Limit theorems for singular Skorohod integrals
Denis Bell, Raul Bolanos, David Nualart

TL;DR
This paper establishes limit theorems for sequences of Itô and Skorohod integrals with singular asymptotic integrands, extending previous examples to broader classes of stochastic convolutions.
Contribution
It introduces new limit theorems for singular Skorohod integrals, generalizing earlier specific cases like the Peccati-Yor example.
Findings
Proves convergence in distribution for sequences of singular integrals.
Includes stochastic convolutions as a special case.
Extends classical results to more general integrand behaviors.
Abstract
In this paper we prove the convergence in distribution of sequences of It\^o and Skorohod integrals with integrands having singular asymptotic behavior. These sequences include stochastic convolutions and generalize the example first studied by Peccati and Yor in 2004.
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Limit theorems for singular Skorohod integrals
Denis Bell
University of North Florida, Department of Mathematics, Jacksonville, Florida, USA
,
Raul Bolaños
University of Kansas, Department of Mathematics, Lawrence, Kansas, USA
and
David Nualart
University of Kansas, Department of Mathematics, Lawrence, Kansas,USA
Abstract.
In this paper we prove the convergence in distribution of sequences of Itô and Skorohod integrals with integrands having singular asymptotic behavior. These sequences include stochastic convolutions and generalize the example first studied by Peccati and Yor in 2004.
D. Nualart is supported by NSF Grant DMS 1811181.
Mathematics Subject Classifications (2010): 60H05, 60H07.
Keywords: Skorohod integral, Malliavin calculus, convergence in law, stochastic convolution.
1. Introduction
The main objective of this paper is to study the limit in law of sequences of random variables defined by Skorohod integrals
[TABLE]
Here is a continuous process, is fractional Brownian motion with Hurst parameter lying in the range , and is a sequence of deterministic kernels converging (in some sense) to a delta function based at 1 (hence the “singular” in the title of the paper). We show, under suitable conditions on the and , that the limit law of the couple has the form , where is a random variable, independent of .
Our study of limit problems of this type was motivated by the special case , and , introduced in Proposition 2.1 of [8], and studied in Proposition 18 of [7], and Example 4.2 in [3]. Then, is standard Brownian motion and the integrals are of classical Itô type. Quantitative bounds for such integrals in the case have been established by Nourdin, Nualart & Peccati [5] using estimates derived from Malliavin calculus and, more recently, by Pratelli & Rigo in [6] for , using more a elementary (but nonetheless intricate) argument.
In this article, we provide a new approach to this problem, valid for a more general class of integrands exhibiting singular asymptotic behavior at the right-hand endpoint. The approach is based on the following observation. The singular behavior of the kernels in (1.1) as implies that the limit law of is determined by the behavior of integrals over arbitrarily time intervals . This makes it possible to study the limit law via the more tractable sequence of random variables
[TABLE]
where is a sequence of times chosen such that at a carefully chosen rate. We show that and have that same limit in , and hence in law. Furthermore, the Skorohod integrals in (1.2) are Gaussian, as is the process . It thus suffices to show that the sequence has a convergent variance and is asymptotically uncorrelated with .
In Section 3.1, we implement this argument in the case where is standard Brownian motion (denoted here by ) and the process is progressively measurable. In this case the integrals are Itô integrals and the argument is technically easier. The basic result in this section is Theorem 3.1. As a special case of this theorem, we obtain the limit in law of the aforementioned sequence
[TABLE]
Theorem 3.1 is extended in Theorems 3.3 and 3.4 to double, and multiple, integrals respectively. In Section 3.2 we discuss the problem for stochastic convolutions.
In Section 4, we study the case of fractional Brownian motion (). Here it turns out to be more convenient to work with the approximating sequence
[TABLE]
As is usual in this subject, the cases and seem to require slightly different hypotheses and analyses, with the latter proving more involved. Analogues of Theorem 3.1 are presented in Theorems 4.1 and 4.3 for these two cases. The proof involves the use the divergence operator on Wiener space and thus has the flavor of Malliavin calculus. As an example of these theorems, we obtain the main result of [6] concerning the limit law of the sequence , for in the range .
2. Preliminaries
Fractional Brownian motion with Hurst parameter , is a zero mean Gaussian process with a covariance function given by
[TABLE]
where . The Hilbert space is defined as the closure of the space of step functions on endowed with the scalar product
[TABLE]
Then the mapping can be extended to a linear isometry between and the Gaussian space spanned by . When , is just a standard Brownian motion and .
