Editing Random Fibonacci sequence (section)

==Description==
A random Fibonacci sequence is an [[integer]] random sequence given by the numbers <math>f_n</math> for [[natural number]]s <math>n</math>, where <math>f_1=f_2=1</math> and the subsequent terms are chosen randomly according to the random recurrence relation
<math display=block>
f_n = \begin{cases}
f_{n-1}+f_{n-2}, & \text{ with probability } \tfrac12; \\
f_{n-1}-f_{n-2}, & \text{ with probability } \tfrac12.
\end{cases}
</math>
An instance of the random Fibonacci sequence starts with 1,1 and the value of the each subsequent term is determined by a [[fair coin]] toss: given two consecutive elements of the sequence, the next element is either their sum or their difference with probability 1/2, independently of all the choices made previously. If in the random Fibonacci sequence the plus sign is chosen at each step, the corresponding instance is the [[Fibonacci sequence]] (''F''<sub>''n''</sub>),
<math display=block> 1,1,2,3,5,8,13,21,34,55,\ldots. </math>
If the signs alternate in minus-plus-plus-minus-plus-plus-... pattern, the result is the sequence
<math display=block> 1,1,0,1,1,0,1,1,0,1,\ldots.</math>

However, such patterns occur with vanishing probability in a random experiment. In a typical run, the terms will not follow a predictable pattern:
<math display=block> 1, 1, 2, 3, 1, -2, -3, -5, -2, -3, \ldots
\text{ for the signs } +, +, +, -, -, +, -, -, \ldots.</math>

Similarly to the deterministic case, the random Fibonacci sequence may be profitably described via [[matrix (mathematics)|matrices]]:
<math display=block>{f_{n-1} \choose f_{n}} = \begin{pmatrix} 0 & 1 \\ \pm 1 & 1 \end{pmatrix} {f_{n-2} \choose f_{n-1}},</math>

where the signs are chosen independently for different ''n'' with equal probabilities for + or −. Thus
<math display=block>{f_{n-1} \choose f_{n}} =  M_{n}M_{n-1}\ldots M_3{f_{1} \choose f_{2}},</math>
where (''M''<sub>''k''</sub>) is a sequence of [[Independent and identically-distributed random variables|independent identically distributed random matrices]] taking values ''A'' or ''B'' with probability 1/2:
<math display=block> A=\begin{pmatrix} 0 & 1 \\ 1 & 1 \end{pmatrix}, \quad
B=\begin{pmatrix} 0 & 1 \\ -1 & 1 \end{pmatrix}. </math>