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A characterization related to the equilibrium distribution associated with a polynomial structure. (English) Zbl 1190.60085

Let \(A\) denote a random variable with density \(f\) and concentrated on the interval \((a,b)\) with \(0\leq a<b\leq \infty \). Suppose that \(E(A^{n})<\infty \) for all integers \(n\). In the paper the authors prove the following result.
Suppose that for any polynomial \(P(x)\), there exists a polynomial \(Q_{P}(x)\) such that \(\int_{x}^{b}P(t)f(t)dt=Q_{P}(x)f(x)\) for all \(x\in (a,b)\). Then there exist constants \(C,\lambda >0\) such that either \(b=\infty \) and \( f(x)=C\exp (-\lambda x)\), or \(b<\infty \) and \(f(x)=C(b-x)^{\lambda -1}\).
The proof of the result is based on solving a differential equation and by using ideals of polynomials. The result is motivated by a renewal theoretic question.

MSC:

60K05 Renewal theory
13P10 Gröbner bases; other bases for ideals and modules (e.g., Janet and border bases)
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References:

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