Consider a sequence X_k=\sum_{j=0}^{\infty}c_jξ_{k-j}, k\geq 1, where c_j, j\geq 0, is a sequence of constants and ξ_j, -\infty <j<\infty, is a sequence of independent identically distributed (i.i.d.) random variables (r.v.s) belonging to the domain of attraction of a strictly stable law with index 0<α\leq 2. Let S_k=\sum_{j=1}^kX_j. Under suitable conditions on the constants c_j it is known that for a suitable normalizing constant γ_n, the partial sum process γ_n^{-1}S_{[nt]} converges in distribution to a linear fractional stable motion (indexed by αand H, 0<H<1). A fractional ARIMA process with possibly heavy tailed innovations is a special case of the process X_k. In this paper it is established that the process n^{-1}β_n\sum_{k=1}^{[nt]}f(β_n(γ_n^{-1}S_k+x)) converges in distribution to (\int_{-\infty}^{\infty}f(y) dy)L(t,-x), where L(t,x) is the local time of the linear fractional stable motion, for a wide class of functions f(y) that includes the indicator functions of bounded intervals of the real line. Here β_n\to \infty such that n^{-1}β_n\to 0. The only further condition that is assumed on the distribution of ξ_1 is that either it satisfies the Cramér's condition or has a nonzero absolutely continuous component.
The results have motivation in large sample inference for certain nonlinear time series models.
In this paper we present multivariate space-time fractional Poisson processes by considering common random time-changes of a (finite-dimensional) vector of independent classical (non-fractional) Poisson processes. In some cases we also consider compound processes. We obtain some equations in terms of some suitable fractional derivatives and fractional difference operators, which provides the extension of known equations for the univariate processes.
Central limit theorems for squared increment sums of fractional Brownian fields based on a Delaunay triangulation in $2D$
Université du Littoral Côte d’Opale（法国北部海岸大学）；Université de Lyon, Université Lyon 1, Institut de Science Financière et d’Assurances（里昂大学，里昂第一大学，金融与保险科学学院）；Laboratory in Finance and Insurance - LFA CREST - Center for Research in Economics and Statistics, ENSAE（金融与保险实验室-LFA CREST-经济与统计研究中心，巴黎国立高等经济学院）
In this paper we consider the relation between random sums and compositions of different processes. In particular, for independent Poisson processes $N_α(t)$, $N_β(t)$, $t>0$, we show that $N_α(N_β(t)) \overset{\text{d}}{=} \sum_{j=1}^{N_β(t)} X_j$, where the $X_j$s are Poisson random variables. We present a series of similar cases, the most general of which is the one in which the outer process is Poisson and the inner one is a nonlinear fractional birth process. We highlight generalisations of these results where the external process is infinitely divisible. A section of the paper concerns c…
We study the persistence probability for some two-sided discrete-time Gaussian sequences that are discrete-time analogs of fractional Brownian motion and integrated fractional Brownian motion, respectively. Our results extend the corresponding ones in continuous-time in [Molchan, Commun. Math. Phys., 1999] and [Molchen, J. Stat. Phys., 2017] to a wide class of discrete-time processes.
We study the asymptotic behaviour of partial sums of long range dependent random variables and that of their counting process, together with an appropriately normalized integral process of the sum of these two processes, the so-called Vervaat process. The first two of these processes are approximated by an appropriately constructed fractional Brownian motion, while the Vervaat process in turn is approximated by the square of the same fractional Brownian motion.
Limit theorems for squared increment sums of the maximum of two isotropic fractional Brownian fields under fixed-domain asymptotics
Université du Littoral Côte d’Opale（滨海大学）；Université de Lyon, Université Lyon 1, Institut de Science Financière et d’Assurances（里昂大学，里昂第一大学，金融与保险科学学院）；Laboratory in Finance and Insurance - LFA CREST - Center for Research in Economics and Statistics, ENSAE（金融与保险实验室-LFA CREST-经济与统计研究中心，国家经济统计学院）
We consider stochastic integration with respect to fractional Brownian motion (fBm) with $H < 1/2$. The integral is constructed as the limit, where it exists, of a sequence of Riemann sums. A theorem by Gradinaru, Nourdin, Russo & Vallois (2005) holds that a sequence of Simpson's rule Riemann sums converges in probability for a sufficiently smooth integrand $f$ and when the stochastic process is fBm with $H > 1/10$. For the case $H = 1/10$, we prove that the sequence of sums converges in distribution. Consequently, we have an Itô-like formula for the resulting stochastic integral. The converge…
We consider complete convergence and closely related Hsu-Robbins-Erdos-Spitzer-Baum-Katz series for sums whose terms are elements of a linear 2-nd order autoregressive sequences of random variables and prove sufficient conditions for the convergence of this series.
