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Let $\{p_i : i \in \mathbb{Z}\}$ be an i.i.d sequence with $p_i \in (0, 1)$. Fix this random environment, then consider the random walk $$P[ X_{n+1} = i + 1 | X_n = i] = 1 − P[ X_{n+1} = i − 1 | X_n = i] = p_i.$$

(a) Let $p_i$ be $\frac13$ or $\frac23$, with probability $\frac12$ each. Find a deterministic sequence $a_n$ such that $\frac{X_n}{a_n}$ seems to be converging in distribution to a non-degenerate variable.

(b) Find a distribution with $\mathbb{E}[p_i] = \frac12$ such that $X_n$ is transient.

(c) In both cases, draw pictures of the space-time trajectorie

I need to simulate the above,I have started working on (a) but I don't get how to find such a sequence. here is my code for the random walk:

import random
import numpy as np 
import matplotlib.pyplot as plt 

prob = [0.333333333333333333,0.666666666666666666]

start = 0 
positions = [start] 

rr = np.random.random(1000) 
downp = rr < prob[0] 
upp = rr > prob[1]

for idownp, iupp in zip(downp, upp):
    down = idownp and positions[-1] > -100
    up = iupp and positions[-1] < 100
    positions.append(positions[-1] - down + up)

plt.plot(positions) 
plt.show() 

how can I proceed??

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    $\begingroup$ I'm voting to close this question as off-topic because it belongs to cross validated site stats.stackexchange.com $\endgroup$ – Tasos Apr 26 '19 at 10:13

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