Limit distributions for polynomials with independent and identically distributed entries
We characterize the limiting distributions of random variables of the form $P_n\left( (X_i)_{i \ge 1} \right)$, where: (i) $(P_n)_{n \ge 1}$ is a sequence of multivariate polynomials, each potentially involving countably many variables; (ii) there exists a constant $D \ge 1$ such that for all $n \ge...
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Zusammenfassung: | We characterize the limiting distributions of random variables of the form
$P_n\left( (X_i)_{i \ge 1} \right)$, where: (i) $(P_n)_{n \ge 1}$ is a sequence
of multivariate polynomials, each potentially involving countably many
variables; (ii) there exists a constant $D \ge 1$ such that for all $n \ge 1$,
the degree of $P_n$ is bounded above by $D$; (iii) $(X_i)_{i \ge 1}$ is a
sequence of independent and identically distributed random variables, each with
zero mean, unit variance, and finite moments of all orders. More specifically,
we prove that the limiting distributions of these random variables can always
be represented as the law of $P_\infty\left( (X_i, G_i)_{i \ge 1} \right)$,
where $P_\infty$ is a polynomial of degree at most $D$ (potentially involving
countably many variables), and $(G_i)_{i \ge 1}$ is a sequence of independent
standard Gaussian random variables, which is independent of $(X_i)_{i \ge 1}$.
We solve this problem in full generality, addressing both Gaussian and
non-Gaussian inputs, and with no extra assumption on the coefficients of the
polynomials. In the Gaussian case, our proof builds upon several original tools
of independent interest, including a new criterion for central convergence
based on the concept of maximal directional influence. Beyond asymptotic
normality, this novel notion also enables us to derive quantitative bounds on
the degree of the polynomial representing the limiting law. We further develop
techniques regarding asymptotic independence and dimensional reduction. To
conclude for polynomials with non-Gaussian inputs, we combine our findings in
the Gaussian case with invariance principles. |
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DOI: | 10.48550/arxiv.2412.06749 |