Example 2.1

Let T(x) = |x|^−1/2 for x ≠ 0 and T(0) = 0 and let Γ(t) = t²ξ, 0 ≤ t ≤ 1, where ξ is a standard normal random variable. (Or, alternatively, let T be as before and Γ(t) = t²W(t), 0 ≤ t ≤ 1, where W(t), 0 ≤ t ≤ 1 is a Brownian motion.) Then

does not exist.

Our first result gives a sufficient condition for the existence of

.

THEOREM 2.1. If (1.2) and (1.4) hold,

and

then

exists with probability one.

We note if Γ(t) has stationary increments, Γ(0) = 0 and

then the local time of Γ exists with probability one (cf. Geman and Horowitz, 1980, Thm. 22.1). Geman and Horowitz (1980) also point out that under conditions like (2.4) the trajectories of Γ(t) oscillate wildly. Hence (2.4) is never satisfied for smooth (e.g., differentiable) Gaussian processes.

Remark 2.1. Although Theorem 2.1 covers a very large class of functions T, from a purely mathematical point of view the question arises what happens if instead of (1.4) we assume only that T is Lebesgue measurable, i.e., the level sets {T < c} are Lebesgue measurable for any real c. (A set on the real line is called Lebesgue measurable if it is the union of a Borel set and a subset of a Borel set with measure 0. Unlike the class

of Borel sets, the class

of Lebesgue measurable sets has the property that given any set

with measure 0, all subsets of A belong to

.) If instead of (1.4) we assume only that T is Lebesgue measurable, the pathwise existence of

becomes a delicate problem, because the composition of a Lebesgue measurable and a continuous function is in general not Lebesgue measurable. (See Halmos, 1950, p. 83.) If Γ(t) has a continuous local time, this problem disappears because, as an argument in Pötscher (2004, p. 5) shows, in this case the integrand in (1.3) differs from a Borel-measurable function only on a set of Lebesgue measure 0, and hence it is Lebesgue measurable. However, the assumptions of Theorem 2.1 do not, in general, imply the existence of a local time for Γ, and thus the measurability of T(Γ(t)) remains open. To handle this case, let T_n be a sequence of functions such that T_n is continuous in the interval

. (Such a sequence T_n exists by Luzin's theorem (cf. Hewitt and Stromberg, 1969, Thm. 11.36).) As a trivial modification of the proof of Theorem 2.1 shows, in this case

is a Cauchy sequence of random variables in L₁ norm and hence convergent in L₁. Defining its limit (which is easily shown to be independent of the approximating sequence T_n) as

, all results of our paper remain valid under the Lebesgue measurability of T. Note that in this case the definition of the integral

is not pathwise.

Next we show an example where Theorem 2.1 can be used to establish the existence of

.

Example 2.2

Let T(x) = |x|^−1/2, x ≠ 0, T(0) = 0, and

where W(t) is Brownian motion. Clearly, Γ(t) is differentiable and σ²(t) = t³/3. Theorem 2.1 can be used with

to establish the existence of

in this case.

After giving conditions for the existence of

, we turn to generalizations of Theorem 1.2.

THEOREM 2.2. Assume that (1.1), (1.2), (1.4), (2.1), (2.2), and (2.3) are satisfied and there exist numbers c_i,k > 0, 1 ≤ k ≤ n, n ≥ 1 such that

and

Then (1.3) holds.

We now give some applications of Theorem 2.2. Let e_i, 1 ≤ i < ∞ be a stationary sequence of Gaussian random variables with Ee_i = 0 and r(i − j) = Ee_i e_j such that

where

Let

for any 1 ≤ k ≤ n. According to Lemma 5.1 of Taqqu (1975),

where B_H(t) is a fractional Brownian motion with parameter H. This means that B_H(t) is a continuous Gaussian process with EB_H(t) = 0 and

Thus conditions (1.1), (1.2), (2.1), and (2.4) are satisfied, because σ(t) = (EB_H²(t))^1/2 = t^H, 0 ≤ t ≤ 1. Clearly x_k,n /c_k,n is standard normal and

By 0 < H < 1 we have for any fixed k

provided that [sum ]_1≤i,j≤k r(i − j) > 0. On the other hand, if [sum ]_1≤i,j≤k r(i − j) = 0, then P(x_k,n = 0) = 1. By (2.8) this can happen only for finitely many k's, independently of n. Drop these 0 terms and use Theorem 2.2 for the rest of the array only. By (2.8) there is a constant k₀ such that

Using the properties of slowly varying functions (cf. Bingham, Goldie, and Teugels, 1987, p. 26) we obtain

Hence (2.6) holds with α = 1, and therefore all conditions of Theorem 2.2 are established with α = 1.

