If sample average converges in an iid sample, must it converge to the mean?

Question

WLLN tells us that if $X_1,...,X_n$ are iid, with $X_1$ having finite mean $\mu$, then their sample average converges in probability to $\mu$.

Suppose instead we know that $X_1,...,X_n$ are iid and their sample average converges in probability to a constant. Can we argue that $X_1$ has a finite mean and hence that such a constant must be the mean of $X_1$?

Answer 1

For counter-examples, you might want to consider cases where the weak law of large numbers applies but the strong law of large numbers does not. These cases must have $E[X_1]$ undefined.

For example, adapting the second example in Wikipedia,

• suppose $\mathbb P\left(X_1=\frac{(-2)^n}{n} \right)=\frac1{2^n}$ for positive integer $n$
• this does not have a mean since $\sum\limits_{n=1}^\infty \frac{(-2)^n}{n} \frac1{2^n}=\sum\limits_{n=1}^\infty \frac{(-1)^n}{n}$ does not converge absolutely
• but the sum does converge conditionally to $-\log_e(2)\approx -0.693$ and the sample averages converge in probability to this

In this example, you need a large sample to see much convergence. For example with $10^4$ simulations each with sample sizes of $10^2$ (red), $10^3$ (green) and $10^4$ (blue), the following R code

Xbar <- function(cases){
  Y <- rgeom(cases, 1/2) + 1 # R's geometric distribution starts at 0
  X <- (-2)^Y / Y
  mean(X)
  }
            
set.seed(2023)
sims4 <- replicate(10^4, Xbar(10^4))
plot(density(sims4, from=-2, to=1), col="blue")
sims3 <- replicate(10^4, Xbar(10^3))
lines(density(sims3, from=-2, to=1), col="green")
sims2 <- replicate(10^4, Xbar(10^2))
lines(density(sims2, from=-2, to=1), col="red")
abline(v = -log(2))

shows the increasing concentration of the sample mean as the sample size increases