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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)


Maxima of random algebraic curves

Authors: M. Das and S. S. Bhatt
Journal: Trans. Amer. Math. Soc. 243 (1978), 195-212
MSC: Primary 60G99
MathSciNet review: 0494482
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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\vert Z \right\vert^\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{displaymath}\begin{array}{*{20}{c}} {{c_1}(\alpha )\, = \,\frac{1} {{{\pi... ...,\alpha )}}\,\exp \,( - z)\,dz} \right\}} \,dv.} } \end{array} \end{displaymath}

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PII: S 0002-9947(1978)0494482-X
Keywords: Random variables, distribution, normally distributed random variables, sequence, mathematical expectation, variance, characteristic function, random algebraic curves
Article copyright: © Copyright 1978 American Mathematical Society