arXiv subjects

Jean-François Chamayou

Publications and source records attributed to Jean-François Chamayou.

3 recordsLinked to original sources

arXiv2014

A Random Difference Equation with Dufresne Variables revisited

The Dufresne laws (laws of product of independent random variables with gamma and beta distributions) occur as stationary distribution of certain Markov chains $ X_n $ on $ R$ defined by: \begin{equation} X_n = A_n ( X_{n-1} + B_n ) \end{equation} where $ X_0 , (A_1,B_1),...,(A_n,B_n) $ are independent and the $(A_i,B_i)'$s are identically distributed. This paper generalizes an explicit example where $A$ is the product of two independent $β_{a,1} , β_{b,1} $ and $B \sim γ_1 $ or $ γ_2 $. Keywords: beta, gamma and Dufresne distributions,Markov chains, stationary distributions, hypergeometric differential equations, Poisson process.

math.PR↗

arXiv2013

Shot Noise with Random Parameters

We give the description of the following model: $$ U_{n}=X_{n}(Y_{n}+U_{n-1})$$ for $n>1$ in the case where the $X_{n}$ are i.d.d. random variables with probability density: $$ A x^{A-1} , x \in [0,1] ,$$ $A$ is also a random variable distributed according to a Gamma law. The $ Y_{n}$ are or deterministic and equal to $1$ or independent Gamma random variables. We use this model to compute the shot noise with random parameters. Keywords: Random difference equations, Shot noise, Volterra functions, differential-difference equations, Monte-carlo simulation.

math.PR↗

arXiv2013

Computational Tools for the Shot Noise with Random Amplitude

The following random recurrency: $$ W_{n+1} = U_{n+1} ( W_n + Λ_{n+1} ) $$ where $W_{-1}=0.$ is known to be associated with the shot noise : $$ W_{t} = \sum_{0<t_k<t} Λ_{k} e^{-(t-t_k)} $$ where the $t_k$ are the dates of a Poisson process, the $ U_i ;\ i = 0,1,... $ are independent uniform variables. This can be extended to the triggered shot noise : $$ W_{t} = \sum_{0<t_{lk}<t} Λ_{lk} e^{-(t-t_{lk})} , l=1,2,3,... $$ The random amplitudes $Λ_i,\ i = 0,1,... $ are positive Dufresne independent variables or also even density variables which characteristic function given by generalized hypergeometric function. Keywords: Random Difference Equations,Triggered Shot Noise, Iterated Cosine-Bessel functions, Logarithmic Distribution Functions,Saddle Point Method, Asymptotic Calculations, Generalized Hypergeometric Functions, Psi Functions.

math.PR↗