The next two examples are from Taqqu (1975).

Example 2.3

If the covariance functions satisfy

where L₁(x) and L₂(x) are slowly varying function at infinity, then (2.8) and (2.9) hold. For the proof we refer to Taqqu (1975).

Example 2.4

Let {ε_k,−∞ < k < ∞} be a sequence of independent, identically distributed standard normal random variables and define

It is easy to see that {e_j, 1 ≤ j < ∞} is a stationary Gaussian sequence with Ee_j = 0 and covariance function r satisfying

Thus (2.8) and (2.9) hold.

We note that the convergence relation (1.3) was also established by Jeganathan (2004) under the conditions of Example 2.4 assuming that T and T² are both Lebesgue integrable on the real line. The limit in Jeganathan (2004) is given as a functional of the local time of fractional Brownian motion.

Our next example is from Horváth and Kokoszka (1997).

Example 2.5

Consider the fractional ARIMA (p,d,q) process, which is a parametric model frequently used in modeling of long-memory time series (see, e.g., Brockwell and Davis, 1991, Sec. 13.2). Let {ε_k,−∞ < k < ∞} be a sequence of independent, identically distributed normal random variables with Eε_k = 0 and σ² = Eε_k² > 0. Define the polynomials

with real coefficients φ_j, θ_j. As usual, we assume that Φ_p and Θ_q have no common roots and no roots in the closed unit disk. The fractional ARIMA (p,d,q) process is defined as the unique solution {e_n} of the equations

where

denotes the backward shift operator defined by

is a linear time-invariant filter defined by

with {b_j, 0 ≤ j < ∞} being the coefficients in the series expansion of (1 − z)^−d, |z| < 1. If d < ½, then the infinite sum in (2.12) converges with probability one, and (2.11) has a unique moving-average solution

with the weights c_j tending to zero at the rate j^d−1.

Theorem 13.2.2 of Brockwell and Davis (1991) and Theorem 4.10.1 of Bingham et al. (1987) yield

where r(k) = Ee_j e_j+k. It is clear that (2.8) and (2.9) hold with H = d + ½ and L(x) = 1.

Example 2.6

Let e_i, −∞ < i < ∞ be defined as in Example 2.4. It is easy to see that

By (2.10) there is a constant c₁ > 0 such that

Let

and define

Because x_k,n /c_k,n is standard normal, (2.7) holds for 0 < α ≤ 1. Also,

and hence (1.1) and (1.2) are satisfied. Using the covariance of B_H(s) we get

and therefore (2.2) holds for all 0 ≤ α < 1/(1 + H). By (2.13) we have (2.5) for all k₀ large enough. If

then nc_k,n^α → 0 for any 0 ≤ α < 1/(H + 1). (We note that if (2.14) fails, then P(x_k,n = 0) = 1, so T(x_k,n) = T(0) has no effect on the limit.) So all the conditions of Theorem 2.2 are satisfied, and therefore

if |x|^α−1T(x) is locally Lebesgue integrable with some 0 ≤ α < 1/(H + 1).

In all our applications so far, the random variables x_k,n in (1.3) were normal. The following example, which extends Examples 2.4 and 2.5, shows that the long memory linear processes

with weights a_k ∼ ck^−β, ½ < β < 1 satisfy the assumptions of Theorem 2.2 even if the generating random variables ε_j are not Gaussian. This extends the results of Pötscher (2004, Sec. 3) to long memory processes.

Example 2.7

Let {ε_k,−∞ < k < ∞} be a sequence of independent, identically distributed random variables with Eε₀ = 0, Eε₀² = 1, Eε₀⁴ < ∞. Let {a_k,k ≥ 0} be a positive sequence satisfying

