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An example of a process which does not follow the central limit theorem is a multiplicative process. Program MultiplicativeProcess explores the distribution of the quantity $\Pi = x_1^nx_2^{N-n}$, where $x_1$ is chosen $n$ times with probability $p$ and $x_2$ is chosen $N-n$ times with probability $q = 1-p$. You can choose $x_1$, $x_2$, $N$, and $p$. The distribution of $\Pi$ is plotted, and the mean value and most probable value of $\Pi$ are computed.

Problem: Simulation of a multiplicative process

Use Program MultiplicativeProcess to simulate the distribution of values of the product ${x_1}^n {x_2}^{N-n}$. Choose $x_1=2$, $x_2=1/2$, and $p=q=1/2$.

1. Choose $N=4$ and estimate $\overline{\Pi}$ and $\widetilde{\Pi}$. Do your estimated values converge more or less uniformly to the analytical values as the number of measurements becomes large?
2. Do a similar simulation for $N=40$. Compare your results with a similar simulation of a random walk and discuss the importance of extreme events for random multiplicative processes.

Resource

Problem 3.56 in Statistical and Thermal Physics: With Computer Applications, 2nd ed., Harvey Gould and Jan Tobochnik, Princeton University Press (2021).