11_RA

Random walk is a mathematical object, known as a stochastic or random process, that describes a path that consists of a succession of random steps on some mathematical space such as the integers [1, 2]. An elementary example of a random walk is the random walk on the integer number line, Z, which starts at 0 and at each step moves +1 or −1 with equal probability.

The random walk that is defined as Yt=Yt−1+et, where et is white noise [3]. Denotes that the current position is the sum of the previous position + an unpredicted term.

The variance increases linearly with time and the mean function μt=0, since

In short because it keeps adding the variance of the next increments to the variability we have in getting to where we are now.

and we can see that tσ^2 increases linearly with t.

The mean is zero at each time point; if you simulated the series many times and averaged across series for a given time, that would average to something near 0

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References

1. https://en.wikipedia.org/wiki/Random_walk

2. https://www.statisticshowto.com/random-walk/

3. https://stats.stackexchange.com/questions/159650/why-does-the-variance-of-the-random-walk-increase