arXiv subjects

T. T. Diu Tran

Publications and source records attributed to T. T. Diu Tran.

3 recordsLinked to original sources

arXiv2018

Statistical inference for Vasicek-type model driven by Hermite processes

Let $Z$ denote a Hermite process of order $q \geq 1$ and self-similarity parameter $H \in (\frac{1}{2}, 1)$. This process is $H$-self-similar, has stationary increments and exhibits long-range dependence. When $q=1$, it corresponds to the fractional Brownian motion, whereas it is not Gaussian as soon as $q\geq 2$. In this paper, we deal with a Vasicek-type model driven by $Z$, of the form $dX_t = a(b - X_t)dt +dZ_t$. Here, $a > 0$ and $b \in \mathbb{R}$ are considered as unknown drift parameters. We provide estimators for $a$ and $b$ based on continuous-time observations. For all possible values of $H$ and $q$, we prove strong consistency and we analyze the asymptotic fluctuations.

math.PR↗

arXiv2017

Asymptotic behavior for quadratic variations of non-Gaussian multiparameter Hermite random fields

Let $(Z^{q, H}_t)_{t \in [0, 1]^d}$ denote a $d$-parameter Hermite random field of order $q \geq 1$ and self-similarity parameter $H = (H_1, \ldots, H_d) \in (\frac{1}{2}, 1)^d$. This process is $H$-self-similar, has stationary increments and exhibits long-range dependence. Particular examples include fractional Brownian motion ($q=1$, $d=1$), fractional Brownian sheet $(q=1, d \geq 2)$, Rosenblatt process ($q=2$, $d=1$) as well as Rosenblatt sheet $(q=2, d \geq 2)$. For any $q \geq 2, d\geq 1$ and $H \in (\frac{1}{2}, 1)^d$ we show in this paper that a proper normalization of the quadratic variation of $Z^{q, H}$ converges in $L^2(Ω)$ to a standard $d$-parameter Rosenblatt random variable with self-similarity index $H" = 1+ (2H-2)/q$.

math.PR↗

arXiv2017

Non-Central Limit Theorem for Quadratic Functionals of Hermite-Driven Long Memory Moving Average Processes

Let $(Z_t^{(q, H)})_{t \geq 0}$ denote a Hermite process of order $q \geq 1$ and self-similarity parameter $H \in (\frac{1}{2}, 1)$. Consider the Hermite-driven moving average process $$X_t^{(q, H)} = \int_0^t x(t-u) dZ^{(q, H)}(u), \qquad t \geq 0.$$ In the special case of $x(u) = e^{-θu}, θ> 0$, $X$ is the non-stationary Hermite Ornstein-Uhlenbeck process of order $q$. Under suitable integrability conditions on the kernel $x$, we prove that as $T \to \infty$, the normalized quadratic functional $$G_T^{(q, H)}(t)=\frac{1}{T^{2H_0 - 1}}\int_0^{Tt}\Big(\big(X_s^{(q, H)}\big)^2 - E\Big[\big(X_s^{(q, H)}\big)^2\Big]\Big) ds , \qquad t \geq 0,$$ where $H_0 = 1 + (H-1)/q$, converges in the sense of finite-dimensional distribution to the Rosenblatt process of parameter $H' = 1 + (2H-2)/q$, up to a multiplicative constant, irrespective of self-similarity parameter whenever $q \geq 2$. In the Gaussian case $(q=1)$, our result complements the study started by Nourdin \textit{et al} in arXiv:1502.03369, where either central or non-central limit theorems may arise depending on the value of self-similarity parameter. A crucial key in our analysis is an extension of the connection between the classical multiple Wiener-Itô integral and the one with respect to a random spectral measure (initiated by Taqqu (1979)), which may be independent of interest.

math.PR↗