In the following we're talking about a simple random walk, $X_i=\pm 1$ with equal probability, and we are calculating running averages,
\[
\frac{S_n}{n}=\frac{1}{n}\sum_{i=1}^{n}X_i, n=1,2,\ldots
\]
The SLLN (convergence almost surely) says that we can be 100% sure that this curve stretching off to the right will eventually, at some finite time, fall entirely within the bands forever afterward (to the right).
The R code used to generate this graph is below (plot labels omitted for brevity).
n <- 1000; m <- 50; e <- 0.05
s <- cumsum(2*(rbinom(n, size=1, prob=0.5) - 0.5))
plot(s/seq.int(n), type = "l", ylim = c(-0.4, 0.4))
abline(h = c(-e,e), lty = 2)
The WLLN (convergence in probability) says that a large proportion of the sample paths will be in the bands on the right-hand side, at time n
n
(for the above it looks like around 48 or 9 out of 50). We can never be sure that any particular curve will be inside at any finite time, but looking at the mass of noodles above it'd be a pretty safe bet. The WLLN also says that we can make the proportion of noodles inside as close to 1 as we like by making the plot sufficiently wide.
The R code for the graph follows (again, skipping labels).
x <- matrix(2*(rbinom(n*m, size=1, prob=0.5) - 0.5), ncol = m)
y <- apply(x, 2, function(z) cumsum(z)/seq_along(z))
matplot(y, type = "l", ylim = c(-0.4,0.4))
abline(h = c(-e,e), lty = 2, lwd = 2)
