I am reading from this book and I want to make sure I understand what is going on.
What I get from the book
Consider a population of $N$ individuals. The population size ($N$) is constant. select two individuals randomly in the population and ask the question, when did the most recent common ancestor (MRCA) lived? Let's denote this time to the MRCA by the random variable $T_2$. Whenever, looking backward in time, we see two individuals having a common ancestor, then we refer to this event as an event of coalescence. In other words, $T_2$ is the random variable of the time (in generations) for coalescence to occur between two randomly chosen individuals in a population of constant size $N$.
The probability of not coalescing in the previous generations (that is the probability that the two randomly sampled individuals are not siblings) is $1-\frac{1}{N}$ and the probability of coalescing (probability of being siblings) is $\frac{1}{N}$. The probability of the coalescence event to occur $t$ generations ago is the probability of not coalescing for $t-1$ generations and coalescing then. Therefore $T$ has the distribution
$P(T_2=t) = \left(1-\frac{1}{N}\right)^{t-1}\frac{1}{N}$
More generally, let's denote $M_n$ the time for $n$ individuals to coalesce. If $T_n$ is the time until $n$ individuals coalesce into $n-1$ individuals (the time for one pair of individuals among $n$ individuals to coalesce), then $$M_n = \sum_{i=2}^n T_i$$
Question
I think that $M_n = \sum_{i=2}^n T_i$ is true only if all $T_i$ are independent variables. Are they independent? Why?