On the rate of convergence for Takagi class functions

Abstract

We consider a generalized version of the Takagi function, which is one of the most famous example of nowhere differentiable continuous functions. We investigate a set of conditions to describe the rate of convergence of Takagi class functions from the probabilistic point of view: The law of large numbers, the central limit theorem, and the law of the iterated logarithm. On the other hand, we show that the Takagi function itself does not satisfy the law of large numbers in the usual sense.

Appendix

Once one recognizes that (8) should hold, it can be quickly obtained by summing up the asymptotics

$$\begin{aligned} N^{1-\beta } e^{-KN^{\beta }} - (N+1)^{1-\beta } e^{-K(N+1)^{\beta }} = (1+o(1))K\beta e^{-KN^{\beta }} \quad \mathrm{as }\; N \rightarrow \infty , \end{aligned}$$

which can be proved by differentiation of $N^{1-\beta } e^{-KN^{\beta }}$ in N. For the sake of completeness, we give a constructive proof of (8).

Lemma A1

For $K>0$ and $0<\beta < 1$, there exist positive constants $C_i=C_i(K,\beta )$$(i=1,2)$ such that

$$\begin{aligned} \int _a^{\infty } e^{-K x^{\beta }}\,dx&\ge C_1 a^{1-\beta } e^{-Ka^{\beta }} \qquad \mathrm{for\,any}\;a>0, \mathrm{and} \\ \int _a^{\infty } e^{-K x^{\beta }}\,dx&\le C_2 a^{1-\beta } e^{-Ka^{\beta }} \qquad \mathrm{for}\, a\ge 1 \,\mathrm{satisfying }\,K a^{\beta } \ge 1. \end{aligned}$$

Proof

By changing the variable to $t=Kx^{\beta }$,

$$\begin{aligned} \int _a^{\infty } e^{-K x^{\beta }}\,dx&= \dfrac{1}{\beta K^{\frac{1}{\beta }}}\int _{Ka^{\beta }}^{\infty } t^{\frac{1}{\beta }-1} e^{-t}\,dt. \end{aligned}$$

Using integration by parts, we have

$$\begin{aligned} \int _{Ka^{\beta }}^{\infty } t^{\frac{1}{\beta }-1} e^{-t}\,dt&=\left[ -t^{\frac{1}{\beta }-1} e^{-t}\right] _{Ka^{\beta }}^{\infty } + \int _{Ka^{\beta }}^{\infty } \left( \frac{1}{\beta }-1\right) t^{\frac{1}{\beta }-2}e^{-t}\,dt \\&\ge (Ka^{\beta })^{\frac{1}{\beta }-1} e^{-Ka^{\beta }} = K^{\frac{1}{\beta }-1} a^{1-\beta } e^{-Ka^{\beta }}. \end{aligned}$$

For the other direction, let $s:= \left\lceil \dfrac{1}{\beta }-1 \right\rceil$. By repeated use of integration by parts,

$$\begin{aligned} \int _{Ka^{\beta }}^{\infty } t^{\frac{1}{\beta }-1} e^{-t}\,dt&=(Ka^{\beta })^{\frac{1}{\beta }-1} e^{-Ka^{\beta }} + \dfrac{\varGamma \left( \tfrac{1}{\beta }\right) }{\varGamma \left( \tfrac{1}{\beta }-1\right) }\int _{Ka^{\beta }}^{\infty } t^{\frac{1}{\beta }-2}e^{-t}\,dt \\&= \cdots \\&= \sum _{i=1}^s \dfrac{\varGamma \left( \tfrac{1}{\beta }\right) }{\varGamma \left( \tfrac{1}{\beta }+i-1\right) } (Ka^{\beta })^{\frac{1}{\beta }-i} e^{-Ka^{\beta }} \\&\quad +\, \dfrac{\varGamma \left( \tfrac{1}{\beta }\right) }{\varGamma \left( \tfrac{1}{\beta }-s\right) }\int _{Ka^{\beta }}^{\infty } t^{\frac{1}{\beta }-s-1}e^{-t}\,dt. \end{aligned}$$

For $a \ge 1$, we have $a^{1-i\beta } \le a^{1-\beta }$ for $i=1,\ldots ,s$. Noting that $(1/\beta )-s-1 \le 0$, for $a \ge 1$ satisfying $Ka^{\beta } \ge 1$, the second term in the right hand side is not more than

$$\begin{aligned} \dfrac{\varGamma \left( \tfrac{1}{\beta }\right) }{\varGamma \left( \tfrac{1}{\beta }-s\right) }\int _{Ka^{\beta }}^{\infty } e^{-t}\,dt = \dfrac{\varGamma \left( \tfrac{1}{\beta }\right) }{\varGamma \left( \tfrac{1}{\beta }-s\right) } \cdot e^{-Ka^{\beta }}. \end{aligned}$$

Thus we have

$$\begin{aligned} \int _{Ka^{\beta }}^{\infty } t^{\frac{1}{\beta }-1} e^{-t}\,dt \le \left( \sum _{i=1}^s \dfrac{\varGamma \left( \tfrac{1}{\beta }\right) }{\varGamma \left( \tfrac{1}{\beta }+i-1\right) }K^{\frac{1}{\beta }-i} + \dfrac{\varGamma \left( \tfrac{1}{\beta }\right) }{\varGamma \left( \tfrac{1}{\beta }-s\right) }\right) a^{1-\beta } e^{-Ka^{\beta }}. \end{aligned}$$

$\square$

By this lemma, we have

$$\begin{aligned} \sum _{n=N}^{\infty } e^{-K n^{\beta }}&\ge \int _N^{\infty } e^{-K x^{\beta }}\,dx \ge C_1 N^{1-\beta } e^{-KN^{\beta }} \end{aligned}$$

for any N, and

$$\begin{aligned} \sum _{n=N}^{\infty } e^{-K n^{\beta }}&\le e^{-KN^{\beta }} +\int _N^{\infty } e^{-K x^{\beta }}\,dx \le (1+C_2) N^{1-\beta } e^{-KN^{\beta }} \end{aligned}$$

for sufficiently large N.