In the paper, upper bounds for the rate of convergence in laws of large numbers for mixed Poisson random sums are constructed. As a measure of the distance between the limit and pre-limit laws, the Zolotarev $ζ$-metric is used. The obtained results extend the known convergence rate estimates for geometric random sums (in the famous R{é}nyi theorem) to a considerably wider class of random indices with mixed Poisson distributions including, e. g., those with the (generalized) negative binomial distribution.
We investigate the complete $p$-th moment convergence for weighted sums of independent, identically distributed random variables under sublinear expectations space. Using moment inequality and truncation methods, we prove the equivalent conditions of complete $p$-th moment convergence of weighted sums of independent, identically distributed random variables under sublinear expectations space, which complement the corresponding results obtained in Guo and Shan (2020).
The complete convergence for weighted sums of sequences of independent, identically distributed random variables under sublinear expectations space was studied. By moment inequality and truncation methods, we establish the equivalent conditions of complete convergence for weighted sums of sequences of independent, identically distributed random variables under sublinear expectations space. The results extend the corresponding results obtained by Guo (2012) to those for sequences of independent, identically distributed random variables under sublinear expectations space.
In this paper we study the asymptotic behaviour of weighted random sums when the sum process converges stably in law to a Brownian motion and the weight process has continuous trajectories, more regular than that of a Brownian motion. We show that these sums converge in law to the integral of the weight process with respect to the Brownian motion when the distance between observations goes to zero. The result is obtained with the help of fractional calculus showing the power of this technique. This study, though interesting by itself, is motivated by an error found in the proof of Theorem 4 in…
For moving average processes with random coefficients and heavy-tailed innovations that are weakly dependent in the sense of strong mixing and local dependence condition $D'$ we study joint functional convergence of partial sums and maxima. Under the assumption that all partial sums of the series of coefficients are a.s. bounded between zero and the sum of the series we derive a functional limit theorem in the space of $\mathbb{R}^{2}$-valued càdlàg functions on $[0, 1]$ with the Skorokhod weak $M_{2}$ topology.
In this work we prove an asymptotic result, that under some conditions on the involved distribution functions, is valid for any Oppenheim expansion, extending a classical result proven by W. Vervaat in 1972 for denominators of the Luroth case. Furthermore, we study the convergence in distribution of weighted sums of a sequence of independent random variables. Although the result is of its own interest, in the present setting it is used to prove convergence in distribution of specific sequences of random variables generalizing known results obtained for Luroth random variables.
We show, how the classical Berry-Esseen theorem for normal approximation may be used to derive rates of convergence for random sums of centerd, real-valued random variables with respect to a certain class of probability metrics, including the Kolmogorov and the Wasserstein distances. This technique is applied to several examples, including the approximation by a Laplace distribution of a geometric sum of centered random variables with finite third moment, where a concrete Berry-Esseen bound is derived. This bound reduces to a bound of the supposedly optimal order $\sqrt{p}$ in the i.i.d. case.
In this work, conditions are provided under which a normed double sum of independent random elements in a real separable Rademacher type $p$ Banach space converges completely to $0$ in mean of order $p$. These conditions for the complete convergence in mean of order $p$ are shown to provide an exact characterization of Rademacher type $p$ Banach spaces. In case the Banach space is not of Rademacher type $p$, it is proved that the complete convergence in mean of order $p$ of a normed double sum implies a strong law of large numbers.
Let $\{X, X_n, n\geq 1\}$ be a sequence of independent identically distributed non-degenerate random variables. Put $S_0=0, S_n = \sum^n_{i=1} X_i$ and $V_n^2=\sum^n_{i=1} X_i^2, n\ge 1.$ A weak convergence theorem is established for the self-normalized partial sums processes $\{S_{[nt]}/V_n, 0\le t\le 1\}$ when $X$ belongs to the domain of attraction of a stable law with index $α\in (0,2]$. The respective limiting distributions of the random variables ${\max_{1\le i\le n}|X_i|}/{S_n}$ and ${\max_{1\le i\le n}|X_i|}/{V_n}$ are also obtained under the same condition.