When , the inner product of two step functions can be expressed as
[TABLE]
where . The space of measurable functions on , such that
[TABLE]
denoted by , is a Banach space and we have the continuous embeddings .
When , the covariance of the fractional Brownian motion can be expressed as
[TABLE]
where is a square integrable kernel defined as
[TABLE]
for , with being a constant depending on . The kernel satisfies the following estimates
[TABLE]
and
[TABLE]
for all and for some constants . Define a linear operator from to as follows
[TABLE]
The operator can be extended to a linear isometry between the Hilbert space and , that is, for any , we have
[TABLE]
The space of Hölder continuous functions of order is included in .
Next, we introduce the derivative operator and its adjoint, the divergence. Consider a smooth and cylindrical random variable of the form , where ( and its partial derivatives are all bounded). We define its Malliavin derivative as the -valued random variable given by
[TABLE]
For any real number , we define the Sobolev space as the closure of the space of smooth and cylindrical random variables with respect to the norm given by
[TABLE]
Similarly, if is a general Hilbert space, we can define the Sobolev space of -valued random variables .
The adjoint of the Malliavin derivative operator , denoted as , is called the divergence operator. A random element belongs to the domain of , denoted as , if there exists a positive constant depending only on such that
[TABLE]
for any . If , then the random variable is defined by the duality relationship
[TABLE]
for any . We make use of the notation and call the Skorohod integral of with respect to the fractional Brownian motion . The Skorohod integral satisfies the following isometry property for any element :
[TABLE]
where is the adjoint of . As a consequence, we have
[TABLE]
We will make use of the following result.
Lemma 2.1**.**
Let and let . Then the process belongs to the domain of and
[TABLE]
We refer to [5] and the references therein for a more detailed account of the properties of the fractional Brownian motion and its associated Malliavin calculus (and to [1] for an introduction to the latter subject).
We will make use of the following property of the Gamma function.
Lemma 2.2**.**
For any positive
[TABLE]
Proof.
This is a direct application of Stirling’s formula. ∎
Throughout the paper we will make use of the notion of stable convergence provided in the next definition. Suppose that the fractional Brownian motion is defined in a probability space , where is the -completion of the -field generated by .
Definition 2.3**.**
Fix . Let be a sequence of random variables with values in , all defined on the probability space . Let be a -valued random variable defined on some extended probability space . We say that converges stably to , if
[TABLE]
for every and every bounded –measurable random variable .
Condition (2.7) is equivalent to saying that the couple converges in law to in the space (see, for instance, [2, Chapter 4]).
3. Singular limits of sequences of Itô integrals
Let be a standard Brownian motion. Denote by the natural filtration generated by . In this section we will study the asymptotic behavior of two types of sequences of Itô integrals. First, we discuss a class of integrals on that include a sequence of deterministic kernels converging to a delta function based at . Secondly, we apply our argument to stochastic convolutions with this type of asymptotic behavior.
3.1. Stochastic integrals concentrating at
Consider a sequence of bounded nonnegative functions on , that satisfies the following conditions:
(h1): There is a sequence such that
[TABLE]
(h2): For any , as .
The aim of this section is to study the asymptotic behavior of the sequence of Itô integrals
[TABLE]
where is a progressively measurable process such that .
Theorem 3.1**.**
Suppose that the process is continuous in at and the sequence satisfies conditions (h1) and (h2). Then, the sequence introduced in (3.1) converges stably, as to , where is a random variable independent of the process .
Proof.
Define
[TABLE]
where is the sequence appearing in condition (h1). Then, as ,
[TABLE]
Moreover, as ,
[TABLE]
because and
[TABLE]
by the -continuity of at . On the other hand, for any fixed ,
[TABLE]
As , the third term in (3.4) has limit and the first term converges to zero in view of hypothesis (h2), because
[TABLE]
Moreover, the second converges to zero as , uniformly in , because we can write
[TABLE]
Therefore,
[TABLE]
Now, (3.2), (3.3) and (3.5) imply that in , and hence also in law. Finally, notice that the sequence converges in law in the space to , where is a random variable independent of . This completes the proof. ∎
An example of a sequence of functions satisfying conditions (h1) and (h2) with is
[TABLE]
Indeed condition (h2) holds trivially and condition (h1) holds taking, for instance, , because and, therefore, as ,
[TABLE]
Thus we have proved the following.
Proposition 3.2**.**
The sequence of Itô integrals
[TABLE]
converges stably, as , to , where is a random variable independent of the process .
