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Proceedings of the American Mathematical Society

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On the lower bound of the number of real roots of a random algebraic equation with infinite variance. III


Authors: G. Samal and M. N. Mishra
Journal: Proc. Amer. Math. Soc. 39 (1973), 184-189
MSC: Primary 60G99
MathSciNet review: 0315786
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Abstract: Let $ {N_n}$ be the number of real roots of a random algebraic equation $ \Sigma _0^n{a_v}{\xi _v}{x^v} = 0$, where $ {\xi _v}$'s are independent random variables with common characteristic function $ \exp ( - C\vert t{\vert^\alpha }),C$ being a positive constant, $ \alpha \geqq 1$ and $ {a_0},{a_1}, \cdots ,{a_n}$ are nonzero real numbers. Let $ {\kappa _n} = {\max _{0 \leqq v \leqq n}}\vert{a_v}\vert,{t_n} = {\min _{0 \leqq v \leqq n}}\vert{a_v}\vert$. If $ {\Lambda _n} = \mu \log n/\log (({\kappa _n}/{t_n})\log n)$, then

$\displaystyle ($i$\displaystyle )\quad \Pr \{ ({N_n}/{\Lambda _n}) \geqq \mu \} < \frac{\mu }{{\log (({\kappa _n}/{t_n})\log n){{(\log n)}^{\alpha - 1}}}}$

if $ 1 \leqq \alpha < 2$ and $ < \mu \log \{ ({\kappa _n}/{t_n})\log n\} /\log n$ if $ \alpha > 2$, provided $ ({\kappa _n}/{t_n}) = o(\log n)$.

$\displaystyle ($ii$\displaystyle )\quad \Pr \left\{ {\mathop {\sup }\limits_{n > {n_0}} \frac{{{N_... ...{\log (({\kappa _{{n_0}}}/{t_{{n_0}}})\log n}}} \right)} \right\}^{\alpha - 1}}$

if $ \alpha > 1$, provided $ \log ({\kappa _n}/{t_n}) = o(\log n)$.

It may be remarked that the coefficients $ {a_v}{\xi _v}$'s in the above equation are not identically distributed.


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DOI: https://doi.org/10.1090/S0002-9939-1973-0315786-8
Keywords: Random variables, stable distribution, infinite variance, random algebraic equations
Article copyright: © Copyright 1973 American Mathematical Society