SLLN for Total Claim Process
> library(VGAM) |
> cols <- rainbow(10)
> plot.new()
> for (i in 1:10) {
+ n <- 1000
+ claims <- rexp(n)
+ arrivals <- cumsum(rexp(n))
+ df <- cbind(cumsum(claims), arrivals)
+ plot(df[, 1]/df[, 2], type = "l", xlim = c(0, 1000), col = cols[i], ylim = c(0,
+ 2), ylab = "S(t)/t")
+ par(new = T)
+ }
> par(new = F) |
The above distribution uses an exponential claim size distribution and standard homogeneous poisson process.
> cols <- rainbow(10)
> plot.new()
> for (i in 1:10) {
+ n <- 1000
+ claims <- rpareto(n, 2, 4)
+ arrivals <- cumsum(rexp(n))
+ df <- cbind(cumsum(claims), arrivals)
+ plot(df[, 1]/df[, 2], type = "l", xlim = c(0, 1000), col = cols[i], ylim = c(0,
+ 6), ylab = "S(t)/t")
+ par(new = T)
+ }
> par(new = F) |
The above distribution uses an pareto claim size distribution and standard homogeneous poisson process.
As one can see that the SLLN holds good.