What does weakly dependent events mean?

Question

From what I know, Poisson distribution can be applied to calculate probability for independent Bernoulli trails. However, I also read from a textbook that it can be applied to calculate the probability for situations that have weakly dependent events such as the birthday paradox. Now, this got got me confused. What does weakly dependent mean? I did a search on Google but could not find anything that explains the definition of weakly dependent events. Could someone please enlighten me?

Answer 1

Illustration: Assume that you have $N$ different (numbered) balls in a jar. You are sampling at random without replacement $2$ times and you are interested to get balls of the first $n$, $(N>>n)$. The probability to get one of this special balls in the first try is $n/N$ while in the second is $$ \frac{n-1}{N-1}\times\frac{n}{N} + \frac{n}{N-1}\times\frac{N-n}{N}, $$ for large enough $N$ and $n$ this quantity equals approximately $n/N$ (Actually, if $\lim_{N,n\to \infty} n/N=p$). So these events can be seen as weekly dependent.

Remark: Small or zero Covariance does not imply independence (except for the multivariate normal case). Assume that $X$ equals $1$ and $-1$ with prob. $1/2$. Define $Y=X^2$. Hence, $$ cov(X,Y) = EXY - EXEY = 0 - 0=0, $$
although the variables "strongly" dependent. On the other hand, weekly correlated can mean that the correlation coefficient $$ \rho_{X,Y} = \frac{cov(X,Y)}{\sigma_X \sigma_Y} $$ is small. I guess that in many practical aspects it would count as a sufficient assumption/restriction.