For a sequence $\{X_{n}, \, n \geqslant 1 \}$ of nonnegative random variables where $\max[\min(X_{n} - s,t),0]$, $t > s \geqslant 0$, satisfy a moment inequality, sufficient conditions are given under which $\sum_{k=1}^n (X_k - \mathbb{E} \, X_k)/b_n \overset{\mathrm{a.s.}}{\longrightarrow} 0$. Our statement allows us to obtain a strong law of large numbers for sequences of pairwise negatively quadrant dependent random variables under sharp normalising constants.
Two counterexamples, addressing questions raised in \cite{AD} and \cite{PZ}, are provided. Both counterexamples are related to chaoses. Let $F_n=Y_n+Z_n$. It may be that $F_n\overset{a.s.}\longrightarrow 0$, $F_n\overset{L_{2+δ}}\longrightarrow 0$ and $E\bigl\{\sup_n\,\abs{F_n}^δ\bigr\}<\infty$, where $δ>0$ and $Y_n$ and $Z_n$ belong to chaoses of uniformly bounded degree, and yet $Y_n$ fails to converge to 0 a.s.
We prove that if a rectangular matrix with uniformly small entries and approximately orthogonal rows is applied to the independent standardized random variables with uniformly bounded third moments, then the empirical CDF of the resulting partial sums converges to the normal CDF with probability one. This implies almost sure convergence of empirical periodograms, almost sure convergence of spectra of circulant and reverse circulant matrices, and almost sure convergence of the CDF's generated from independent random variables by independent random orthogonal matrices. For special trigonometric …
Given a triangular array $\left\{X_{n,k}, \, 1 \leqslant k \leqslant n, n \geqslant 1 \right\}$ of random variables satisfying $\mathbb{E} \lvert X_{n,k} \rvert^{p} < \infty$ for some $p \geqslant 1$ and sequences $\{b_{n} \}$, $\{c_{n} \}$ of positive real numbers, we shall prove that $\sum_{n=1}^\infty c_n \mathbb{E} \left[ |\sum_{k=1}^n (X_{n,k} - \mathbb{E} \, X_{n,k})| / b_n - \varepsilon \right]_+^p < \infty$, where $x_+ = \max(x,0)$. Our results are announced in a general setting, allowing us to obtain the convergence of the series in question under various types of dependence.
We present some product representations for random variables with the Linnik, Mittag-Leffler and Weibull distributions and establish the relationship between the mixing distributions in these representations. Based on these representations, we prove some limit theorems for a wide class of rather simple statistics constructed from samples with random sized including, e. g., random sums of independent random variables with finite variances, maximum random sums, extreme order statistics, in which the Linnik and Mittag-Leffler distributions play the role of limit laws. Thus we demonstrate that the…
This paper deals with rates of convergence in the strong law of large numbers, in the Baum-Katz form, for partial sums of Banach space valued random variables. The results are then applied to solve similar problems for weighted partial sums of conditional expectations. They are further used to treat partial sums of powers of a reversible Markov chain operator. The method of proof is based on martingale approximation. The conditions are expressed in terms moments of the individual summands.
Let $\{X_n;n\ge 1\}$ be a sequence of independent random variables on a probability space $(Ω, \mathcal{F}, P)$ and $S_n=\sum_{k=1}^n X_k$. It is well-known that the almost sure convergence, the convergence in probability and the convergence in distribution of $S_n$ are equivalent. In this paper, we prove similar results for the independent random variables under the sub-linear expectations, and give a group of sufficient and necessary conditions for these convergence. For proving the results, the Levy and Kolmogorov maximal inequalities for independent random variables under the sub-linear ex…
For a sequence of identically distributed negatively associated random variables $\{X_n; n\geq 1\}$ with partial sums $S_n=\sum_{i=1}^nX_i, n\geq 1$, refinements are presented of the classical Baum-Katz and Lai complete convergence theorems. More specifically, necessary and sufficient moment conditions are provided for complete moment convergence of the form $$ \sum_{n \ge n_0} n^{r -2 -\frac{1}{pq}} a_n E(\max_{1 \le k \le n}|S_k|^{\frac{1}{q}} - εb_n^{\frac{1}{pq}})^+ < \infty $$ to hold where $r>1, q>0$ and either $n_0=1, 0<p<2, a_n=1, b_n=n$ or $n_0=3, p=2, a_n=(\log n)^{-\frac{1}{2q}}, b_…
In this paper we obtain Berry-Esseén bounds on partial sums of functionals of heavy-tailed moving averages, including the linear fractional stable noise, stable fractional ARIMA processes and stable Ornstein-Uhlenbeck processes. Our rates are obtained for the Wasserstein and Kolmogorov distances, and depend strongly on the interplay between the memory of the process, which is controlled by a parameter $α$, and its tail-index, which is controlled by a parameter $β$. In fact, we obtain the classical $\sqrt{1/ n}$ rate of convergence when the tails are not too heavy and the memory is not too stro…
Weak convergence of various general functionals of partial sums of dependent random variables to stochastic integral now play a major role in the modern statistics theory. In this paper, we obtain the weak convergence of various general functionals of partial sums of casual process by means of the method which was introduced in Jacod and Shiryaev (2003).
