In the previous post, we considered a simple symmetric random walk in two dimensions. We said that when we change the co-ordinates say by rotating the axis and scaling, the random walks along the single dimensions are independent. To prove this, we start with the random walk in two dimensions and express it as a sequence of vectors where each vector is an incremental displacement along the rotated and scaled axis in cartesian co-ordinate pairs. This sequence is also i.i.d because it is taken with some function on the elements of sequence consisting of incremental vectors along the original axis. When we look at this function, we see that the rotation and the scaling implies taking the sum and the difference of the incremental displacements along the original axis. This kind of sum and difference is independent for each of the elements in the sequence of the two dimension walk. Moreover, the possible values for each incremental in the two dimension walk are either positive or negative 1. So we can write the probabilites for state transtions of the two dimension walk as follows:
Probability to get to (1,1) in the two dimension walk = Probability to get to (1,0) in the single dimension walk = 1/4
Probability to get to (-1, -1) in the two dimension walk = Probability to get to (-1,0) in the single dimension walk = 1/4
Probability to get to (1, -1) in the two dimension walk = Probability to get to (0,1) in the single dimension walk = 1/4
Probability to get to (-1,1) in the two dimension walk = Probability to get to (0, -1) in the single dimension walk = 1/4
This implies that the probability for each of the incremental displacements along the rotated axis is 1/2. And so we can decompose the random walk in two dimensions with probability for the final state to a pair (n1 to be some value e1 , n2 to be some value e2) as a combination of the probability for the final state to be (n1 to be value e1) and the probability of the final state to be (n2 to be some value e2) for all choices of the value e1 and e2 to be in positive or negative unit distance. Thus they are independent.
We will next look at a few more lemmas in this case.
Probability to get to (1,1) in the two dimension walk = Probability to get to (1,0) in the single dimension walk = 1/4
Probability to get to (-1, -1) in the two dimension walk = Probability to get to (-1,0) in the single dimension walk = 1/4
Probability to get to (1, -1) in the two dimension walk = Probability to get to (0,1) in the single dimension walk = 1/4
Probability to get to (-1,1) in the two dimension walk = Probability to get to (0, -1) in the single dimension walk = 1/4
This implies that the probability for each of the incremental displacements along the rotated axis is 1/2. And so we can decompose the random walk in two dimensions with probability for the final state to a pair (n1 to be some value e1 , n2 to be some value e2) as a combination of the probability for the final state to be (n1 to be value e1) and the probability of the final state to be (n2 to be some value e2) for all choices of the value e1 and e2 to be in positive or negative unit distance. Thus they are independent.
We will next look at a few more lemmas in this case.
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