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I am interested in the following question. Consider $n$ independent standard normal random variables $g_i$. Cosider a linear combination $w_1g_1+\cdots+w_ng_n$. Can one give a "decent" upper bound for \begin{equation} \min_{w_i \in \left\{-1,1\right\}}\mathbb{E}|w_1g_1+\cdots+w_ng_n| \text{?} \end{equation}\begin{equation} \mathbb{E}\min_{w_i \in \left\{-1,1\right\}}|w_1g_1+\cdots+w_ng_n| \text{?} \end{equation}

Basically, I am asking about the minimum expected absolute value of a family of correlated gaussian random variables.

If a good bound can be obtained, what about the same question for more general linear combinations, such as $w_1a_1g_1+\cdots+w_na_ng_n$ in term of $n$ and some norm of $a_i$, say $l_{2}$?

I am interested in the following question. Consider $n$ independent standard normal random variables $g_i$. Cosider a linear combination $w_1g_1+\cdots+w_ng_n$. Can one give a "decent" upper bound for \begin{equation} \min_{w_i \in \left\{-1,1\right\}}\mathbb{E}|w_1g_1+\cdots+w_ng_n| \text{?} \end{equation}

Basically, I am asking about the minimum expected absolute value of a family of correlated gaussian random variables.

If a good bound can be obtained, what about the same question for more general linear combinations, such as $w_1a_1g_1+\cdots+w_na_ng_n$ in term of $n$ and some norm of $a_i$, say $l_{2}$?

I am interested in the following question. Consider $n$ independent standard normal random variables $g_i$. Cosider a linear combination $w_1g_1+\cdots+w_ng_n$. Can one give a "decent" upper bound for \begin{equation} \mathbb{E}\min_{w_i \in \left\{-1,1\right\}}|w_1g_1+\cdots+w_ng_n| \text{?} \end{equation}

Basically, I am asking about the minimum expected absolute value of a family of correlated gaussian random variables.

If a good bound can be obtained, what about the same question for more general linear combinations, such as $w_1a_1g_1+\cdots+w_na_ng_n$ in term of $n$ and some norm of $a_i$, say $l_{2}$?

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TOM
TOM
  • 2.3k
  • 13
  • 23

Minimal expected absolute value of linear combinations of Gaussian random variables

I am interested in the following question. Consider $n$ independent standard normal random variables $g_i$. Cosider a linear combination $w_1g_1+\cdots+w_ng_n$. Can one give a "decent" upper bound for \begin{equation} \min_{w_i \in \left\{-1,1\right\}}\mathbb{E}|w_1g_1+\cdots+w_ng_n| \text{?} \end{equation}

Basically, I am asking about the minimum expected absolute value of a family of correlated gaussian random variables.

If a good bound can be obtained, what about the same question for more general linear combinations, such as $w_1a_1g_1+\cdots+w_na_ng_n$ in term of $n$ and some norm of $a_i$, say $l_{2}$?