Properties of cumulative sum of a random walk?

Question

A random walk can be generated by computing the cumulative sum of a list of random numbers.

import random
import itertools
import seaborn as sns
N=1000000
x = range(N)
series = [random.randrange(-1,2) for i in range(N)]   # random integer numbers -1,0,1
walk = list(itertools.accumulate(series))
sns.lineplot(x=x, y=walk)

But why stop here? What happens if we compute the cumulative sum of the random walk itself

walk_of_walk = list(itertools.accumulate(walk))
sns.lineplot(x=x, y=walk_of_walk)

We can even go further and compute the cumulative sum of the cumulative sum of a random walk

walk_of_walk_of_walk = list(itertools.accumulate(walk_of_walk))
sns.lineplot(x=x, y=walk_of_walk_of_walk)

To my surprise the cumulative sum and the cumulative sum of the cumulative sum of a random walk looks smoother and smoother than the random walk. Is this to be expected? I couldn't find anything regarding the cumulative sums of random walks. Is there anything known about its mathematical properties? Or am I doing nonsense here?

Answer 1

Define $$S_{i0}=X_i\qquad i=1,\ldots,n$$ and $$S_{i(t+1)}=\sum_{j=1}^i S_{jt}\qquad t=0,1,\ldots$$ then $$S_{i2}=\sum_{j=1}^i \sum_{k=1}^j X_{k}=\sum_{k=1}^i \sum_{j=k}^i X_{k}=\sum_{k=1}^i (k-i+1) X_{k}=\sum_{k=1}^i kX_k - (i-1)\bar X_i$$ and $$S_{i(t+1)}=\sum_{j=1}^i \sum_{k=1}^j S_{k(t-1)}=\sum_{k=1}^i \sum_{j=k}^i S_{k(t-1)}=\sum_{k=1}^i (k-i+1) S_{k(t-1)}$$ meaning that the last terms are more and more heavily weighted and that the scale of the cumulated sums increases as well

Answer 2

Re: smoothness of a Random Walk compared to that of a cumulative sum of a Random Walk.

Smoothness of these functions is determined by their uncertainties :

S = Σ ht(n) ∆n, where ht(n) is a coin toss,{+ /- 1};

∆ S = Σ∆ ht(n) ∆n == ht(n) or +/- 1;

sS = Σ S(n) ∆n,

∆sS= Σ ∆ S(n) ∆n == S(n);

an average S ~ sqrt(n), whereas an average sS= n*sqrt(n).

Thus, ∆S/S = (+/- 1) /sqrt(n);

while . ∆sS /sS = S / ~ sqrt(n) / n*sqrt(n)~ (+/-1) / n.

The uncertainty in S is relatively larger than that of sS.

The uncertainty of S is always (+/- 1) per S, but is (+/-1) /n for sS.

The same argument would apply to the cumulative sum of sS.

{ if S = Σ ht(n) ∆n is called a Random Walk, Then, sS = Σ S(n) ∆n, may be called a Random Absement. }