Let
n
n
be a positive integer, let
X
=
(
X
1
,
…
,
X
n
)
\boldsymbol {X}=(X_1,\dots ,X_n)
be a random vector in
R
n
\mathbb {R}^n
with bounded entries, and let
(
θ
1
,
…
,
θ
n
)
(\theta _1,\dots ,\theta _n)
be a vector in
R
n
\mathbb {R}^n
. We show that the subgaussian behavior of the random variable
θ
1
X
1
+
⋯
+
θ
n
X
n
\theta _1 X_1+\dots +\theta _n X_n
is essentially determined by the subgaussian behavior of the random variables
∑
i
∈
H
θ
i
X
i
\sum _{i\in H} \theta _i X_i
where
H
H
is a random subset of
{
1
,
…
,
n
}
\{1,\dots ,n\}
.