In this monograph, we prove an asymptotic approximation for integrals of probability densities over sets in finite dimensional euclidean space, which are far away from the origin (asymptotic sets). We use this approximation to investigate tails of quadratic forms of random vectors, supremum of random linear forms among others. Applications to the study of finite size random matrices, finite sample statistics of autoregressive processes, and supremum of some stochastic processes.
We consider the spherical integral of real symmetric or Hermitian matrices when the rank of one matrix is one. We prove the existence of the full asymptotic expansions of these spherical integrals and derive the first and the second term in the asymptotic expansion. Using asymptotic expression of the spherical integral, we derive the asymptotic freeness of Wigner matrices with (deterministic) Hermitian matrices.
We present the product formula for complex multiple Wiener-Ito integrals. As applications, we show the Ustunel-Zakai independent criterion, the Nourdin-Rosinski asymptotic moment-independent criterion and joint convergence criterion for complex multiple Wiener-Ito integrals.
The expected number of false inflection points of kernel smoothers is evaluated. To obtain the small noise limit, we use a reformulation of the Leadbetter-Cryer integral for the expected number of zero crossings of a differentiable Gaussian process.
In this article we study the Dyson Bessel process, which describes the evolution of singular values of rectangular matrix Brownian motions, and prove a large deviation principle for its empirical particle density. We then use it to obtain the asymptotics of the so-called rectangular spherical integrals as $m,n$ go to infinity while $m/n$ converges.
We study a Edgeworth-type refinement of the central limit theorem for the discretizacion error of Itô integrals. Towards this end, we introduce a new approach, based on the anticipating Itô formula. This alternative technique allows us to compute explicitly the terms of the corresponding expansion formula.
In this note, we study the asymptotics of a spherical integral that is a multiplicative counterpart to the well-known Harish-Chandra Itzykson Zuber integral. This counterpart can also be expressed in terms the Heckman-Opdam hypergeometric function. When the argument of this spherical integral is of finite support and of order $N$, these asymptotics involve a modified version of the $S$-transform of the limit measure of the matrix argument and its largest eigenvalue. To prove our main result, we are leveraging a technique of successive conditionning. In particular we prove in a "mathematically …
We consider Riemann sum approximations of stochastic integrals with respect to the fractional Browian motion of index $H\geq \frac12$. We show the convergence of these schemes at first and second order. The processes obtained in the limit in the second case are stochastic integrals with respect to the Rosenblatt process if $H >\frac34$ and the standard Brownian motion otherwise. These results are obtained under the assumption that the integrand is a `controlled' process. We provide many examples of such processes, in particular fractional semimartingales and multiple Wiener-Itô integrals
We continue the approach in Part I \cite{duchong19} to study stationary states of controlled differential equations driven by rough paths, using the framework of random dynamical systems and random attractors. Part II deals with driving paths of finite $ν$ - Hölder norms with $ν\in (\frac{1}{3},\frac{1}{2})$ so that the integrals are interpreted in the Gubinelli sense for controlled rough paths. We prove sufficient conditions for the attractor to be a singleton, thus the pathwise convergence is in both pullback and forward senses.
On a Szego Type Limit Theorem, the Holder-Young-Brascamp-Lieb Inequality, and the Asymptotic Theory of Integrals and Quadratic Forms of Stationary Fields
We provide a unified analytic approach to study stationary states of controlled differential equations driven by rough paths, using the framework of random dynamical systems and random attractors. Part I deals with driving paths of finite $p$-variations with $1 \leq p <2$ so that the integrals are interpreted in the Young sense. Our method helps to generalize recent results \cite{GAKLBSch2010}, \cite{ducGANSch18}, \cite{duchongcong18} on the existence of the global pullback attractors for the generated random dynamical systems. We also prove sufficient conditions for the attractor to be a sing…
Asymptotic expansion of the distribution of a perturbation $Z_n$ of a Skorohod integral jointly with a reference variable $X_n$ is derived. We introduce a second-order interpolation formula in frequency domain to expand a characteristic functional and combine it with the scheme developed in the martingale expansion. The second-order interpolation and Fourier inversion give asymptotic expansion of the expectation $E[f(Z_n,X_n)]$ for differentiable functions $f$ and also measurable functions $f$. In the latter case, the interpolation method connects the two non-degeneracies of variables for fini…
We obtain an asymptotic series $\sum_{j=0}^\infty\frac{I_j}{n^j}$ for the integral $\int_0^1[x^n+(1-x)^n]^{\frac1{n}}dx$ as $n\to\infty$, and compute $I_j$ in terms of alternating (or "colored") multiple zeta value. We also show that $I_j$ is a rational polynomial the ordinary zeta values, and give explicit formulas for $j\le 12$. As a byproduct, we obtain precise results about the convergence of norms of random variables and their moments. We study $\Vert(U,1-U)\Vert_n$ as $n$ tends to infinity and we also discuss $\Vert(U_1,U_2,\dots,U_r)\Vert_n$ for standard uniformly distributed random var…
