Simulation Beta Distribution

Question

Suppose a function such as:

$\phi^{\alpha} (x,y) = \left\{ \begin{array}{l l} \alpha x & \quad \text{if $x < y$}\\ x & \quad \text{if $x = y$}\\ \alpha x + (1 -\alpha) & \quad \text{if $x > y$}\\ \end{array} \right.$

I wish to run a simulation for 5 random walks (each with length 100) on $\phi$ where $x$ is drawn from a beta distribution with parameters (3,1) and $\alpha = 0.5$.

I don't know how to deal with $\phi$ in the random walk process, especially given its 3 conditions.

EDIT: $y$ is the mean of all the previous $x$. For instance, suppose we are at time $t=3$ and observations start at $t=0$. Then, $y_3 = \frac{x_0 + x_1 + x_2}{3}$.

---

ϕ[x_, y_] := Piecewise[{
  {α*x, x < y}, 
  {x, x == y}, 
  {α*x + (1 - α), x > y}}]
ϕ[x, y]

Answer 1

With the function

f[a_, {x_, y_}] := Piecewise[{{a x, x < y}, {x, x == y}, {1 - a + a x, x > y}}, 0]

you can define a new function, that will perform a single simulation

sim[a_Real, n_Integer] := 
 Module[{data = Partition[Riffle[#, Accumulate[#]/Range[n]], 2] &@ RandomVariate[BetaDistribution[3, 1], n]},
  f[a, #] & /@ data
 ]

where the first argument is the value of alpha and the second argument is length of the random walk.
Than 5 random walks of length 100 with alpha = 0.5 can be plotted using

ListPlot[Table[sim[0.5, 100], {n, 5}], PlotRange -> All, Joined -> True, Frame -> True]

Answer 2

Here is an approach using MovingMap and f from Karsten 7:

f[a_, {x_, y_}] := 
 Piecewise[{{a x, x < y}, {x, x == y}, {1 - a + a x, x > y}}, 0]
s[lst_] := MovingMap[f[0.5, {#[[4]], Mean[#[[1 ;; 3]]]}] &, lst, 3]

Visualizing:

ListPlot[s /@ RandomVariate[BetaDistribution[3, 1], {5, 100}], 
 Joined -> True, Frame -> True]