I assume an infinite population.

I'll focus on Pearson's linear correlations.

The core sample (CS) has size Nc. It's sampled from the infinite population.

The bootstrap samples (BS) are sampled with replacement from the CS

BS mean and variance are not independent, even if the population distribution is normal, because the CS has its own distribution (different from the population distribution due to sampling fluctuation).

As the CS is always slightly skewed (unless the population distribution is discrete, in that case, it might sometimes be symmetric), there will always be a correlation (different from one CS to one CS) between mean and variance of BS. This correlation is rho(mean,var)(BS)

Asymptotically, as Nc tends towards infinite, rho(mean,var)(BS) tends towards a constant rho(mean,var)(pop) dependent on the population distribution. If the population distribution is normal, this is zero. Basically, this constant depends on the skewness of the population distribution.

Then, you can compute r(mean,var)(BS) on a finite set (Nb) of BS. It's an unbiased estimator of rho(mean,var)(BS), and as Nb tends towards infinite, r(mean,var)(BS) tends towards rho(mean,var)(BS).

If the population is finite (size Np) and the CS is sampled from the population without replacement, then, sampling fluctuation are smaller, and as Nc tends towards Np, rho(mean,var)(BS) tends towards rho(mean,var)(pop). If Nc = Np, the whole population is taken and rho(mean,var)(BS) = rho(mean,var)(pop).

Trends with Nb hold true for r(mean,var)(BS).

Non linear dependence :

There will always be a dependence (sometimes with rho(Pearson)=0) between mean and variance, even for a symmetric population distribution.

Example : Population has a binary variable with P(X=0)=0.5 et P(X=1)=0.5

Then, assume the CS has the same size than the population.

Assume an infinite BS size

Then, variance and mean of the BS will be linked by a quadratic relation : variance = mean*(1-mean)

Hello,

The key thing is that your sample of 100 is representative to the population you are interested in. 

Regards,

Thanks for your responses, they have tremendously helped me to understand the situation.