Today I will discuss simple symmetric random walk in two dimensions. We can consider the same drunkard problem in two dimensions. Consider a drunkard in a city whose roads are arranged in a rectangular grid. The corners have coordinates (x,y) and the origin is (0,0). He moves east, west, north or south with equal probability of 1/4. When he reaches a corner, he moves again in the same manner.
This random walk can be considered in the form Z^2 = Z * Z. Let E1, E2 be i.i.d random numbers such that the incremental relative displacements are each 1/4.
Because the relative displacements are all the same, we move the start of all random walks to be the origin.
We further describe the horizontal and the vertical displacements with separate notations say x1 and x2 for horizontal and vertical respectively such that they belong to the grid and the final state is the sum of the incrementals in both directions taken together.
One of the lemmas now is that if both the horizontal and the vertical displacements are random walks, the two are not independent.
Since E1, E2 are independent their x and y projections are also independent and they are identically distributed with a transition probability to zero as 1/2 and to either 1 or -1 as 1/4.
So the horizontal states are a sum of iid random variables and hence a random walk. But it is not a simple random walk because the transition to zero has a different probability.
The same argument applies to vertical states.
Thus we see that the two random walks are not independent. Further, if we change the co-ordinates by rotating the axis by 45 degrees and scaling them then we can suppose the final state is described by the sum and the difference of their horizontal and vertical final states. In such a case the random walks along those axis is now considered independent.
This random walk can be considered in the form Z^2 = Z * Z. Let E1, E2 be i.i.d random numbers such that the incremental relative displacements are each 1/4.
Because the relative displacements are all the same, we move the start of all random walks to be the origin.
We further describe the horizontal and the vertical displacements with separate notations say x1 and x2 for horizontal and vertical respectively such that they belong to the grid and the final state is the sum of the incrementals in both directions taken together.
One of the lemmas now is that if both the horizontal and the vertical displacements are random walks, the two are not independent.
Since E1, E2 are independent their x and y projections are also independent and they are identically distributed with a transition probability to zero as 1/2 and to either 1 or -1 as 1/4.
So the horizontal states are a sum of iid random variables and hence a random walk. But it is not a simple random walk because the transition to zero has a different probability.
The same argument applies to vertical states.
Thus we see that the two random walks are not independent. Further, if we change the co-ordinates by rotating the axis by 45 degrees and scaling them then we can suppose the final state is described by the sum and the difference of their horizontal and vertical final states. In such a case the random walks along those axis is now considered independent.
No comments:
Post a Comment