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




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    up vote
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    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




    share|cite|improve this question






















      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




      share|cite|improve this question












      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





      share|cite|improve this question











      share|cite|improve this question




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      asked Aug 16 at 7:20









      Andreas

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