sRandom  :=  2*(Random[Integer] - 1/2)

Plot[
{Abs[
          Sqrt[Pi/2] Sum[sRandom, {i, n}]], 
Sqrt[
          n]}, 
{n, 0, 
        10000}, 
PlotStyle -> {{Hue[0.2]}, {Hue[
              1.0]}}]

aRandom  :=  
      If[(r = Random[]) < 0.5, 0, If[r < 0.75, -1, 1]]

n = 0;
a = {0};
v = {0};
x = {0};
While[
        n ≤ 100, 
a = 
          Append[a, aRandom]; 
v = 
          Append[v, Last[v] + Last[a]]; 
x = 
          Append[x, 
            Last[x] + 
              Last[v]]; 
++n];
ListPlot[a];
ListPlot[v]; ListPlot[x];

i = 1;
While[
      i <= 100, 
n = 
        1; 
a = {aRandom}; 
v = {0}; 
x = {0}; 
While[
        n ≤ 1000, 
a = 
          Append[a, aRandom]; 
v = 
          Append[v, Last[v] + Last[a]]; 
x = 
          Append[x, Last[x] + Last[v]]; 
++n];  (* 
        end of While[n  ... ] *) 
If[
        i == 1, 
aa = a^2; vv = v^2; 
        xx = x^2, 
aa += a^2; vv += v^2; 
        xx += x^2]; 
++i];  (* 
      end of While[i  ... ] *) 
ListPlot[
      Sqrt[aa/(i - 1)]];
ListPlot[Sqrt[vv/(i - 1)]];
ListPlot[Sqrt[xx/(i - 1)]];