorem for -statistics of dependent data. As mentioned in Example 3.8, the case of function of i.i.d. processes can be treated, since the convergence of empirical processes of functions of an i.i.d. sequence has been treated in [24]. However, in the later paper, the condition on the weak dependence relies on the physical dependence measure of and this does not seem to be comparable with our assumptions.
1.7. Applications
In this Subsection, we give examples of kernels for which the measure of dependence defined by (1.9) can be estimated.
- (1)
Uniformly continuous kernel
Let be a measurable function. We assume that there exists a non-negative function which is increasing, satisfies and for each ,
[TABLE]
Then the following inequality holds:
[TABLE]
Since has the same distribution as , the two terms of the right hand side of (1.53) are equal hence
[TABLE]
In particular, if is -Hölder continuous for some , we can choose for some constant and in this case, the estimate (1.54) becomes
[TABLE]
which can be rewritten as
[TABLE] 2. (2)
Variance estimation. Consider the kernel defined by . The associated U-statistic is (after normalization) the classical variance estimator. For this choice of kernel, one can estimate the measure of dependence defined by (1.9). One needs to control the -norm of the difference of the square of two random variables and . Since
[TABLE]
we derive that
[TABLE]
We use this for a fixed to and . Accounting the following bounds (which are a consequence of stationarity):
[TABLE]
we get that
[TABLE]
2. Proofs
2.1. Proof of the generalised Hoeffding’s decomposition
Let us explain the idea of proof of Proposition 1.1. First, we write as a series (index by ) of U-statistics. For the term of index , the data of the corresponding U-statistic is a function of i.i.d. random variables. We then decompose this a sum of U-statistics of independent data.
We divide the proof in two steps:
- (1)
First we treat the case where and a symmetric function. 2. (2)
Then we write the mentioned U-statistic a sum of U-statistics tractable with the work of step 1, plus a remainder term.
Step 1: we decompose a U-statistic whose data is a function of i.i.d. random variables as a sum of U-statistics of i.i.d. data plus remainder terms.
Lemma 2.1**.**
Let be an integer, be a measurable function, let be an i.i.d. real-valued sequence and an independent copy of . Define , where and .
Then
[TABLE]
where
- •
the function is defined for by
[TABLE]
and for by
[TABLE]
- •
the random vectors are defined by
[TABLE]
[TABLE]
- •
the remainder terms are defined (with the convention that , ) by
[TABLE]
[TABLE]
[TABLE]
[TABLE]
[TABLE]
[TABLE]
Proof.
Before going into the details of the proof, let us explain the general idea, which will also explain the origin of the remainder terms. First, it turns out that it would be more convenient that is a multiple of . However, it has no reason to be the case and we should also take into account the difference between and where is a multiple of close to . The control of the difference of these two terms is precisely . Now that we are reduced to the case where is a multiple of , we look that the terms and the remainder of and by the Euclidian division by . If these remainder are closed enough, then we can write the corresponding sum as a U-statistic whose data are independent vectors of length . If the remainders are too far way, we have to add and substract a term to be reduced to the previous case, and this leads to the definition of , . If the quotient in the Euclidian division by of and are the same, then the corresponding sum is .
From the equality we get by (2.6) that
[TABLE]
For simplicity, let us denote by the set and . With these notations and in view of (2.7), the following equality takes place
[TABLE]
Let us treat the first term. For such that , in view of the definitions (2.2) and (2.4) and the symmetry of , the following equaliy holds
[TABLE]
hence
[TABLE]
Let us treat the second term in the right hand side of (2.13). Adding and stubstracting the terms and gives, after having rewriten the corresponding sums as double sums and exploited a telescoping of the inside sum,
[TABLE]
where
[TABLE]
We express as a U-statistic of independent data. Noticing that , we are in a similar situation as in the case , with replaced by . Therefore, in view of (2.5), (2.3) and (2.9), we obtain that
[TABLE]
Collecting these terms gives
[TABLE]
We end the proof of Lemma 2.1 by combining the equalities (2.13), (2.15) and (2.19). ∎
Now, we can apply the Hoeffding decomposition to each U-statistic of i.i.d. data and rewrite the remainder term in a more tractable form.
