  The Lasso is a method for high-dimensional regression, which is now commonly
used when the number of covariates $p$ is of the same order or larger than the
number of observations $n$. Classical asymptotic normality theory does not
apply to this model due to two fundamental reasons: $(1)$ The regularized risk
is non-smooth; $(2)$ The distance between the estimator
$\widehat{\boldsymbol{\theta}}$ and the true parameters vector
$\boldsymbol{\theta}^*$ cannot be neglected. As a consequence, standard
perturbative arguments that are the traditional basis for asymptotic normality
fail.
  On the other hand, the Lasso estimator can be precisely characterized in the
regime in which both $n$ and $p$ are large and $n/p$ is of order one. This
characterization was first obtained in the case of Gaussian designs with i.i.d.
covariates: here we generalize it to Gaussian correlated designs with
non-singular covariance structure. This is expressed in terms of a simpler
``fixed-design'' model. We establish non-asymptotic bounds on the distance
between the distribution of various quantities in the two models, which hold
uniformly over signals $\boldsymbol{\theta}^*$ in a suitable sparsity class and
over values of the regularization parameter.
  As an application, we study the distribution of the debiased Lasso and show
that a degrees-of-freedom correction is necessary for computing valid
confidence intervals.
