Local maxima of a random algebraic polynomial
We present a useful formula for the expected number of maxima of a normal process ξ(t) that occur below a level u. In the derivation we assume chiefly that ξ(t),ξ′(t), and ξ′′(t) have, with probability one, continuous 1 dimensional distributions and expected values of zero. The formula referred to a...
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| Main Authors: | , |
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| Format: | Article |
| Language: | English |
| Published: |
Wiley
2001-01-01
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| Series: | International Journal of Mathematics and Mathematical Sciences |
| Online Access: | http://dx.doi.org/10.1155/S016117120100391X |
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| Summary: | We present a useful formula for the expected number of maxima of a
normal process ξ(t) that occur below a level u. In the
derivation we assume chiefly that ξ(t),ξ′(t), and ξ′′(t) have, with probability one, continuous 1 dimensional
distributions and expected values of zero. The formula referred to
above is then used to find the expected number of maxima below the
level u for the random algebraic polynomial. This result
highlights the very pronounced difference in the behaviour of the
random algebraic polynomial on the interval (−1,1) compared with
the intervals (−∞,−1) and (1,∞). It is also shown
that the number of maxima below the zero level is no longer O(logn) on the intervals (−∞,−1) and (1,∞). |
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| ISSN: | 0161-1712 1687-0425 |