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Let $F_n$ be a real valued function, $$F_n(r) = \frac{r}{n}(\psi(n) - \psi(r)$$ where $\psi$ is a digamma function. Let the sequence of functions $\{g_n\}$ be defined by $g_n(x) := F_n(nx)$.

Show that $\{g_n\}$ converges uniformly on [0,1] and to the function $g(x) := -xlog(x)$.

I am trying to understand the paper, "A variant of the Secretary Problem: the Best or the Worst" by L. Bayon, J. Grau, A. M. Oller-Marcen, M. Ruiz, P.M. Suarez.

It says there that this can be seen with little effort but I am at a loss.

Any hints on how I would prove this would be much appreciated.

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  • $\begingroup$ $\psi(n+1) = \psi(n) + \frac{1}{n}$ then use $\log(x) = \int_1^x \frac{1}{t}dt = \lim_{m \to \infty} \frac{x-1}{m}\sum_{n=1}^m \frac{1}{1+n \frac{x-1}{m}}$ $\endgroup$ – reuns May 6 at 5:38
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We have

$$\tag{1}\begin{align}|F_n(nx) -(-x \log x)| &= |x(\psi(n) - \psi(nx)) + x\log x| \\&= |x(\psi(n) - \log n) - x(\psi(nx) - \log(nx)| \\ &\leqslant|x||\psi(n) - \log n| + |x (\psi(nx) - \log(nx))|\end{align}$$

There are many well-known representations of the digamma function. Using a recurrence relation we get following representation for integer arguments:

$$\psi(n) = \sum_{k=1}^{n-1} \frac{1}{k} - \gamma,$$

where $\gamma$ is the Euler-Mascheroni constant. With $0 \leqslant x \leqslant 1$, we then have $$\tag{2}|x||\psi(n) - \log n| \leqslant |\psi(n) - \log n| = \left| \sum_{k=1}^{n-1}\frac{1}{k} - \log n - \gamma\right| $$

From the definition of $\gamma$, the RHS of (2) converges to $0$ as $n \to \infty$ and , thus, for any $\epsilon > 0$ there exists $N_1 \in \mathbb{N}$, independent of $x$, such that for all $n \geqslant N_1$, we have for the first term on the RHS of (1),

$$|x||\psi(n) - \log n| < \epsilon/2$$

Using Gauss' integral representation we have

$$\psi(z) = \log z - \frac{1}{2z} - \int_0^\infty \left(\frac{1}{2} - \frac{1}{t} + \frac{1}{e^{t} + 1} \right)e^{-tz} \, dt,$$

and it can be shown (see here) that for all $z > 0$,

$$\frac{1}{2z} \leqslant \log z - \psi(z) \leqslant \frac{1}{z}$$

Thus, for the second term on the RHS of (1) we have

$$\frac{1}{2n} = \frac{x}{2nx} \leqslant |x (\psi(nx) - \log(nx))| = x(\log (nx) - \psi(nx))\leqslant \frac{x}{nx} = \frac{1}{n},$$

and so for any $\epsilon > 0$ there exists $N_2 \in \mathbb{N}$, independent of x, such that for all $n \geqslant N_2$ and all $x \in (0,1]$ we have

$$|x (\psi(nx) - \log(nx))| < \epsilon /2$$

Thus, for all $n \geqslant (N_1,N_2)$ and all $x \in (0,1]$, we have

$$|F_n(nx) - (-x\log x)| < \epsilon$$

This establishes uniform convergence on $(0,1]$. To completely finish here, we would need to establish that $F_n(0) = 0$ by extension as is the case for $x\log x$ where we have $x \log x \to 0$ as $x \to 0$.

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