library(ggplot2)
library(latex2exp)

set.seed(2018)

# Number of samples:
N <- 3000
# Three series of indepedent N samples from the standard Cauchy distribution:
x1 <- rcauchy(N); x2 <- rcauchy(N); x3 <- rcauchy(N)
# Sample means:
m <- function(x) {
  return(sapply(1:N, function(n) mean(x[1:n])))
}
m1 <- m(x1); m2 <- m(x2); m3 <- m(x3)

d1 <- data.frame(x=1:N, y=m1, w=sapply(1:N, function(n) "1"))
d2 <- data.frame(x=1:N, y=m2, w=sapply(1:N, function(n) "2"))
d3 <- data.frame(x=1:N, y=m3, w=sapply(1:N, function(n) "3"))
data <- rbind(d1, d2, d3)

fig <- ggplot(data) +   
  geom_hline(yintercept=0, alpha=.5) +
  geom_line(aes(x=x, y=y, color=w)) + 
  labs(x=TeX("$n$"), y=TeX("$(x_1 + \\cdots + x_n)/n$")) +
  guides(color=FALSE)

svg(filename="LawOfLargeNumbersCauchy.svg", width=16/2, height=9/2)
plot(fig)
dev.off()