And yes, the result is true when taking the limit to infinity. In the proof of Theorem 1, we establish the following
--- where E[tau_M] / (M-k+1) is the expected specific density of a random sequence of length M (and then M->infty)

You can derive actually an interval for a finite sequence from this

But as you can see, numerically with Monte Carlo, we obtain that E[Z_i] are somehow around (w+1)/2, so all good.

(and the values E[Z_i] - (w+1)/2 are, equivalently, somewhat around 0)

They're not, actually ! For instance, this is what we (formally) obtain for Z1 and Z2, with random minimizers. For w=10, E[Z1] = 5.5 whereas E[Z2] = 5.87 (there is a good reason for this, I can explain more if you're curious, but basically it is a border effect).

Remember that for random minimizers, d = 2/(w+1) ? Well, for d*, we have this lovecraftian thing.

21/

The specific d* of a k-mer set X is defined as the number of distinct minimizers chosen divided by |X|.

For a given local scheme, the (expected) d* is computed as the average specific d* of all possibles k-mer sets, taken uniformly at random.

- a straightforward parallel to standard density

19/

So basically, we assimilate the sequence of m-mers of the base sequence S as a sequence of integers X.

And define Y as the sequence of positions selected in each window.

Since density is usually defined on random sequences, we assimilate X and Y as random sequences, of unknow distribution.

8/

Both are true but waking up this morning was a real challenge haha!

somehow I live by this motto from an old Absinthe commercial

Last week I got to present my work on lexicographical minimizers at DSB, in Pisa. It was a wonderful moment, thanks again to everyone involved!