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Functional Equations and Characterization of Probability Distributions · 1974-pde-359

Difference and Functional Equations · ICM 1974: Functional Equations and Characterization of Probability Distributions

MSC 35 · Partial differential equations

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Problem statement

Let r,pr,p be positive integers, and let m1,…,mrm_1,\ldots,m_r be positive integers such that m1+⋯+mr=pm_1+\cdots+m_r=p. Let P,Q∈Rp×pP,Q\in\mathbb R^{p\times p} be orthogonal matrices, partitioned into column blocks

P=(P1 ⋯ Pr),Q=(Q1 ⋯ Qr), P=(P_1\ \cdots\ P_r),\qquad Q=(Q_1\ \cdots\ Q_r),

where Pi,Qi∈Rp×miP_i,Q_i\in\mathbb R^{p\times m_i}, and let δ1,…,δr>0\delta_1,\ldots,\delta_r>0. Determine all continuous functions hi:Rmi→Ch_i:\mathbb R^{m_i}\to\mathbb C, i=1,…,ri=1,\ldots,r, such that, for every ti∈Rmit_i\in\mathbb R^{m_i}, i=1,…,ri=1,\ldots,r,

∑i=1rhi ⁣(∑j=1rPiTQjtj)=∑i=1rδi2hi ⁣(δi−1ti). \sum_{i=1}^{r}h_i\!\left(\sum_{j=1}^{r}P_i^{\mathsf T}Q_jt_j\right) = \sum_{i=1}^{r}\delta_i^2h_i\!\left(\delta_i^{-1}t_i\right).

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Verdict: unresolved.

The exact problem asks for all continuous solutions for every orthogonal pair P,QP,Q, every positive block partition, and every choice of positive δi\delta_i. No citable complete classification for those unrestricted data was found. Rao states the equation as Problem 5 and explicitly says that it was not solved in general: solutions were known for p=1,2p=1,2, and for general pp only when the blocks PiTQjP_i^{\mathsf T}Q_j satisfy the additional hypotheses of Kagan--Linnik--Rao, Theorem A.5.3. The available functional-equation results below therefore establish important special cases but do not cover all zero, singular, repeated, or otherwise degenerate block configurations permitted by the statement.

  • C. Radhakrishna Rao, "Functional Equations and Characterization of Probability Distributions," Proceedings of the International Congress of Mathematicians, Vancouver 1974, vol. 2, Canadian Mathematical Congress, 1975, pp. 163-168, official proceedings PDF. Problem 5 is the exact equation. The paragraph immediately preceding it records the p=1,2p=1,2 solutions and the conditional general-pp result, then identifies the unrestricted case as unsolved.
  • A. M. Kagan, Yu. V. Linnik, and C. R. Rao, Characterization Problems in Mathematical Statistics, Wiley Series in Probability and Mathematical Statistics, Wiley, 1973, ISBN 978-0-471-45421-2, bibliographic record. Theorem A.5.3 (p. 476) supplies the coefficient-conditioned result invoked by Rao; it is not a theorem for arbitrary block matrices PiTQjP_i^{\mathsf T}Q_j.
  • C. G. Khatri and C. Radhakrishna Rao, "Functional Equations and Characterization of Probability Laws through Linear Functions of Random Variables," Journal of Multivariate Analysis 2(2) (1972), 162-173, doi:10.1016/0047-259X(72)90024-3. The paper solves related sums of functions of linear maps under hypotheses on their coefficient matrices and yields the polynomial/normal-distribution cases used in this line of work. Those hypotheses do not provide the missing unrestricted classification.

The remaining gap is precisely the classification when the cross-block maps PiTQjP_i^{\mathsf T}Q_j are allowed to have arbitrary ranks and algebraic relations, as the problem statement permits. Searches of later functional-equation and probability-characterization literature found no result closing that gap.

References
  1. [1]ICM Conjectures · 1974-pde-359 link
  2. [2]ICM 1974: Functional Equations and Characterization of Probability Distributions link

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