Affiliation:
1. Department of Mathematics and Statistics, Concordia University , Montreal , Quebec, H3G 1M8 , Canada
Abstract
Abstract
The long-standing Gaussian product inequality (GPI) conjecture states that
E
[
∏
j
=
1
n
∣
X
j
∣
y
j
]
≥
∏
j
=
1
n
E
[
∣
X
j
∣
y
j
]
E\left[{\prod }_{j=1}^{n}{| {X}_{j}| }^{{y}_{j}}]\ge {\prod }_{j=1}^{n}E\left[{| {X}_{j}| }^{{y}_{j}}]
for any centered Gaussian random vector
(
X
1
,
…
,
X
n
)
\left({X}_{1},\ldots ,{X}_{n})
and any non-negative real numbers
y
j
{y}_{j}
,
j
=
1
,
…
,
n
j=1,\ldots ,n
. In this study, we describe a computational algorithm involving sums-of-squares representations of multivariate polynomials that can be used to resolve the GPI conjecture. To exhibit the power of the novel method, we apply it to prove new four- and five-dimensional GPIs:
E
[
X
1
2
m
X
2
2
X
3
2
X
4
2
]
≥
E
[
X
1
2
m
]
E
[
X
2
2
]
E
[
X
3
2
]
E
[
X
4
2
]
E\left[{X}_{1}^{2m}{X}_{2}^{2}{X}_{3}^{2}{X}_{4}^{2}]\ge E\left[{X}_{1}^{2m}]E\left[{X}_{2}^{2}]E\left[{X}_{3}^{2}]E\left[{X}_{4}^{2}]
for any
m
∈
N
m\in {\mathbb{N}}
, and
E
[
∣
X
1
∣
y
X
2
2
X
3
2
X
4
2
X
5
2
]
≥
E
[
∣
X
1
∣
y
]
E
[
X
2
2
]
E
[
X
3
2
]
E
[
X
4
2
]
E
[
X
5
2
]
E\left[{| {X}_{1}| }^{y}{X}_{2}^{2}{X}_{3}^{2}{X}_{4}^{2}{X}_{5}^{2}]\ge E\left[{| {X}_{1}| }^{y}]E\left[{X}_{2}^{2}]E\left[{X}_{3}^{2}]E\left[{X}_{4}^{2}]E\left[{X}_{5}^{2}]
for any
y
≥
1
10
y\ge \frac{1}{10}
.