Maxima of random algebraic curves

Abstract:

Let ${X_1},{X_2}, \ldots ,{X_n}$ be a sequence of independent and identically distributed random variables with common characteristic function ${\exp }( - {\left | Z \right |^\alpha })$ where $0 < \alpha \leqslant 2$, and $P(x) = \sum \nolimits _1^n {{X_k}{x^k}}$. Then we show that the numbers ${M_n}$ of maxima of the curves $y = P(x)$ have expectation $E{M_n} \sim c \log n$, as $n \to \infty$, where $c = c(\alpha ) = {c_1}(\alpha ) + {c_2}(\alpha )$ and \[ \begin {array}{*{20}{c}} {{c_1}(\alpha ) = \frac {1} {{{\pi ^2}{\alpha ^2}}}\int _{ - \infty }^\infty {dv \log \int _0^\infty {\left [ {\frac {{{{\left | {v - y} \right |}^\alpha }}} {{{{\left | {v - 1} \right |}^\alpha }}}} \right ] \exp ( - y) dy,} } } {{c_2}(\alpha ) = \frac {1} {{{\pi ^2}{\alpha ^2}}}\int _{ - \infty }^\infty {\log \int _0^\infty {\left \{ {\left ( {\frac {{{{\left | {v - z} \right |}^\alpha }}} {{{{\left | {v - \alpha - 1} \right |}^\alpha }}}} \right ) \frac {{{z^\alpha }}} {{\Gamma (1 + \alpha )}} \exp ( - z) dz} \right \}} dv.} } \end {array} \]