TBan-I

Nei seguente esercizio useremo le notazioni della statistica dei valori estremi usate nel corso.

Exercise 1: The Gumbel Distribution

In the spirit of the central limit theorem, you look for a scaling form:

$E_{\min }=a_{M}+b_{M}z$

The constants $a_{M}$ and $b_{M}$ absorb the dependence on $M$, while the random variable $z$ is distributed according to a probability distribution that does not depend on $M$.

In the Gaussian case, expand $A(E)$ around $a_{M}$:

By setting

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a_M = E_{\min}^{\text{typ}} = -\sigma \sqrt{2 \log M} + \ldots \qquad\text{and}\qquad b_M = \frac{1}{A'(a_M)} = \frac{\sigma}{\sqrt{2 \log M}} }

you find

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \exp(A(a_M)) = \frac{1}{M} \qquad\text{and}\qquad Q_M(E) \equiv \text{Prob}(E_{\min} > E) \sim \exp\!\!\left[-\exp\!\!\left(\frac{E - a_M}{b_M}\right)\right] }

Therefore, the variable $z=(E-a_{M})/b_{M}$ is distributed according to an M-independent distribution.

It is possible to generalize this result and classify the scaling forms into the Gumbel universality class:

• Characteristics:
  • Applies when the tails of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p(E)} decay faster than any power law.
  • Examples: the Gaussian case discussed here or exponential distributions Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p(E) = \exp(E) \quad \text{with} \quad E \in (-\infty, 0)} .

• Scaling Form:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle P(z) = \exp(z)\,\exp(-e^{z}) }

Exercise 3: number of states above the minimum

Definition of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n(x) } :Given a realization of the random energies Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {E_1, E_2, \ldots, E_M}} , define

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n(x) = \#\{ i \mid E_{\min} < E_i < E_{\min}+x \} }

that is, the number of random variables lying above the minimum Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle E_{\min}} but less than Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle E_{\min}+x} . This is itself a random variable. We are interested in its mean value:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \overline{n(x)} = \sum_{k=0}^{M-1} k \, \text{Prob}[n(x)=k] }

The Final goal is to show that, for large M (when the extremes are described by the Gumbel distribution), you have:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \overline{n(x)} = e^{x/b_M}-1 }

Step 1: Exact manipulations: You start from the exact expression for the probability of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle k} states in the interval:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \text{Prob}[n(x)=k] = M \binom{M-1}{k} \int_{-\infty}^\infty dE \; p(E)\,[P(E+x)-P(E)]^{k}\,[1-P(E+x)]^{M-k-1} }

To compute Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \overline{n(x)}} , you must sum over Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle k} . Use the identity

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sum_{k=0}^{M-1} k \binom{M-1}{k} (A-B)^k B^{M-1-k} = (A-B)\frac{d}{dA} \sum_{k=0}^{M-1} \binom{M-1}{k} (A-B)^k B^{M-1-k} = (M-1)(A-B)A^{M-2} }

to arrive at the form:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \overline{n(x)} = M (M-1) \int_{-\infty}^\infty dE \; p(E) \left[P(E+x) - P(E)\right] (1-P(E))^{M-2}= M \int_{-\infty}^\infty dE \; \left[P(E+x) - P(E)\right] \frac{d Q_{M-1}(E)}{dE} }

where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Q_{M-1}(E) = [1-P(E)]^{M-1}} .

Step 2: the Gumbel limit So far, no approximations have been made. To proceed, we use Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Q_{M-1}(E)\approx Q_M(E)} and its asymptotics Gumbel form:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{d Q_{M-1}(E)}{dE} \, dE \;\sim\; \exp\!\!\left(\frac{E-a_M}{b_M}\right) \exp\!\!\left[-\exp\!\!\left(\frac{E-a_M}{b_M}\right)\right] \frac{dE}{b_M} = e^{z} e^{-e^{z}} dz }

where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle z = (E-a_M)/b_M} .

The main contribution to the integral comes from the region near $E\approx a_{M}$, where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle P(E) \approx e^{(E-a_M)/b_M}/M} .

Compute the integral and verify that you obtain:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \overline{n(x)} = \bigl(e^{x/b_M}-1\bigr) \int_{-\infty}^{\infty} dz \, e^{2z - e^z} = e^{x/b_M}-1 }