Maxima of random algebraic curves
HTML articles powered by AMS MathViewer
- by M. Das and S. S. Bhatt PDF
- Trans. Amer. Math. Soc. 243 (1978), 195-212 Request permission
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 \[ \begin {array}{*{20}{c}} {{c_1}(\alpha ) = \frac {1} {{{\pi ^2}{\alpha ^2}}}\int _{ - \infty }^\infty {dv \log \int _0^\infty {\left [ {\frac {{{{\left | {v - y} \right |}^\alpha }}} {{{{\left | {v - 1} \right |}^\alpha }}}} \right ] \exp ( - y) dy,} } } {{c_2}(\alpha ) = \frac {1} {{{\pi ^2}{\alpha ^2}}}\int _{ - \infty }^\infty {\log \int _0^\infty {\left \{ {\left ( {\frac {{{{\left | {v - z} \right |}^\alpha }}} {{{{\left | {v - \alpha - 1} \right |}^\alpha }}}} \right ) \frac {{{z^\alpha }}} {{\Gamma (1 + \alpha )}} \exp ( - z) dz} \right \}} dv.} } \end {array} \]References
- Minaketan Das, The average number of maxima of a random algebraic curve, Proc. Cambridge Philos. Soc. 65 (1969), 741–753. MR 239669, DOI 10.1017/s0305004100003583
- Minaketan Das, Real zeros of a class of random algebraic polynomials, J. Indian Math. Soc. (N.S.) 36 (1972), 53–63. MR 322960
- B. V. Gnedenko and A. N. Kolmogorov, Limit distributions for sums of independent random variables, Addison-Wesley Publishing Co., Inc., Cambridge, Mass., 1954. Translated and annotated by K. L. Chung. With an Appendix by J. L. Doob. MR 0062975
- G. H. Hardy, J. E. Littlewood, and G. Pólya, Inequalities, Cambridge, at the University Press, 1952. 2d ed. MR 0046395
- M. Kac, On the average number of real roots of a random algebraic equation, Bull. Amer. Math. Soc. 49 (1943), 314–320. MR 7812, DOI 10.1090/S0002-9904-1943-07912-8 K. Knopp, Theory and application of infinite series, Blackie, London, 1957.
- B. F. Logan and L. A. Shepp, Real zeros of random polynomials, Proc. London Math. Soc. (3) 18 (1968), 29–35. MR 234512, DOI 10.1112/plms/s3-18.1.29
- B. F. Logan and L. A. Shepp, Real zeros of random polynomials. II, Proc. London Math. Soc. (3) 18 (1968), 308–314. MR 234513, DOI 10.1112/plms/s3-18.2.308
Additional Information
- © Copyright 1978 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 243 (1978), 195-212
- MSC: Primary 60G99
- DOI: https://doi.org/10.1090/S0002-9947-1978-0494482-X
- MathSciNet review: 0494482