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If $m=2$ then this is a Laplace distribution. Equivalently, the distribution of the determinant of a $2\times2$ matrix with IID centered normal entries is a Laplace distribution. See whuber's comment.

A Laplace distribution is also the difference of two IID exponentials. So, if $m$ is even, then the inner product can be written as a sum of $m/2$ IID Laplace distributions, or the difference of two IID gamma distributions. See "tight bounds on probability of sum of laplace random variables""tight bounds on probability of sum of laplace random variables" for the density function as a single sum.

If $m=2$ then this is a Laplace distribution. Equivalently, the distribution of the determinant of a $2\times2$ matrix with IID centered normal entries is a Laplace distribution. See whuber's comment.

A Laplace distribution is also the difference of two IID exponentials. So, if $m$ is even, then the inner product can be written as a sum of $m/2$ IID Laplace distributions, or the difference of two IID gamma distributions. See "tight bounds on probability of sum of laplace random variables" for the density function as a single sum.

If $m=2$ then this is a Laplace distribution. Equivalently, the distribution of the determinant of a $2\times2$ matrix with IID centered normal entries is a Laplace distribution. See whuber's comment.

A Laplace distribution is also the difference of two IID exponentials. So, if $m$ is even, then the inner product can be written as a sum of $m/2$ IID Laplace distributions, or the difference of two IID gamma distributions. See "tight bounds on probability of sum of laplace random variables" for the density function as a single sum.

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If $m=2$ then this is a Laplace distribution. Equivalently, the distribution of the determinant of a $2\times2$ matrix with IID centered normal entries is a Laplace distribution. See whuber's commentwhuber's comment.

A Laplace distribution is also the difference of two IID exponentials. So, if $m$ is even, then the inner product can be written as a sum of $m/2$ IID Laplace distributions, or the difference of two IID gamma distributions. See "tight bounds on probability of sum of laplace random variables" for the density function as a single sum.

If $m=2$ then this is a Laplace distribution. Equivalently, the distribution of the determinant of a $2\times2$ matrix with IID centered normal entries is a Laplace distribution. See whuber's comment.

A Laplace distribution is also the difference of two IID exponentials. So, if $m$ is even, then the inner product can be written as a sum of $m/2$ IID Laplace distributions, or the difference of two IID gamma distributions. See "tight bounds on probability of sum of laplace random variables" for the density function as a single sum.

If $m=2$ then this is a Laplace distribution. Equivalently, the distribution of the determinant of a $2\times2$ matrix with IID centered normal entries is a Laplace distribution. See whuber's comment.

A Laplace distribution is also the difference of two IID exponentials. So, if $m$ is even, then the inner product can be written as a sum of $m/2$ IID Laplace distributions, or the difference of two IID gamma distributions. See "tight bounds on probability of sum of laplace random variables" for the density function as a single sum.

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Douglas Zare
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If $m=2$ then this is a Laplace distribution. Equivalently, the distribution of the determinant of a $2\times2$ matrix with IID centered normal entries is a Laplace distribution. See whuber's comment.

A Laplace distribution is also the difference of two IID exponentials. So, if $m$ is even, then the inner product can be written as a sum of $m/2$ IID Laplace distributions, or the difference of two IID gamma distributions. See "tight bounds on probability of sum of laplace random variables" for the density function as a single sum.