13. Pareto tail. Continuous time random walk.
LAST TIME

• Geordie Rose argued that quantum theory describes the evolution of the probability amplitude.
• In our treatment based on the master equation we only discussed the evolution of the probability.
• Neglect of phase coherence justified if decoherence time short compared with other time scales of interest.
• In this limit we obtained a description where the different participating
  processes took place at Poisson rates.
• Today we will allude to, but not show in detail, that there are many other processes correlations between successive steps are important.
• We will continue our discussion of the scaling properties of the Levy-stable distributions.
• We also start relaxing the assumption of Poisson rates.

NON-GAUSSIAN BEHAVIOR

• In the last but one lecture we showed how non-Gaussian behavior could be obtained from jump distributions that had infinite variance
  $\Rightarrow$ Lévy flights
• In physics Lévy flights with uncorrelated increments are hard to find because they involve discontinuous processes. We will, however, later in this course discuss superdiffive processes where typical displacements increase with time faster than $\sqrt{t}$.
• Possibly best physical example of levy flights is diffusion in worm-like surfactant micelles Ott et al. [1990], Cates and Candeau [1990].
• Discontinuous random walks more common in finance, economics and evolution and will be discussed further in Lecture 15.
• Earliest and possibly most best known example of Lévy stable distributions in finance $\Rightarrow$ Pareto tail.

PARETO TAIL FOR INCOMES

• The income distribution in most
  societies is roughly lognormal for most people who do have an income, up to approximately the 97th percentile.
• We will later discuss how lognormal
  distributions arise in random multiplicative processes.
• The lognormal distributions does not to hold for the 2-3% who are extremely rich.

Vilfredo Pareto tried, during the last part of the 19th and the early part of the 20th century, to make economics and sociology into an "exact" science by pursuing analogies with physics and mechanics. He was particularly interested into the dynamics of business cycles and the rise and fall of empires and elites, and his work remains somewhat controversial. He wrote in 1897 that

In all places and at all times the distribution of income in a stable economy, when the origin of measurement is at a sufficiently high income level, will be given approximately by the empirical formula y=ax^-v, where y is the number of people having an income x or greater and v is approximately 1.5.

Income tax data in several countries agree qualitatively with Pareto's observation also in more recent times.

MECHANISM PROPOSED BY LYDALL[1959]
Arrange employees within an enterprise in a hierarchy. Let y_i be the number of people at the i'th level and let i+1 be the level above.
The ratio of personnel at the two levels are

n=y_i/y_i+1

Suppose each operator on the i'th level earn their income x_i from a commission of a fraction $\lambda$ of the income of the people in the level below. In return a fraction $\lambda$ of the income is paid to the immediate boss above.
The income at the i+1 level is thus

$$x_{i+1}=n(1-\lambda )\lambda x_i.$$

If p[x] is the probability distribution for
income x we have

$$n p[n(1-\lambda )\lambda x]=p[x]$$

We find a power law distribution of the Pareto form with

$$v+1={{\ln n}\over{\ln[n\lambda (1-\lambda )]}}$$

In order for amplification to take place we must impose the restriction

$$n\lambda (1-\lambda )> 1.$$

v is the exponent of the cumulative distribution i.e.

$$n\sim x^{-v-1}$$

Note that the exponent v is non-universal, and depends on the parameters n and $\lambda$ which must be expected to vary from society to society.


Montroll and Shlesinger [1982][1983] builds a model to explain the data from the observation that

The leverage people in the investment business have their style of amplification. During certain periods of prosperity easy money become available for investment, sometimes in stock, sometimes in real estate or perhaps in silver or Rembrandts. A common feature of such times is that the daring may exploit the easy money to acquire some speculative commodity through margin payment, say, 10% with a promise to pay the remainder. If the commodity doubles in price a 10% margin is amplified into a ninefold profit.

Let $g(x/\hat{x})$ be the basic distribution without amplification.
With a small probability $\lambda$ the income is amplified by leverage N times. The income distribution becomes

$$G(x)=(1-\lambda)[g(x)+\frac{\lambda}{N}g(\frac{y}{N})+ \frac{\lambda^2}{N^2}g(\frac{y^2}{N^2})+\cdots]$$

$$=\frac{\lambda}{N}G(\frac{x}{N})+(1-\lambda)g(x)$$

Asymptotically

$$G(x)=\frac{\lambda}{N}G(\frac{x}{N})$$

Substituting $G\propto x^{-1-\nu}$ we find

$$\nu=\frac{-\ln\lambda}{\ln N}$$

The two approaches are mathematically equivalent. A less tongue in cheek discussion of the income distribution can be found in Mandelbrot [1962]. This article reprinted commented upon in Mandelbrot [1997]. Pareto's own thoughts on the subject can be found in Pareto[1968] and [1984].

RELAXING ASSUMPTION OF INDEPENDENT POISSON RATES:

• Consider cases where waiting time
  distribution in "traps" non-Poisson.
• In financial markets one observes periods with high volatility and enormous trading volumes interspersed with slow periods.
• Look at situations where randomness is "quenched" or "frozen". If system comes back to where it has been before, it finds the same "randomness" as it found the last time $\Rightarrow$ long range correlations.
• In physical systems examples abound in heterogeneous systems with either
  • Wide distribution of trap energies $\Rightarrow$ amorphous solds.
  • Frozen disorder $\Rightarrow$ glasses.
  • Irregular flow fields $\Rightarrow$ hydrodynamics in heterogeneous media.

CONTINUOUS TIME RANDOM WALKS
Consider a walk on a regular lattice in d-dimensions
Three probability distributions of interest:

• P_n(x)= probability that walker has arrived at x after n steps, starting at the origin i.e
  
  $$P_0(x)=\delta_{x,0}$$
  
  

• $\tau=$ waiting time at each site. $\psi(\tau)=$ waiting time distribution
  
• P( x,t)= probability that a walker is at x at time t.

Assume that probability distribution p for jumps in different directions is given:

$$P_{n+1}(x)=\sum_{x'}p( x,x') P_n( x')$$

For a translationally invariant lattice equation can be solved using methods of generating functions and Fourier transforms.

$$G( x,z)=\sum_{n=0}^\infty P_n(x)z^n$$

Substituting into recursion relation

$$G( x,z)=\delta_{x,0}+z\sum_{x'}p( x-x')G( x',z)$$

g(k) Fourier transform of G(x,z) over lattice.
$\lambda( k)$ Fourier transform of p( x). Get

$$g( k,z)=\frac{1}{1-z\lambda( k)}$$

In the limit of an infinite lattice we then find the lattice Green's function

$$G( x,z)=\frac{1}{(2\pi)^d}\int_{BZ}d^dk\frac{\exp(- ix\cdot k)} {1-z\lambda(k)}$$

Where BZ stands for Brillouin Zone. (Readers unfamiliar with solid state physics can find an excellent introduction in Ascroft and Mermin [1976]. We will not need to get into details about reciprocal lattices and lattice sums in this course).

SUMMARY

• Discussed Pareto tail of income distribution.
• Started to discuss situations where individual processes in master equation were not Poisson distributed.
• Formulated continuous time random walk problem.
• Separated probability distribution after n steps from distribution after time t.
• The former distribution could be found by lattice Greens function methods.

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14. Return time distribution. Continuous time random walk(continued)