A sequence of functions on the real line is defined as
$$f_0=\chi_{[-1,1]},\qquad f_{n+1}=f_n*f_0, n=0,1,2,\dots $$ Here * means convolution. I tried to draw the graphs of the functions and see what happens to the graphs. But, it is too hard to draw the graph of each $f_n$. Could anyone tell me what kind of changes to the graphs as $n$ increases?
It will tend towards a (scaled) Gaussian distribution function; see the Central Limit Theorem. If we scale $\chi$ so that $\int\chi(x)dx=1$ then we get the following sequence of functions (for the first 10)