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

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 $\exp ( - C|t{|^\alpha }),C$ being a positive constant, $\alpha \geqq 1$ and ${a_0},{a_1}, \cdots ,{a_n}$ are nonzero real numbers. Let ${\kappa _n} = {\max _{0 \leqq v \leqq n}}|{a_v}|,{t_n} = {\min _{0 \leqq v \leqq n}}|{a_v}|$. If ${\Lambda _n} = \mu \log n/\log (({\kappa _n}/{t_n})\log n)$, then \[ (\text {i} )\quad \Pr \{ ({N_n}/{\Lambda _n}) \geqq \mu \} < \frac {\mu }{{\log (({\kappa _n}/{t_n})\log n){{(\log n)}^{\alpha - 1}}}}\] if $1 \leqq \alpha < 2$ and $< \mu \log \{ ({\kappa _n}/{t_n})\log n\} /\log n$ if $\alpha > 2$, provided $({\kappa _n}/{t_n}) = o(\log n)$. \[ (\text {ii} )\quad \Pr \left \{ {\sup \limits _{n > {n_0}} \frac {{{N_n}}}{{{\Lambda _n}}} \geqq \mu } \right \} < \mu ’ / {\left \{ {\log \left ( {\frac {{\log {n_0}}}{{\log (({\kappa _{{n_0}}}/{t_{{n_0}}})\log n}}} \right )} \right \}^{\alpha - 1}}\] 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.