Abstract:
Recall Lebesgue's singular function. Imagine flipping a biased coin with probability $p$ of heads and probability $q=1-p$ of tails. Let the binary expansion of $\xi\in[0,1]$:
$ \xi = \sum_{k=1}^{\infty}c_k2^{-k}$ be determined
by flipping the coin infinitely many times, that is,
$c_k =1$ if the $k$-th toss is heads and $c_k =0$ if it is tails. We define Lebesgue's singular function $L(t)$ as the distribution function of the random variable $\xi$:
$$
L(t) = Prob\{\xi < t\}.
$$
It is well-known that $L(t)$ is strictly increasing
and its derivative is zero almost everywhere ($p\ne q$).
The moments of Lebesque' singular function are defined as
$$
M_n = \mathsf{E}\xi^n.
$$
The main result of this paper is the following:
$$
M_n = O(n^{\log_2 p}).
$$
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