Compound Poisson Process - r

Question

My task is to simulate a compound Poisson process defined as:
where
is a Poisson process and Y_i are Gamma(shape,scale) distributed. This is my R code:
# parameter for Poisson distribution.
lambda = 1
# parameters for Gamma distribution.
shape = 7.5
scale = 1
comp.pois = function(t.max, lambda) {
stopifnot(t.max >= 0 && t.max %% 1 == 0)
# offset ns by 1 because first y is 0.
# generate N(t), that is number of arrivals until time t.
ns = cumsum(rpois(n = t.max, lambda = lambda)) + 1
# generate gamma distributed random variables Y_i.
ys = c(0, rgamma(n = max(ns), shape = shape, scale = scale))
# generate all X(t) for t <= t.max.
return(c(0, cumsum(x = ys[ns])))
}
Compute a random sample of X(10) and compare means and variances.
# sample size.
size = 1000
t = 10
# ts is a vector of sample values for X(10).
ts = sapply(1:size, function(i) comp.pois(t, lambda)[t])
# sample mean and variance:
(mean.s = mean(ts))
(var.s = var(ts))
# theoretical mean and variance:
(mean.t = lambda * t * shape * scale)
(var.t = (shape + 1) * shape * scale^2)
output:
> # sample:
> (mean.s = mean(ts))
[1] 63.38403
> (var.s = var(ts))
[1] 184.3264
> # theoretical:
> (mean.t = lambda * t * shape * scale)
[1] 75
> (var.t = (shape + 1) * shape * scale^2)
[1] 63.75
This variance is gigantic, but I cannot spot my mistake. Please help. Thank you.
EDIT:
I used the following algorithm to generate the N(t). I don't know why it is supposed to be better. I took it from Rizzo, Maria L. Statistical computing with R. CRC Press, 2007. The mean is good, but the variance is even worse. I tried sampling from the Gamma distribution only once for the entire simulation (although I'm pretty sure this does not reflect the problem very well) and the mean was off by around 10-40 for t = 10. When resampling for every X(t) (which is what the following code does), the mean is very exact. As pointed out, the variance is horrifying. This is probably not a good solution, but I suppose it is as good as it gets.
lambda = 3
shape = 6
scale = 2
size = 10000
eps = 1e-8
t = 10
# with probability 1-eps, n or less gamma distributed random variables are needed.
n = qpois(1-eps, lambda = lambda * t)
# sample from the gamma distribution. Not sure if it's ok to use the same sample every time.
# with this, the mean is of by about 10%.
# ys = c(rgamma(n = n, shape = shape, scale = scale))
# the interarrival times are exponentially distributed with rate lambda.
pp.exp = function (t0) {
# not sure how many Tn are needed :/
Tn = rexp(1000, lambda)
Sn = cumsum(Tn)
return(min(which(Sn > t0)) - 1)
}
# generate N(t) which follow the poisson process.
ns = sapply(1:size, function (i) pp.exp(t))
# generate X(t) as in the problem description.
xs = sapply(ns, function (n) {
ys = c(rgamma(n = n, shape = shape, scale = scale))
sum(ys[1:n])
})
output (t=10) in this case:
> # compare mean and variance of 'size' samples of X(t) for verification.
> # sample:
> (mean.s = mean(xs))
[1] 359.864
> (var.s = var(xs))
[1] 4933.277
> # theoretical:
> (mean.t = lambda * t * shape * scale)
[1] 360
> (var.t = (shape + 1) * shape * scale^2)
[1] 168