Lemma 2.2**.**
Let be an integer, be a measurable function, let be an i.i.d. real-valued sequence and an independent copy of . Define , where , and .
Then
[TABLE]
where the function is defined
[TABLE]
and
- •
the function is defined for by
[TABLE]
and for by
[TABLE]
- •
the random vectors are defined by
[TABLE]
[TABLE]
- •
the remainder terms are defined (with the convention that , ) by
[TABLE]
[TABLE]
[TABLE]
[TABLE]
[TABLE]
[TABLE]
[TABLE]
To sum up, the centered U-statistic whose data is a function of i.i.d. random variables can be decomposed as a partial sum of a strictly stationary sequence, a sum of degenerated U-statistics plus a remainder term.
Proof of Proposition 1.1.
By the assumption (1.17), the following equality holds almost surely:
[TABLE]
Since is a U-statistic of independent identically distributed data, it can be treated by the classical Hoeffding’s decomposition written in (1.14).
To proceed, we need to decompose for a fixed the term . To this aim, we apply Lemma 2.2 in the following setting: the function is replaced by , which is defined by
[TABLE]
In this way,
[TABLE]
and with the notation , Lemma 2.1 gives
[TABLE]
Observe that
[TABLE]
and using Lemma A.5, this equality becomes
[TABLE]
Moreover,
[TABLE]
where
[TABLE]
We conclude by collecting all the terms. ∎
2.2. Proof of the results of Subsection 1.4
Proof of Proposition 1.2.
It will be more convenient to work with dyadics, since the martingale property will be useful to handle the maximums. First observe that , where
[TABLE]
For a fixed integer , consider the event
[TABLE]
It suffices to prove that there exists a constant (depending only on ) such that
[TABLE]
Indeed, item 1 follows from an application of (2.43) to for a positive and the Borel-Cantelli lemma. In order to prove item 2, we notice that and we apply (2.43) to for each positive . Consequently, we focus on establishing a satisfactory bound for .
Define for the random variable . Let denote the -algebra generated by the random variables . Define
[TABLE]
[TABLE]
Since , it follows that hence , where
[TABLE]
[TABLE]
Let us bound . Markov’s inequality entails
[TABLE]
and since is a martingale differences sequence, we obtain by Doob’s inequality and orthogonality of increments that
[TABLE]
Now, we use
[TABLE]
and after the substitution , we get
[TABLE]
We thus obtained the estimate
[TABLE]
In order to bound , we start by Markov’s inequality to get
[TABLE]
Since there exists a constant depending only on such that
[TABLE]
we derive after the substitution that
[TABLE]
The combination of (2.54) with (2.57) yields
[TABLE]
We are thus reduced to control the tail of , which will be done by using Proposition A.1. Our particular setting permits some simplification of the involved terms.
We first observe that has the same distribution as (since the vectors and are identically distributed). Define . Since for all , the sequence is a martingale differences sequence for the filtration where is the -algebra generated by , . We apply Proposition A.1 to for a fixed positive and . let . By Lemma A.5 applied to , and , we have
[TABLE]
Using Lemma A.6 with , and , we derive that
[TABLE]
Using this equality combined with the fact that the random variables , have the same distribution as , one gets
[TABLE]
In view of (2.58), we derive that
[TABLE]
Summing over , we get (2.43) in view of the inequality
[TABLE]
for a non-negative random variable and the convergence of the integrals and . This ends the proof of Proposition 1.2. ∎
2.2.1. Treatment of and
Recall that
[TABLE]
and
[TABLE]
[TABLE]
For a fix , we evaluate the contribution of of and .
Lemma 2.3**.**
Let . The following inequalities hold:
[TABLE]
[TABLE]
where depends only on .
Proof.
