  We tackle estimating sparse coefficients in a linear regression when the
covariates are sampled from an $L$-subexponential random vector. This vector
belongs to a class of distributions that exhibit heavier tails than Gaussian
random vector. Previous studies have established error bounds similar to those
derived for Gaussian random vectors. However, these methods require stronger
conditions than those used for Gaussian random vectors to derive the error
bounds. In this study, we present an error bound identical to the one obtained
for Gaussian random vectors up to constant factors without imposing stronger
conditions, when the covariates are drawn from an $L$-subexponential random
vector. Interestingly, we employ an $\ell_1$-penalized Huber regression, which
is known for its robustness against heavy-tailed random noises rather than
covariates. We believe that this study uncovers a new aspect of the
$\ell_1$-penalized Huber regression method.
