Jacobi sequences of squares of random variables

Abstract

We express the Jacobi sequences of the square of a real valued random variable with all moments, not necessarily symmetric, as functions of the corresponding sequences of the random variable itself. In the symmetric case, the result is known and, we give a short, purely algebraic proof of it. We apply our result to the square of the Gamma distribution, i.e. the 4th power of the standard Gaussian. The result confirms the conjecture that belongs to the polynomial class, but its principal Jacobi sequence grows like , not as expected.

Keywords:

• Jacobi sequences
• orthogonal polynomials
• random variable
• orthogonal projections

Acknowledgements

LA acknowledges support by the Russian Science Foundation N. RSF 14-11-00687, Steklov Mathematical Institute.

Notes

No potential conflict of interest was reported by the authors.

Additional information

Funding

This work was supported by the Russian Science Foundation N. [RSF 14-11-00687], Steklov Mathematical Institute.