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Contraction map of geometric mean

  1. Feb 25, 2010 #1
    I have the following mapping (generalized geometric mean):

    [tex]y(i)=exp\left[{\sum_j p(j|i)\log x(j)}\right]\\ ,\ i,j=1..N[/tex]

    where p(j|i) is a normalized conditional probability.

    my question is - is this a contraction mapping?

    in other words, does the following equation have a unique solution:

    [tex]x(i)=exp\left[{\sum_j p(j|i)\log x(j)}\right]\\ ,\ i,j=1..N[/tex]

    thanks in advance,
  2. jcsd
  3. Apr 1, 2010 #2
    Would you specify what x(i) stands for & what p(i|j) is the probability of?
    ( Lokks familiar, though. if it's something like a Markov chain or transition of probability vectors, use the Brower's fixed point theorem to get a solution. The mapping may not be contractive.)
  4. Apr 1, 2010 #3
    the p(i|j) does specify the transition in a Markov chain, and the x(i) stands for some 'value' function over the states. i know that a solution exists, but i'd like to show it's also unique. I think that for this aim, i must introduce some discount parameter [tex]0<\gamma<1[/tex]:

    x(i)=\gamma\exp\left[{\sum_j p(j|i)\log x(j)}\right]\\ ,\ i,j=1..N

    can that be shown to have a unique solution ?
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