Let X be a random variable with distribution function F. Take a transformation on X, say Y = T(X), given that mth moment of Y and mth moment of X are different. Then, does it implies that remaining (m-1) moments are also different from X's (m-1) moments? Do we need to have some additional conditions on T ?
i.e., after transformation, preserving mth moment does it implies that remaining (m-1) moments are also preserved ?
It appears to me that if you take the distribution N(0,1), and the function Y=2X, you'll have the 2nd moments of Y and X differ, but their first moments will both be 0 by symmetry. Similarly, their third moments are the same, but that doesn't imply that their second moments are equal.
I imagine it will be simple to come up with more complicated examples.
Hello Neil,
Yes, your counterexample is right.
Indeed, currently I am trying to understand the concept of moments preserving transformations by projecting them from to lower order.
You might find a paper I just uploaded interesting: titled "Biased Coin Tossing and Moment Inequalities".
Hello Neil,
Thank you very much for the article. I will have a look into it.
Hello Neil,
It is very interesting problem. Indeed, if we start with very large n the convergence of the process to win or loss is interesting to know, of course bias factor also play a role .
Introducing parameters to \mu, i.e., drawing bias factor from different distribution for each iteration is also one of the extension of the result
See the earlier paper I wrote with Canfield and Wilf on "Averaging Sequences and a problem of Lampert and Slater" (or some similar title....)
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