Remarks:
- (i)
We note that Proposition 3.2 was obtained by Nourdin, Nualart & Peccati in [2, Proposition 3.7] as a corollary of a theorem proved by integration by parts on Wiener space.
- (ii)
If we assume that is bounded on , then it is easy to show that we can remove condition (h2) in Theorem 3.1.
The next result is an extension of Theorem 3.1 to the case of double stochastic Itô integrals, which is proved by similar arguments. We need the following condition on the sequence , which is stronger than (h2):
(h3): For any , as .
Theorem 3.3**.**
Let be a two-parameter process satisfying the following conditions:
- (i)
* is -measurable for .*
- (ii)
.
- (iii)
* is continuous at in the sense.*
Consider the sequence of iterated Itô integrals
[TABLE]
where the sequence satisfies conditions (h1) and (h3). Then converges stably, as to , where is independent of the process , and is the second Hermite polynomial.
Proof.
Define
[TABLE]
where is the sequence appearing in condition (h1). By (h1) we have, as ,
[TABLE]
Also
[TABLE]
As in the proof of Theorem 3.1, fix and consider the decomposition
[TABLE]
The first term of (3.7) converges to zero as by condition (h3) and the third term converges to . For the second term we have the estimate
[TABLE]
which shows that this term converges to zero as , due to the continuity of and , uniformly in . In this way, we obtain
[TABLE]
Also
[TABLE]
At this point similar calculations to (3.6) show that the second term in (3.9) has limit , while the first term of (3.9) converges to [math] due again to Cauchy-Schwartz inequality, condition (h1) and the continuity of at in . Consequently, as ,
[TABLE]
Thus (3.6), (3.8) and (3.10) imply that in , and hence also in law. Finally to see that the limit of has the desired form note that
[TABLE]
where denotes the double Itô-Wiener integral. Then by using the fact that multiple stochastic integrals of this form can be written in terms of Hermite polynomials, we can write
[TABLE]
Then the conclusion follows because as , converges to in as , and the sequence converges in law in the space to , where is a random variable independent of . This implies that the limit law of has the stated form and completes the proof. ∎
Remarks:
- (i)
The previous theorem applies to the particular case , as before.
- (ii)
One can consider the more general situation of a sequence of bounded symmetric functions on , satisfying the following conditions:
(h12): There is a sequence such that
[TABLE]
(h22): For any , as .
In this case we need to compute the limit in law of , which is a more complicated problem that requires additional conditions on the sequence . We will not treat this problem here.
Theorem 3.3 can be extended to higher dimensions. The proof is similar and omitted. We need the following condition on the sequence , which is stronger than (h2):
(h3m): For any , as , where is the number of parameters.
Theorem 3.4**.**
Let be an -parameter stochastic process satisfying the following properties.
- i)
* is -measurable.*
- ii)
.
- iii)
* is continuous at in the sense.*
Consider the sequence of iterated Itô integrals
[TABLE]
where the sequence satisfies conditions (h1) and (h3m). Then, converges stably, as , to , where is independent of the process and is the th Hermite polynomial.
3.2. Asymptotic behavior of stochastic convolutions
As before, let be a standard Brownian motion and set . Consider a nonnegative continuous and bounded function on , such that . Let . Then is an approximation of the identity.
Let be an adapted and square integrable process. Define the stochastic convolution
[TABLE]
In this subsection we are interested in the asymptotic behavior of as tends to infinity. The limit in law will have the form , where is a Gaussian process independent of .
The following theorem is the main result of this subsection.
Theorem 3.5**.**
Assume is an adapted, square integrable process, continuous at a fixed time in the sense. Consider a nonnegative continuous and bounded function on , such that and as . Then, the stochastic convolution converges stably to as , where is a standard Gaussian random variable independent of .
Proof.
Let be a sequence decreasing to [math] so that . For , set
[TABLE]
where with . Then we can write
[TABLE]
Moreover, since is measurable we can write
[TABLE]
and therefore
[TABLE]
On the other hand, by Ito’s isometry property we can write
[TABLE]
That means, is the convolution of with , and by Theorem 9.9 in [9], we deduce
[TABLE]
Finally, by Itô’s isometry and the -continuity of at
[TABLE]
as . Thus as and hence in law. Finally, note that for each , and are independent random variables such that converges to and . This implies that the limit law of has the stated form and completes the proof. ∎
As in the proof of Theorem 3.5, if is a sequence decreasing to [math] so that , we can consider for each the sequence of random variables
[TABLE]
The next lemma establishes the asymptotic behavior of the sequence of processes .