First observe that is a sum of at most random variables whose weak--norm does not exceed hence by cutting the supremum where is between two consecutive multiples of gives
[TABLE]
In order to treat , we decompose it as , where
[TABLE]
[TABLE]
Using (1.32), it follows that
[TABLE]
and for all such that ,
[TABLE]
hence
[TABLE]
For all fixed , we notice using Lemma A.5 that is a martingale differences sequence with respect to the filtration where
[TABLE]
hence by Burkholder’s inequality,
[TABLE]
and plugging this estimate into (2.77) gives
[TABLE]
In order to treat the contribution of , we observe that is independent of hence
[TABLE]
Consequently, the control of the contribution of can be done thanks to Proposition A.2.
This ends the proof of Lemma 2.3. ∎
2.2.2. Treatment of terms of the form and
Lemma 2.4**.**
For all and all , the following inequality holds
[TABLE]
Lemma 2.5**.**
For all and all , the following inequality holds
[TABLE]
These two lemmas are the consequence of the following observations.
- (1)
We first assume that ; if not we add and substract and use telescoping to treat instead (where we will apply the previous case to and ). A similar method can be used to treat the sum involved in Lemma 2.5. 2. (2)
The supremum involved in the statement can be restricted to the integers with are a multiple of . 3. (3)
For all integer and all sequence of random variables , , we write
[TABLE]
With the choice , the following equality holds:
[TABLE]
when . Since and can be both expressed as U-statistics of i.i.d. data, we can use Corollary 1.3 to treat these terms.
Lemma 2.5 can be done in a similar way: we express this time the involved sum as .
2.2.3. Treatment of terms of the form
Lemma 2.6**.**
For all and all , the following inequality holds
[TABLE]
This follows from the fact that forms a two-dependent sequence.
2.2.4. Proof of Theorem 1.4 and
Theorem 1.5
Theorem 1.4 is a consequence of the combination of Proposition 1.1, the estimates of Lemmas 2.3, 2.4, 2.6 and 2.5.
In order to prove Theorem 1.5, we start from the decomposition
[TABLE]
where for a fixed , is the sum for the indexes smaller or equal to (viewing the terms associated to as the corresponding ones for ) and the remaining term. We have to prove that for each positive ,
[TABLE]
Using (2.86) and the fact that consists of sums of terms which can be treated by Lemmas 2.3, 2.4, 2.6 and 2.5, we derive that for all fixed ,
[TABLE]
Bounding the latter probability by and using Lemmas 2.3, 2.4, 2.6 and 2.5, we can see that for some constant depending only on ,
[TABLE]
which can be made arbitrarily small.
2.3. Proof of Theorem 1.6
Like for the results on the strong law of larger numbers, we have to control the contribution of the extra-terms in the decomposition obtained in Proposition 1.1.
2.3.1. Treatment of and
Recall that
[TABLE]
and
[TABLE]
[TABLE]
For a fix , we evaluate the contribution of of and .
Lemma 2.7**.**
Let . The following inequalities hold.
[TABLE]
[TABLE]
Proof.
First observe that is a sum of at most random variables whose weak--norm does not exceed hence by cutting the supremum where is between two consecutive multiples of gives
[TABLE]
In order to treat , we decompose it as , where
[TABLE]
[TABLE]
We first use (1.32) to get
[TABLE]
and for all such that ,
[TABLE]
hence
[TABLE]
For all fixed , we notice using Lemma A.5 that is a martingale differences sequence with respect to the filtration where
[TABLE]
hence
[TABLE]
and plugging this estimate into (2.103) gives
[TABLE]
We control the contribution of by noticing that is actually independent on .
This ends the proof of Lemma 2.7. ∎
2.3.2. Treatment of terms of the form and
Lemma 2.8**.**
For all and all , the following inequality holds
[TABLE]
Lemma 2.9**.**
For all and all , the following inequality holds
[TABLE]
The proof follows exactly the same idea as the proof of Lemmas 2.4 and 2.5, where the use of Corollary 1.3 is replaced by that of Proposition A.4.
2.3.3. Treatment of terms of the form
Lemma 2.10**.**
For all and all , the following inequality holds
[TABLE]
This follows from the fact that forms a two-dependent sequence.
