Multinomial bivariate integral
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I am struggling to integrate
$$
int_-infty^infty ... int_-infty^infty dx_1 cdots dx_K dy_1 cdots dy_N exp left[ - frac12 sum_k=0^K-1 x_k^2 + i sum_k=0^K-1 x_k y_k a_k - frac12 sum_n=0^N-1 (sum_k=0^K-1 x_k y_k-n)^2 right]
$$
The $a_k$ are real positive constants, and $N <K$. All indices are understood modulo $K$. A solution in the limit of large $K$ (and $N$ proportional to $K$) would also be appreciated.
Whereas it would be no problem to integrate with the first two terms in the exponential, I do not find how to decorrelate the third term. Performing the $x$-integrals would require to find the inverse of a matrix in $y$-terms, so I wonder if there is a solution which performs both the $x$-integrals and $y$-integrals simultaneously.
multivariable-calculus normal-distribution
asked Aug 16 at 7:20
Andreas
6,544935
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up vote
1
down vote
favorite
I am struggling to integrate
$$
int_-infty^infty ... int_-infty^infty dx_1 cdots dx_K dy_1 cdots dy_N exp left[ - frac12 sum_k=0^K-1 x_k^2 + i sum_k=0^K-1 x_k y_k a_k - frac12 sum_n=0^N-1 (sum_k=0^K-1 x_k y_k-n)^2 right]
$$
The $a_k$ are real positive constants, and $N <K$. All indices are understood modulo $K$. A solution in the limit of large $K$ (and $N$ proportional to $K$) would also be appreciated.
Whereas it would be no problem to integrate with the first two terms in the exponential, I do not find how to decorrelate the third term. Performing the $x$-integrals would require to find the inverse of a matrix in $y$-terms, so I wonder if there is a solution which performs both the $x$-integrals and $y$-integrals simultaneously.
multivariable-calculus normal-distribution
asked Aug 16 at 7:20
Andreas
6,544935
add a comment |Â
up vote
1
down vote
favorite
up vote
1
down vote
favorite
I am struggling to integrate
$$
int_-infty^infty ... int_-infty^infty dx_1 cdots dx_K dy_1 cdots dy_N exp left[ - frac12 sum_k=0^K-1 x_k^2 + i sum_k=0^K-1 x_k y_k a_k - frac12 sum_n=0^N-1 (sum_k=0^K-1 x_k y_k-n)^2 right]
$$
The $a_k$ are real positive constants, and $N <K$. All indices are understood modulo $K$. A solution in the limit of large $K$ (and $N$ proportional to $K$) would also be appreciated.
Whereas it would be no problem to integrate with the first two terms in the exponential, I do not find how to decorrelate the third term. Performing the $x$-integrals would require to find the inverse of a matrix in $y$-terms, so I wonder if there is a solution which performs both the $x$-integrals and $y$-integrals simultaneously.
multivariable-calculus normal-distribution
asked Aug 16 at 7:20
Andreas
6,544935
I am struggling to integrate
$$
int_-infty^infty ... int_-infty^infty dx_1 cdots dx_K dy_1 cdots dy_N exp left[ - frac12 sum_k=0^K-1 x_k^2 + i sum_k=0^K-1 x_k y_k a_k - frac12 sum_n=0^N-1 (sum_k=0^K-1 x_k y_k-n)^2 right]
$$
The $a_k$ are real positive constants, and $N <K$. All indices are understood modulo $K$. A solution in the limit of large $K$ (and $N$ proportional to $K$) would also be appreciated.
Whereas it would be no problem to integrate with the first two terms in the exponential, I do not find how to decorrelate the third term. Performing the $x$-integrals would require to find the inverse of a matrix in $y$-terms, so I wonder if there is a solution which performs both the $x$-integrals and $y$-integrals simultaneously.
multivariable-calculus normal-distribution
asked Aug 16 at 7:20
Andreas
6,544935
asked Aug 16 at 7:20
Andreas
6,544935
asked Aug 16 at 7:20
Andreas
6,544935
asked Aug 16 at 7:20
Andreas
6,544935
6,544935
add a comment |Â
add a comment |Â
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