In this paper, we provide two approximations in law of operator fractional Brownian motions. One is constructed by Poisson processes, and the other generalizes a result of Taqqu (1975).
In this paper, we derive the exact rate of convergence of some approximation schemes associated to scalar stochastic differential equations driven by a fractional Brownian motion with Hurst index H.
A multivariate fractional Poisson process was recently defined in Beghin and Macci (2016) by considering a common independent random time change for a finite dimensional vector of independent (non-fractional) Poisson processes; moreover it was proved that, for each fixed $t\geq 0$, it has a suitable multinomial conditional distribution of the components given their sum. In this paper we consider another multivariate process $\{\underline{M}^ν(t)=(M_1^ν(t),\ldots,M_m^ν(t)):t\geq 0\}$ with the same conditional distributions of the components given their sums, and different marginal distributions…
The purpose of this paper is to provide a complete description the convergence in distribution of two subsequences of the signed cubic variation of the fractional Brownian motion with Hurst parameter $H = 1/6$.
In this paper we obtain a rate of convergence in the central limit theorem for high order weighted Hermite variations of the fractional Brownian motion. The proof is based on the techniques of Malliavin calculus and the quantitative stable limit theorems proved by Nourdin, Nualart and Peccati in 2016.
We consider a mixed stochastic differential equation involving both standard Brownian motion and fractional Brownian motion with Hurst parameter $H>1/2$. The mean-square rate of convergence of Euler approximations of solution to this equation is obtained.
Let $\bX=\{X_n\}_{n\geq 1}$ and $\bY=\{Y_n\}_{n\geq 1}$ be two independent random sequences. We obtain rates of convergence to the normal law of randomly weighted self-normalized sums $$ ψ_n(\bX,\bY)=\sum_{i=1}^nX_iY_i/V_n,\quad V_n=\sqrt{Y_1^2+...+Y_n^2}. $$ These rates are seen to hold for the convergence of a number of important statistics, such as for instance Student's $t$-statistic or the empirical correlation coefficient.
We obtain almost optimal convergence rate in the central limit theorem for "nonconevntional" sums of the form $S_N=N^{-\frac12}\sum_{n=1}^N (F(ξ_n,ξ_{2n},...,ξ_{\ell n})-\bar F)$.
We consider stochastic differential equation involving pathwise integral with respect to fractional Brownian motion. The estimates for the Hurst parameter are constructed according to first- and second-order quadratic variations of observed values of the solution. The rate of convergence of these estimates to the true value of a parameter is established.
In this paper, by using Girsanov's transformation and the property of the corresponding reference stochastic differential equations, we investigate weak existence and uniqueness of solutions and weak convergence of Euler-Maruyama scheme to stochastic functional differential equations with Hölder continuous drift driven by fractional Brownian motion with Hurst index $H\in (1/2,1)$.
In this paper we state and prove a central limit theorem for the finite-dimensional laws of the quadratic variations process of certain fractional Brownian sheets. The main tool of this article is a method developed by Nourdin and Nualart based on the Malliavin calculus.
We define weighted fractional Brownian sheets, which are a class of Gaussian random fields with four parameters that include fractional Brownian sheets as special cases, and we give some of their properties. We show that for certain values of the parameters the weighted fractional Brownian sheets are obtained as limits in law of occupation time fluctuations of a stochastic particle model. In contrast with some known approximations of fractional Brownian sheets which use a kernel in a Volterra type integral representation of fractional Brownian motion with respect to ordinary Brownian motion, o…
We study functional convergence of sums of moving averages with random coefficients and heavy-tailed innovations. Under some standard moment conditions and the assumption that all partial sums of the series of coefficients are a.s. bounded between zero and the sum of the series we obtain functional convergence of the corresponding partial sum stochastic process in the space $D[0,1]$ of càdlàg functions with the Skorohod $M_{2}$ topology.
