Model the probability distribution of the nth term given a set of sequence?

Question

Given Data: X Sequences each of n terms. n is small number around 20, While X is greater than 1000.

• Z1 = A11, A12, A13, A14, A15, .. A1n
• Z2 = A21, A22, A23, A24, A25, .. A2n
• .
• .
• ZX = Ax1, Ax2, Ax3, Ax4, Ax5, .. Axn

First term of all X Sequences are 1. A11 = A21 = A31 = .. An1 = 1 You can get the next term in a sequence by adding/subtracting a small amount, definitely less than 0.1, usually even smaller. It is very similar to a random walk starting from 1 each time increasing/decreasing the value by a small percentage of the past value.

From only the data calculate the probability distribution of A(n) given A(1), This is the density plot of (A1n, A2n, ... Axn). In a random walk this would a bell curve with mean at 1.

The data isn't random, there are patterns, the density plot of (A1n, A2n, ... Axn) shows A(n) is more likely to be less than 1, usually by a small amount.

Question:

1. Given A(1) and A(2) calculate the probability distribution of A(n) ?
2. Given A(1), A(2), .. A(x) calculate the probability distribution of A(n) ?