can be thought as a discretized version of a Brownian sample path in $[0,1]$. The multivariate contamination comes from $\epsilon$, with different rates of contamination controlled by the Bernoulli variable $O(\delta)$, independent from $X$ and $\epsilon$, with $\mathbb{P}(O(\delta) =1) =  \delta$ and $\mathbb{P}(O(\delta) =0) =  1-\delta$. Three different scenarios (see Figure \[fig:traj\]) are considered for the distribution of $\epsilon$:

-   The elements of vector $\epsilon$ are $d$ independent realizations of a Student $t$ distribution with one degree of freedom. This means that the first moment of $Y$ is not defined when $\delta>0$.

-   The elements of vector $\epsilon$ are $d$ independent realizations of a Student $t$ distribution with two degrees of freedom. This means that the second moment of $Y$ is not defined when $\delta>0$.

-   The vector $\epsilon$ is distributed as a “reverse time” Brownian motion. It has a Gaussian centered distribution, with covariance matrix $[\Sigma_\epsilon]_{\ell, j} = 2\min (d-\ell,d-j)/d$. The covariance matrix of $Y$ is $(1-\delta) \Sigma + \delta \Sigma_\epsilon$.

![A sample of $n=20$ trajectories when $d=50$ and $\delta=0.10$ for the three different contamination scenarios: Student $t$ with 1 degree of freedom, Student $t$ with 2 degrees of freedom and reverse time Brownian motion (from left to right).[]{data-label="fig:traj"}](GraphesBruit.pdf){height="9cm" 