TY  JOUR
AU1  Harnett, Daniel
AU2  Jaramillo, Arturo
AU3  Nualart, David
AB  We study the asymptotic behavior of the
$$\nu $$
ν
symmetric Riemann sums for functionals of a selfsimilar centered Gaussian process X with increment exponent
$$0<\alpha <1$$
0
<
α
<
1
. We prove that, under mild assumptions on the covariance of X, the law of the weak
$$\nu $$
ν
symmetric Riemann sums converge in the Skorohod topology when
$$\alpha =(2\ell +1)^{1}$$
α
=
(
2
ℓ
+
1
)

1
, where
$$\ell $$
ℓ
denotes the largest positive integer satisfying
$$\int _{0}^{1}x^{2j}\nu (\mathrm{d}x)=(2j+1)^{1}$$
∫
0
1
x
2
j
ν
(
d
x
)
=
(
2
j
+
1
)

1
for all
$$j=0,\dots , \ell 1$$
j
=
0
,
⋯
,
ℓ

1
. In the case
$$\alpha >(2\ell +1)^{1}$$
α
>
(
2
ℓ
+
1
)

1
, we prove that the convergence holds in probability.
TI  Symmetric Stochastic Integrals with Respect to a Class of Selfsimilar Gaussian Processes
JF  Journal of Theoretical Probability
DO  10.1007/s1095901808331
DA  20180530
UR  https://www.deepdyve.com/lp/springerjournals/symmetricstochasticintegralswithrespecttoaclassofselfsimilarPEtP7UWdzw
SP  1105
EP  1144
VL  32
IS  3
DP  DeepDyve