Why is $-cN/z^\alpha$ a good approximation for $\ln(1-F(z))^N$?

Question

Bertin's Statistical Physics of Complex Systems, 3rd ed. p. 65 defines a "complementary cumulative distribution $\tilde{F}(z)$" equal to $\int_z^\infty p(x)\,dx$, where the density $p(z)$ is proportional to $1/x^{1+\alpha}$ (p. 64). (Later in the book, p. 237, Bertin notes what I can work out myself, that $p(x)=\frac{\alpha x_0^\alpha}{x^{1+\alpha}}$, where $x_0$ is a minimum value for $x$.) I'm assuming that $\alpha$ is positive.

In case it's relevant background for my real question below, it's puzzling to me that Bertin then says that

we can approximate $\tilde{F}(z)$ by its asymptotic behavior at large $z$: $$\tilde{F}(z) \approx \frac{c}{z^\alpha}$$

where $c$ is a "proportionality constant". This isn't an approximation, though. For $c=x_0^\alpha$, $\tilde{F}(z) = \frac{c}{z^\alpha}$. I think the claim about this being an approximation for large $z$ concerns the next point that Bertin makes, i.e. the one that I don't understand:

Bertin claims that the following approximation is reasonable when $N$ is large:

$$\ln\left(1-\tilde{F}(z)\right)^N \approx -\frac{cN}{z^\alpha}$$

[The left hand side is as it is in the book; I interpret it as $\ln\left(\left(1-\tilde{F}(z)\right)^N\right)$, since this appears without the log function on the previous page.]

So Bertin is implying that $\ln(1-\tilde{F}(z))$ is a good enough approximation of $(-\tilde{F}(z))$ that they can be substituted for each other when $1-\tilde{F}(z)$ is raised to the power $N$.

If $\tilde{F}(z)$ was very close to $1$ and $N$ was not too large, then in the product $(1-\tilde{F}(z))^N$, the term $\tilde{F}(z)^N$ could dominate, be the major part of the result. Then maybe $-cN/z^\alpha$ would make sense as an approximation? But in the quotation above Bertin said that the approximation would hold for "large $z$", and in which case $c/z^\alpha$ should be small. This is so even though, on the preceding page, Bertin said that he would focus on cases in which $\alpha<1$. If $\alpha$ was small, that would mean that $z^\alpha$ would grow slowly, but $z^\alpha$ still would be large for sufficiently large $z$. And in any event, $N$ is supposed to be large. So it seems as if $(1-\tilde{F}(z))^N$ should be close to $1$. Why is $(-\frac{cN}{z^\alpha})$ a good approximation for the log of this value? Or am I supposed to assume that $\alpha$ is small enough that $z^\alpha$ is close to $c=x_0^\alpha$? But I'm not sure how that would help.

[If you suspect that Bertin's claim depends on physics-specific background intuitions and that as a result it would be better to ask in physics.SE, feel free to suggest that.]

Answer 1

Requested from comments:

You can say

$$\ln\left(\left(1-\tilde{F}(z)\right)^N\right) =N\ln\left(1-\tilde{F}(z)\right) \approx - N \tilde{F}(z) \approx -N \dfrac{c}{z^\alpha}$$

• the equality is a standard property of logarithms
• the first approximation is from $\ln(1-x) \approx -x$ for small $x$, and the complementary CDF $\tilde{F}(z)$ will be small in the tail
• the second approximation is the stated assumption about the shape of the tail of the distribution (essentially saying it is close to a power law for large $z$)