Convolutions to get Gaussian

xmax = 4;
ConvolveAnim[f_, g_, c_, t_] :=
 Module[{\[Tau] = 1.5 xmax - 2.5 xmax t}, Show[

   Plot[f[x], {x, -xmax, xmax}, PlotRange -> {0, 3}, Axes -> None,
    ImageSize -> 300]
   ,
   Plot[g[x + \[Tau]], {x, -xmax, xmax}],
   Plot[c[x], {x, -xmax, -\[Tau]},
    PlotStyle -> Directive[Thick, Black]],
   RegionPlot[
    0 <= y <= g[x + \[Tau]] f[x], {x, -xmax, xmax}, {y, 0, 3},
    PlotPoints -> 50, PlotStyle -> RGBColor[0.8, 0.6, 0.65]]
   ]];
conv[1, x_] := f[x];
conv[n_, x_] :=
  conv[n, x] = (Convolve[f[ttt], conv[n - 1, ttt], ttt, xt]) /. {xt ->
       x};
convs = Table[conv[i, x], {i, 5}];
convs = convs/(convs /. {x -> 0});
anim[n_, t_] :=
  ConvolveAnim[(convs[[n]] /. {x -> #}) &,
   f, (convs[[n + 1]] /. {x -> #}) &, t];
animall[t_] := anim[1 + Floor[t], t - Floor[t]];

frames = Table[animall[t], {t, 0, 3.99, 0.0399}];

Export["convolve.gif", frames];