arXiv subjects
Publications and source records attributed to Safari Mukeru.
In this paper we study the local times of Brownian motion from the point of view of algorithmic randomness. We introduce the notion of effective local time and show that any path which is Martin-Löf random with respect to the Wiener measure has continuous effective local times at every computable point. Finally we obtain a new simple representation of classical Brownian local times, computationally expressed.
In this paper, we define a generalised fractional Cox-Ingersoll-Ross process as a square of singular stochastic differential equation with respect to fractional Brownian motion with Hurst parameter H in (0,1) and continuous drift function. Firstly, we show that this differential equation has a unique solution which is continuous and positive up to the time of the first visit to zero. In addition, we prove that it is strictly positive everywhere almost surely for H > 1/2. In the case where H < 1/2, we consider a sequence of increasing functions and we prove that the probability of hitting zero tends to zero as n goes to infinity. These results are illustrated with some simulations using the generalisation of the extended Cox-Ingersoll-Ross process.
Given a sequence $(ξ_n)$ of standard i.i.d complex Gaussian random variables, Peres and Virág (in the paper ``Zeros of the i.i.d. Gaussian power series: a conformally invariant determinantal process'' {\it Acta Math.} (2005) 194, 1-35) discovered the striking fact that the zeros of the random power series $f(z) = \sum_{n=1}^\infty ξ_n z^{n-1}$ in the complex unit disc $\mathbb{D}$ constitute a determinantal point process. The study of the zeros of the general random series $f(z)$ where the restriction of independence is relaxed upon the random variables $(ξ_n)$ is an important open problem. This paper proves that if $(ξ_n)$ is an infinite sequence of complex Gaussian random variables such that their covariance matrix is invertible and its inverse is a Toeplitz matrix, then the zero set of $f(z)$ constitutes a determinantal point process with the same distribution as the case of i.i.d variables studied by Peres and Virág. The arguments are based on some interplays between Hardy spaces and reproducing kernels. Illustrative examples are constructed from classical Toeplitz matrices and the classical fractional Gaussian noise.
In 1979 Pisier proved remarkably that a sequence of independent and identically distributed standard Gaussian random variables determines, via random Fourier series, a homogeneous Banach algebra $\mathscr{P}$ strictly contained in $C(\mathbb{T})$, the class of continuous functions on the unit circle $\mathbb{T}$ and strictly containing the classical Wiener algebra $\mathbb{A}(\mathbb{T})$, that is, $\mathbb{A}(\mathbb{T}) \subsetneqq \mathscr{P} \subsetneqq C(\mathbb{T}).$ This improved some previous results obtained by Zafran in solving a long-standing problem raised by Katznelson. In this paper we extend Pisier's result by showing that any probability measure on the unit circle defines a homogeneous Banach algebra contained in $C(\mathbb{T})$. Thus Pisier algebra is not an isolated object but rather an element in a large class of Pisier-type algebras. We consider the case of spectral measures of stationary sequences of Gaussian random variables and obtain a sufficient condition for the boundedness of the random Fourier series $\sum_{n\in \mathbb{Z}}\hat f(n) \,ξ_n \exp(2πi n t)$ in the general setting of dependent random variables $(ξ_n)$.
Given a sequence $(X_n)$ of real or complex random variables and a sequence of numbers $(a_n)$, an interesting problem is to determine the conditions under which the series $\sum_{n=1}^\infty a_n X_n$ is almost surely convergent. This paper extends the classical Menshov--Rademacher theorem on the convergence of orthogonal series to general series of dependent random variables and derives interesting sufficient conditions for the almost everywhere convergence of trigonometric series with respect to singular measures whose Fourier transform decays to 0 at infinity with positive rate.
Given a sequence $(X_n)$ of symmetrical random variables taking values in a Hilbert space, an interesting open problem is to determine the conditions under which the series $\sum_{n=1}^\infty X_n$ is almost surely convergent. For independent random variables, it is well-known that if $\sum_{n=1}^\infty \mathbb{E}(\|X_n\|^2) <\infty$, then $\sum_{n=1}^\infty X_n$ converges almost surely. This has been extended to some cases of dependent variables (namely negatively associated random variables) but in the general setting of dependent variables, the problem remains open. This paper considers the case where each variable $X_n$ is given as a linear combination $a_{n,1}Z_1+ \ldots +a_{n,n}Z_n$ where $(Z_n)$ is a sequence of independent symmetrical random variables of unit variance and $(a_{n,k})$ are constants. For Gaussian random variables, this is the general setting. We obtain a sufficient condition for the almost sure convergence of $\sum_{n=1}^\infty X_n$ which is also sufficient for the almost sure convergence of $\sum_{n=1}^\infty \pm X_n$ for all (non-random) changes of sign. The result is based on an important bound of the mean of the random variable $\sup(\|X_1 + \ldots +X_k\|: 1\leq k \leq n)$ which extends the classical Lévy's inequality and has some independent interest.
The study of random polynomials has a long and rich history. This paper studies random algebraic polynomials $P_n(x) = a_0 + a_1 x + \ldots + a_{n-1} x^{n-1}$ where the coefficients $(a_k)$ are correlated random variables taken as the increments $X(k+1) - X(k)$, $k\in \mathbb{N}$, of a fractional Brownian motion $X$ of Hurst index $0< H < 1$. This reduces to the classical setting of independent coefficients for $H = 1/2$. We obtain that the average number of the real zeros of $P_n(x)$ is~$\sim K_H \log n$, for large $n$, where $K_H = (1 + 2 \sqrt{H(1-H)})/π$ (a generalisation of a classical result obtained by Kac in 1943). Unexpectedly, the parameter $H$ affects only the number of positive zeros, and the number of real zeros of the polynomials corresponding to fractional Brownian motions of indexes $H$ and $1-H$ are essentially the same. The limit case $H = 0$ presents some particularities: the average number of positive zeros converges to a constant. These results shed some light on the nature of fractional Brownian motion on the one hand and on the behaviour of real zeros of random polynomials of dependent coefficients on the other hand.
In this paper we study the behaviour at infinity of the Fourier transform of Radon measures supported by the images of fractal sets under an algorithmically random Brownian motion. We show that, under some computability conditions on these sets, the Fourier transform of the associated measures have, relative to the Hausdorff dimensions of these sets, optimal asymptotic decay at infinity. The argument relies heavily on a direct characterisation, due to Asarin and Pokrovskii, of algorithmically random Brownian motion in terms of the prefix free Kolmogorov complexity of finite binary sequences. The study also necessitates a closer look at the potential theory over fractals from a computable point of view.