Exercise 11.2: Random walk with steps of variable length

1. Consider a 1d random walk with jumps of all lengths allowed. The probability that the length of a single step is $l$ is denoted $p(l)$. If the form of $p(l)$ is given by $p(l)=e^{-l}$, what is the form of $P_N(x)$? Suggestions: Use the inverse transform method explained in the previous chapter to generate steps of length $l$ according to $p(l)$. Then generate a walk of $N$ steps and determine the net displacement $x$. Generate many such walks and determine $P_N(x)$. Plot $P_N(x)$ vs. $x$ and confirm that the form of $P_N(x)$ is consistent with a Gaussian distribution.

2. Consider the step probability $p(l)=A/l^2$. Determine the normalization constant $A$ using:
   
   $$\sum_{l=1}^{\infty} \frac{1}{l^2}=\frac{\pi ^2}{6}$$
   
   
   and the requirement that $p(l)$ be normalized to unity. Perform a Monte Carlo simulation as in part 1 and verify that the from of $P_N(x)$ is given by
   
   $$P_N(x)\sim \frac{bN}{x^2+b^2N^2}.$$
   
   
   What is the magnitude of the constant $b$? Does the variance of $P_N(x)$ exist?

The ``central limit theorem'' states that the probability distribution of a large number of measurements will be a Gaussian centered at the mean. The only requirement is that the probability $p(x)$ has finite first and seconds moments, that the measurements are statistically independent, and that $N$ is large. In part 2 of the previous exercise, the probability distribution has no second moment, and that explains the result.