<< Statistics`ContinuousDistributions`
<< Statistics`DiscreteDistributions`
<< Statistics`DescriptiveStatistics`
gdist := NormalDistribution[0, 1]
bdist := BernoulliDistribution[ .5]
PartialSums[
        n_Integer] := {
xg1 = 
          Table[Random[gdist], {n}]; 
sg1 =  
          FoldList[Plus, 0, xg1]/Sqrt[n]; 
xg2 = 
          Table[Random[bdist] -  .5, {n}]; 
sg2 =  
          FoldList[Plus, 0, xg2]/Sqrt[n]; 
zz = 
          Table[k, {k, 0, n}]*(1/n) // N; 
dd = 
          Table[1,  {k, 0, n}]*(1/n)}
PartialSums[1000];
xg1;
xg2;
sg1;
sg2;
J1 = Transpose[{zz, sg1}];
J2 = Transpose[{zz, sg2}];
ListPlot[J1, PlotJoined -> True]
ListPlot[J2, 
      PlotJoined -> True]