An Interesting Class of Non-Kac Random Polynomials

• Published in: Journal of the Indian Society for Probability and Statistics Vol. 24; no. 2; pp. 545 - 564
• Main Authors: Praharaj, Samya, Guha, Suman
• Format: Journal Article
• Language: English
• Published: New Delhi Springer India 01.12.2023
• Subjects: Mathematics and Statistics; Operations Research/Decision Theory; Probability Theory and Stochastic Processes; Research Article; Statistical Theory and Methods; Statistics; Kac–Rice formula; Binomial distribution; Heavy-tailed random coefficients; Random polynomials
• ISSN: 2364-9569; 2364-9569
• DOI: 10.1007/s41096-023-00166-5

As evident from classical results on random polynomials, it is difficult to derive the probability distribution of the number of real roots N n ( R ) of a random polynomial of degree n , and even if derived, the distribution is not of any standard form. In this article, we construct a class of random polynomials of degree 2 ( n + 1 ) such that the distribution of N 2 ( n + 1 ) ( R ) belongs to the scale family of binomial distributions. For the constructed class of random polynomials, we further notice that as n → ∞ , the expected proportion of real roots E N 2 ( n + 1 ) ( R ) 2 ( n + 1 ) need not converge to 0, in contrast to most of the existing literature on random polynomials which show E ( N n ( R ) ) = o ( n ) as n → ∞ that, in turn, implies that asymptotically the majority of the roots of the random polynomial are non-real. The second result of this article shows that in fact for any given p ∈ [ 0 , 1 ] , the construction can be engineered in such a way that the random polynomial has light-tailed coefficients and E ( N 2 ( n + 1 ) ( R ) ) ∼ 2 ( n + 1 ) p as n → ∞ . Hence, for the class of random polynomials, that we have constructed in this article, asymptotically the number of real roots can be arbitrarily large. Compared to Kac polynomials, which consist of light-tailed random coefficients, the amount of research done for random polynomials whose coefficients are non-identical/dependent/heavy-tailed, is relatively scarce. In the final part of the present article, we give the third and final result that concerns random polynomials with heavy-tailed coefficients. We extend the second result to show that for any given p ∈ ( 0 , 1 ] , we can construct non-Kac, random polynomials with heavy-tailed, stochastically dependent coefficients for which E ( N 2 ( n + 1 ) ( R ) ) ∼ 2 ( n + 1 ) p as n → ∞ . All these results are based on the assumption that all the coefficients of the constructed class of random polynomials are continuous random variables. We conclude the article with a discussion of how they would change if instead, we assume that the coefficients are general random variables and how far the results derived in this article can be extended to some higher degree random polynomials of the same structure.