Because [sum ]a_k² < ∞, the sum defining y_j converges a.s. and Ey_j = 0, Ey_j² < ∞. An easy calculation shows that

for some constant A > 0, and thus letting

it follows from Davydov (1970, Thm. 2) that

We now show that if the characteristic function φ of ε₀ satisfies

for some γ > 0, then the random variables x_n,n have uniformly bounded densities, and thus Theorem 2.2 applies with c_k,n = (Ex_k,n²)^1/2. Our argument follows Pötscher (2004). Let Ψ_n denote the characteristic function of x_n,n. As is seen from the proof of Lemma 3.1 of Pötscher (2004) (cf. formulas (3.3) and (B.1) there), we have

where

. By Eε₀ = 0, Eε₀² = 1 we have φ(t) = 1 − t²/2 + o(t²) as t → 0, and thus |φ(t)| ≤ (1 + t²/4)⁻¹ in a neighborhood of 0. We now claim that

with some positive constant c. By the previous remark and the bound |φ(t)| = O(|t|^−γ) (and assuming, without loss of generality, that γ ≤ 1) the claimed inequality holds with c = ¼ for |t| ≤ t₀ and |t| ≥ t₁, provided t₀ is small enough and t₁ is large enough. To prove it for t₀ < |t| < t₁ note that |φ(t)| < 1 for t ≠ 0 (otherwise |φ| would be periodic) and thus by the continuity of φ there exists a constant ϱamp; > 0 such that |φ(t)| ≤ 1 − ϱamp; for t₀ ≤ |t| ≤ t₁. Hence choosing c small enough, the claimed inequality holds also for t₀ ≤ |t| ≤ t₁. Because c_j ∼ const·j^1−β we get, using |φ(t)| ≤ 1,

for some constant a > 0. Thus

Using 1/(1 + u²) ≤ exp(−c₁u²) for |u| ≤ 1 and the substitution v = n^1/2u we see that I₁ ≤ C₁ n^−1/2 where C₁ is a constant depending only on γ. On the other hand, for |u| > 1 we have

where C₂ is a constant depending only on γ. Hence

and thus we proved that

where C is a constant depending only on φ. By a well-known property of characteristic functions (see, e.g., Lukács, 1970, Thm. 3.2.2) it follows that the random variables x_n,n have uniformly bounded densities, as claimed.

A minor variation of the preceding argument shows that the assumption |φ(t)| = O(|t|^−γ), γ > 0 can be weakened to

which is the condition assumed in Pötscher (2004). Indeed, using |φ(t)| ≤ 1 and c_j ∼ const·j^1−β we get, similarly as before,

where the a_n,j are between positive bounds, independent of n,j. The last relation is very close to formula (B.1) in Pötscher (2004), and from there the proof can be completed by following his reasoning with minor changes.

So far we have replaced convergence to a Brownian motion in Pötscher's Theorem 1.2 with the convergence to a continuous Gaussian process. We showed that we still have the convergence in distribution of the integral functionals. Next we consider the case when (1.6) is not satisfied, i.e., if the distribution of x_k,n is not necessarily smooth. The next example shows that (1.3) can fail if only (1.4) and (1.5) are assumed.

Example 2.8

Let e₁,e₂,… be independent, identically distributed random variables with P(e₁ = 1) = P(e₁ = −1) = ½. Let T(x) = 1 if x is irrational and T(x) = 0 if x is rational. If n is a square number, then

However,

where W is a Brownian motion, so (1.3) cannot be true.

Our last result says that without assuming (1.6) or (2.7), the local Lebesgue integrability conditions in Theorems 1.2 and 2.2 should be replaced by the local Riemann integrability of T(x) to have (1.3).

THEOREM 2.3. If (1.1), (1.2), (1.4), (2.1), and (2.2) are satisfied and

then (1.3) holds.

Note that the integral in (2.15) is finite if and only if T is locally bounded, i.e., bounded on bounded intervals. The sufficiency of the last condition is obvious from 0 < α ≤ 1; to see the necessity note that if there exists a point x₀ such that T is unbounded in any neighborhood of x₀, then for any fixed h > 0 the integrand in (2.15) equals +∞ for |x − x₀| < h, and thus the integral is infinite. The integrand is undefined for x = 0, but because we mean (2.15) as a Lebesgue integral, this does not cause any problem.

We would like to point out that (2.15) cannot be replaced by

Indeed, the function T in Example 2.8 is bounded and Lebesgue measurable, and thus it satisfies

(see Hewitt and Stromberg, 1969, p. 199). From 0 < α < 1 and the Hölder inequality it follows that (2.16) is also valid, but according to Example 2.8, (1.3) cannot be true.

Remark 2.2. Condition (2.15) holds if and only if T is locally Riemann integrable, i.e., it is bounded and Riemann integrable on any bounded interval.