Lemma 3.6**.**
The finite-dimensional distributions of the process introduced in (3.11) converge stably to those of a centered Gaussian process independent of and with covariance function given by
[TABLE]
Proof.
Let . We need to prove the convergence in law
[TABLE]
in the space . We can choose large enough so that for , the Gaussian random variable become uncorrelated and hence independent. Then as in the proof of Theorem 3.5, it holds that
[TABLE]
where the random vector has a standard Gaussian distribution on and is independent of . This completes the proof. ∎
Notice that we cannot expect that the convergence in Proposition 3.7 holds in . Indeed, although under some mild conditions the stochastic convolution has a continuous version, the process does not have a continuous version.
The following proposition establishes the convergence of the stochastic convolution as a process in the sense of the finite-dimensional distributions.
Proposition 3.7**.**
Under the assumptions of Theorem 3.5, suppose that the process is continuous in in the sense. Then the finite-dimensional distributions of the process converges stably to those of , where is a Gaussian process independent of with covariance function given by (3.12).
Proof.
Let . We want to show that
[TABLE]
where the random vector has a standard Gaussian distribution on and is independent of . As in the proof of Theorem 3.5, if is a sequence decreasing to [math] such that , we can consider for each the sequence of random variables defined in (3.11). Then, we have that, by the proof of theorem 3.5, for each ,
[TABLE]
Also, by the -continuity of and the Cauchy-Schwartz inequality, we can write
[TABLE]
In particular the above convergence holds also in probability, so that
[TABLE]
for . As a consequence,
[TABLE]
Then by Slutsky’s theorem (3.13) follows from the convergence in law
[TABLE]
which is a consequence of Lemma 3.6. This completes the proof. ∎
4. Skorohod integrals with respect to fractional Brownian Motion
Consider a fractional Brownian motion with Hurst parameter . That is, is a zero mean Gaussian process with covariance function (3.12). In this section we will study the asymptotic behavior as of a sequence of Skorohod integrals of the form
[TABLE]
where is a stochastic process verifying some suitable conditions. We split our study in two cases according to whether or .
Case
We will assume the following conditions on the sequence of nonnegative and bounded functions:
(h4): .
(h5): for some (where here, and in the sequel. denotes the -norm on ).
We are now ready to state and prove the main results of this section.
Theorem 4.1**.**
Assume is a fractional Brownian motion with Hurst parameter . Consider a sequence of nonnegative and bounded functions on satisfying conditions (h3), (h4) and (h5). Let be a stochastic process satisfying the following conditions:
- (i)
For any , and the mapping belongs to .
- (ii)
* is continuous in at .*
- (iii)
* where , and is the number appearing in condition (h5).*
Consider the sequence of Skorohod integrals introduced in (4.1). Then converges stably as to , where is a random variable independent of .
Proof.
Notice first that conditions (i) and (ii) imply that belongs to . Set . Denoting , in view of (2.6) we can write
[TABLE]
Both terms in (4.2) are handled similarly and we will show the details only for the second one. Let . Then, separating the second term in two integrals, yields
[TABLE]
At this step note that by condition (i)
[TABLE]
So there is a constant such that the first term in (4.3) is bounded by
[TABLE]
which converges to [math] as by condition (h3).
On the other hand, for the second term in (4.3), it follows from Cauchy-Schwartz inequality that
[TABLE]
By condition (h4), the sequence is bounded and by condition (ii) the first factor tends to zero as . This shows that tends to zero as . Repeating the same argument, we obtain that tends to zero as .
We have shown that in , and hence also in law. All is left is to show that the limit of has the desired form. To this end note that, applying Lemma 2.1, can also be written as
[TABLE]
Let be as in the statement of the theorem and note that . Applying Hölder’s inequality with , yields
[TABLE]
The second factor is the -norm of the fractional integral of order of the function on . By the Hardy-Littlewood inequality, this factor is bounded by a constant times , where . Taking into account conditions (iii) and (h5), we deduce
[TABLE]
In order to complete the proof of the theorem it suffices to show that converges in law in the space to , where is a random variable independent of . In view of the fact that is a Gaussian process, this will follow from the next two properties:
(a): , which follows from property (h4).
(b): For any , . In fact, using property (h5), we obtain for ,
[TABLE]
as . ∎
Theorem 4.1 can be applied to the example , and in this case, . Indeed, condition (h3) is obvious. Condition (h4) follows from Lemma 4.2 below. Condition (h5) holds for any . This means that in condition (iii) it suffices to show that the integral is bounded for some .