We consider fractional Brownian motion with the Hurst parameters from (1/2,1). We found that the increment of a fractional Brownian motion can be represented as the sum of a two independent Gaussian processes one of which is smooth in the sense that it is differentiable in mean square. We consider fractional Brownian motion and stochastic integrals generated by the Riemann sums. As an example of applications, this results is used to find an optimal pre-programmed strategy in the mean-variance setting for a Bachelier type market model driven by a fractional Brownian motion.
The purpose of this paper is to study the convergence in distribution of two subsequences of the signed cubic variation of the fractional Brownian motion with Hurst parameter $H=1/6$. We prove that, under some conditions on both subsequences, the limit is a two-dimensional Brownian motion whose components may be correlated and we find explicit formulae for its covariance function.
In this paper we study some convergence results concerning the one-dimensional distribution of a time-changed fractional Ornstein-Uhlenbeck process. In particular, we establish that, despite the time change, the process admits a Gaussian limit random variable. On the other hand, we prove that the process converges towards the time-changed Ornstein-Uhlenbeck as the Hurst index $H \to 1/2^+$, with locally uniform convergence of one-dimensional distributions. Moreover, we also achieve convergence in the Skorohod $J_1$-topology of the time-changed fractional Ornstein-Uhlenbeck process as $H \to 1/…
In this article, an uniform discretization of stochastic integrals $\int_{0}^{1} f'_-(B_t)\ud B_t$, with respect to fractional Brownian motion with Hurst parameter $H \in (1/2,1)$, for a large class of convex functions $f$ is considered. In Statistics & Decisions, 27, 129-143, for any convex function $f$, the almost sure convergence of uniform discretization to such stochastic integral is proved. Here we prove $L^r$- convergence of uniform discretization to stochastic integral. In addition, we obtain a rate of convergence. It turns out that the rate of convergence can be brought as closely as …
In this paper we show that the continuous version of the self normalised process $Y_{n,p}(t)= S_n(t)/V_{n,p}+(nt-[nt])X_{[nt]+1}/V_{n,p}$ where $S_n(t)=\sum_{i=1}^{[nt]} X_i$ and $V_{(n,p)}= \sum_{i=1}^{n}|X_i|^p)^{\frac{1}{p}}$ and $X_i$ i.i.d. random variables belong to $DA(α)$, has a non trivial distribution iff $p=α=2$. The case for $2 > p > α$ and $p \le α< 2$ is systematically eliminated by showing that either of tightness or finite dimensional convergence to a non-degenerate limiting distribution does not hold. This work is an extension of the work by Csörgö et al. who showed Donsker's …
In this note we prove an existence and uniqueness result of solution for stochastic differential delay equations with hereditary drift driven by a fractional Brownian motion with Hurst parameter $H > 1/2$. Then, we show that, when the delay goes to zero, the solutions to these equations converge, almost surely and in $L^p$, to the solution for the equation without delay. The stochastic integral with respect to the fractional Brownian motion is a pathwise Riemann-Stieltjes integral.
In this paper, we show an approximation in law, in the space of the continuous functions on $[0,1]^2$, of two-parameter Gaussian processes that can be represented as a Wiener type integral by processes constructed from processes that converge to the Brownian sheet. As an application, we obtain a sequence of processes constructed from a Lévy sheet that converges in law towards the fractional Brownian sheet.
We prove moment inequalities for a class of functionals of i.i.d. random fields. We then derive rates in the central limit theorem for weighted sums of such randoms fields via an approximation by $m$-dependent random fields.
We investigate asymptotic behaviour of probabilities of large deviations for normalized combinatorial sums. We find a zone in which these probabilities are equivalent to the tail of the standard normal law. Our conditions are similar to the classical Bernstein condition. The range of the zone of the normal convergence can be of power order.
We consider the small deviation probabilities (SDP) for sums of stationary Gaussian sequences. For the cases of constant boundaries and boundaries tending to zero, we obtain quite general results. For the case of the boundaries tending to infinity, we focus our attention on the discrete analogs of the fractional Brownian motion (FBM). It turns out that the lower bounds for the SDP can be transferred from the well studied FBM caseto the discrete time setting under the usual assumptions that imply weak convergence while the transfer of the corresponding upper bounds necessarily requires a deeper…
We consider the maximum process of a random walk with additive independent noise in form of $\max_{i=1,\dots,n}(S_i+Y_i)$. The random walk may have dependent increments, but its sample path is assumed to converge weakly to a fractional Brownian motion. When the largest noise has the same order as the maximal displacement of the random walk, we establish an invariance principle for the maximum process in the Skorohod topology. The limiting process is the maximum process of the fractional Brownian notion with shot noise generated by Poisson point processes.
