Distribution of the Maxima of Random Takagi Functions
نویسنده
چکیده
This paper concerns the maximum value and the set of maximum points of a random version of Takagi's continuous, nowhere di erentiable function. Let F (x) := ∑∞ n=1 ( 1 2 )n−1 εnφ(2 n−1x), x ∈ R, where ε1, ε2, . . . are independent, identically distributed random variables taking values in {−1, 1}, and φ is the tent map de ned by φ(x) = 2dist (x,Z). Let p := P (ε1 = 1), M := max { F (x) : x ∈ R} , andM := {x ∈ [0,1) : F (x) = M} . An explicit expression for M is given in terms of the sequence {εn}, and it is shown that the probability distribution μ of M is purely atomic if p < 1 2 , and is singular continuous if p = 1 2 . In the latter case, the Hausdor dimension and the multifractal spectrum of μ are determined. It is shown further that the set M is nite almost surely if p < 1 2 , and is topologically equivalent to a Cantor set almost surely if p = 1 2 . The distribution of the cardinality of M is determined in the rst case, and the almost-sure Hausdor dimension of M is shown to be (2p− 1)/2p in the second case. The distribution of the leftmost point of M is also given. Finally, some of the results are extended to the more general functions ∑ an−1εnφ(2n−1x), where 0 < a < 1.
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