Lemma 4.2**.**
For any and
[TABLE]
In particular for
[TABLE]
Proof.
First of all, note that using yields
[TABLE]
where denotes the Beta function. Then
[TABLE]
The first part of the lemma now follows from the well-known relationship between the Beta and Gamma functions. The second part follows by taking and using Lemma 3.6. ∎
Case
We assume the following conditions on the sequence of nonnegative and bounded functions:
(h6): .
(h7): For any , we have
[TABLE]
(h8): for some .
Theorem 4.3**.**
Assume is a fractional Brownian motion with Hurst parameter . Consider a sequence of nonnegative and bounded functions on satisfying conditions (h3), (h4), (h6), (h7) and (h8). Let be a stochastic process satisfying the following conditions:
- (i)
For all , .
- (ii)
The mapping is Hölder continuous of order from into .
- (iii)
We have
[TABLE]
where and is the exponent appearing in condition (h8).
Consider the sequence of Skorohod integrals introduced in (4.1). Then converges stably as to , where is a random variable independent of .
Proof.
We divide the proof into 3 steps.
Step 1: We need to compute the variance of the random variable . Condition (h4) implies that
[TABLE]
Step 2: Showing , where
[TABLE]
As in the proof of theorem 4.1, we can write
[TABLE]
We only work with , the analysis of being similar by changing and appropriately by and in the argument below. We have, using (2.5) and (2.4),
[TABLE]
Since
[TABLE]
we obtain
[TABLE]
To handle the term we note that, by (2.2), there is a constant such that
[TABLE]
We will denote by a generic constant that may vary from line to line. Then by Minkowski’s inequality and condition (ii) for any we obtain
[TABLE]
The term can be estimated as follows
[TABLE]
Taking into account that , we deduce from condition (h3) that converges to zero as . For we can write
[TABLE]
From (4.5) and condition (h6), we deduce that as . Therefore, we have proved that
[TABLE]
Concerning the term , using Minkowski’s inequality, the estimate (2.3) and condition (ii), we obtain
[TABLE]
Then, for any , the integral converges to zero as due to condition (h3), whereas, by condition (h6),
[TABLE]
as . Therefore, we have proved that
[TABLE]
Finally for taking , it follows from Minkowski’s inequality that
[TABLE]
At this step we study each term separately. For both and note that so condition (ii) gives
[TABLE]
Also so that
[TABLE]
By condition (h4), is bounded uniformly in , and by condition (h6), is bounded as well. Therefore,
[TABLE]
Thus, in order to show that converges to zero as , it suffices to show that, for a fixed ,
[TABLE]
Using Minkowski’s inequality, condition (ii) and the estimate (2.3), we can write
[TABLE]
which converges to [math] as by condition (h7). This completes the proof of step 2.
Step 3: We show that the limit in law of has the desired form. To this end note that can also be written as
[TABLE]
First we will show using condition (iii) that
[TABLE]
as . Fix . We can write, by Hölder’s inequality
[TABLE]
The first factor converges to zero as by property (h8) and the second one is bounded by condition (iii). Therefore, (4.10) holds.
It remains to show that converges in law in the space to , where is a random variable independent of . This claim follows from Step 1 and the fact that for any , . Indeed, we can write
[TABLE]
and
[TABLE]
where . Then the result follows from property (h8) and the fact that
[TABLE]
∎
Theorem 4.3 can be applied to the example , when . Indeed, condition (h4), again with , holds by Lemma 4.4 below. Condition (h3) is obvious. Property (h6) follows from the following computations:
[TABLE]
which is uniformly bounded by Lemma 3.6. In order to show property (h7), we write, for any ,
[TABLE]
which converges to zero as .
To show property (h8), we write
[TABLE]
For the term , we have
[TABLE]
By Lemma 3.6, this term converges to zero as , provided . The same conclusion can be deduced for the term using Young’s inequality.
Lemma 4.4**.**
For any , we have
[TABLE]
Proof.
Using the operator and integrating by parts, we can write
[TABLE]
At this step we work each term in (4.11) separately. Since
[TABLE]
the first term is . Changing the order of integration in the second term yields
[TABLE]
Writing the third term as a triple integral, changing the order of integration and using Lemma 4.2 gives
[TABLE]
Thus (4.11) simplifies to
[TABLE]
which, due to Lemma 3.6, converges to
[TABLE]
∎
The reference list from the paper itself. Each links out to its DOI / PubMed record.
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