2.4. Proof of Theorem 1.7
Let us explain the idea of the proof. The convergence is essentially due to the partial sums of strictly stationnary sequence , where
[TABLE]
and
[TABLE]
We can establish the convergence of by showing the convergence of for a fixed and by controlling the remainder. We have to prove that all the terms in the decomposition obtained in Proposition 1.1 converge to zero in probability. To sum up, we start by writing
[TABLE]
where for each and all , the U-statistic has independent data and is degenerated, and the remainder terms are defined as
[TABLE]
[TABLE]
[TABLE]
[TABLE]
[TABLE]
[TABLE]
[TABLE]
[TABLE]
with
[TABLE]
We then follow the steps :
- (1)
we show that converges to a centered normal distribution with variance 2. (2)
We show the convergence in probability to zero of all the terms , , , .
2.4.1. Convergence of
We use Theorem 4.2 in [7], which states the following. For , , and are real-valued random variables defined on a common probability space . We assume that
- (1)
for all , in distribution as ; 2. (2)
in distribution as , and 3. (3)
for each , .
Then in distribution as .
We will apply the result in the following setting:
[TABLE]
a centered normal variable with variance and a centered normal variable with variance .
The first item follows from the central limit theorem for -dependent random variables; the second one from the convergence of to , which can be seen by writing and using Cauchy-Schwarz inequality:
[TABLE]
and the quantities and are bounded independently of and .
For the third item, we start from Chebytchev’s inequality:
[TABLE]
and
[TABLE]
and does not exceed a constant times hence
[TABLE]
2.4.2. Convergence in probability of ,
and , to [math]
Observe that
[TABLE]
that and that the number of terms in the two inner sums is of order hence
[TABLE]
We use the same method for the terms and , .
2.4.3. Treatment of
First we rewrite as a double sum, namely, as
[TABLE]
where
[TABLE]
with the notation if the assertion holds and [math] otherwise. Write as , where . Then
[TABLE]
hence
[TABLE]
and taking the expectation give that
[TABLE]
Acknowledgement: this research supported by the German National Academic Foundation and Collaborative Research Center SFB 823 "Statistical modelling of nonlinear dynamic processes".
Appendix A Appendix
In this appendix, we collect some fact about partial sums of martingales or functional of independent sequences that we will need in the proof.
The first is a probability inequatlity for martingales; PropositionA.2 andA.3 give a control of the maximal function involved in the strong law of large numbers, respectively for martingales and functionals of a fixed number of i.i.d. random variables. We end the Appendix by two lemmas on conditional expectation.
Proposition A.1** **(Theorem 1.3
in [15]).
Let and . Then there exists constants and depending only on such that if is a martingale differences sequence with respect to a filtration , then for each integer and each positive ,
[TABLE]
The next Proposition gives a control of the maximal function involved in the strong law of large numbers. A control on the -th moment for was obtained in the proof of Theorem 4.1 in [23]. The control on the weak--moment was explicitely established in [9], p. 324, under a stationarity assumption, but the proof work for martingale with identically distributed increments.
Proposition A.2**.**
Let be an identically distributed martingale differences sequence with respect to the filtration . Then for all ,
[TABLE]
Proposition A.3**.**
Let and be a sequence such that , where is i.i.d. and is centered. Then for all ,
[TABLE]
Proof.
We first notice that
[TABLE]
and we apply Proposition A.2 to the sequences for all fixed . ∎
The next Proposition give a control of the weak--norm of the maximum function involved in the bounded law of the iterated logarithms for partial sum of stationary sequences. When the involved sequence is i.i.d. and centered, this reduces to Théorème 1 in [19]. This can be extended to the context of the functional of i.i.d. random variables by the same method as in the proof of Proposition A.4.
Proposition A.4**.**
Let and be a sequence such that , where is i.i.d. and is centered. Then for all ,
[TABLE]
Lemma A.5** (Proposition 2 p. 1693 in [22]).**
Let be a real-valued random variable and let and be two sub--algebra such that is independent of the -algebra generated by and . Then
[TABLE]
Finally, the next lemma is well-known.
Lemma A.6**.**
Let and be two independent random variables with values in . Let be a measurable function. Then
[TABLE]
where is defined by .
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