Illustration of random sequence that convergences in distribution, but not density

convergence_in_distribution.R

library(tidyverse)
library(gganimate)

f <- function(n, x) 1 - cos(2 * pi * n * x)
g <- function(.f, x) map_dbl(x, ~integrate(function(z) .f(z), 0, .x)$value)

df <- cross_df(list(
  x = seq(0, 1, length.out = 100),
  n = c(1, 2, 3, 5, 10, 15, 25)
)) %>% 
  mutate(pdf = f(n, x)) %>% 
  group_by(n) %>% 
  mutate(cdf = map2_dbl(n, x, ~g(partial(f, n = .x), .y))) %>% 
  gather("distribution", "f_x", pdf:cdf) %>% 
  mutate(f_unif = if_else(distribution == "pdf", dunif(x), punif(x)))

ggplot(df, aes(x = x)) +
  geom_line(aes(y = f_unif, linetype = "X ~ Unif[0,1]"), color = "black")  +
  geom_line(aes(y = f_x, color = n, linetype = "Xn"))  +
  facet_wrap(~distribution, scales = "free") +
  transition_states(n, 1, 1) +
  scale_color_gradient(low = "red", high = "blue") +
  scale_linetype_manual(values = c("dashed", "solid")) +
  theme_minimal() +
  theme(legend.position = "bottom") +
  guides(linetype = guide_legend("Random variable", "top", ncol = 1),
         color = guide_colorbar(title.position = "top")) +
  labs(
    title = "Example of random sequence that converges in distribution but not density",
    subtitle = expression(pdf(X[n]): f[n](x) == 1 - cos(2*pi*n*x)*bold(1)[(paste(0, ',', 1))]),
    y = "F(x) / f(x)", linetype = NULL
  )

anim_save("convergence_in_distribution.gif")