They are a lot of ways to compute an "estimated bid-ask spread". The most straightforward one is to sample the bid-ask on a regular time grid (for instance every second), but that for you need all the quote changes (a quote is the best bid and ask, price and quantity).

In this context, the original question is: "is the bid-ask spread sampled just before a trade always lower than the bid-ask spread sampled on a regular time grid?"

• After a trade is before the next trade. It don't get at all what you are saying. Commented Mar 4 at 13:27
• I edited the question , I hope is clearer
– XY0
Commented Mar 4 at 13:47
• Not at all. If I understand correctly this means the bid-ask spread should decrease forever as long as new trades are closed. An absurd claim. Commented Mar 4 at 13:57
• I was not clear ,I am sorry . Is the spread smaller when a trade (X_5) occurs than at the preceding instants? by precendent instants I mean the instants after trading X_4 . this does not mean that the spread observed at the occurrence of trading X_6 (and later) are always smaller and smaller than that sampled spread at trading X_5.
– XY0
Commented Mar 4 at 14:08
• If the spread is smaller at $T_5$ than at $T_4$ because of $T_4<T_5$ then, for the life of me, I don't understand why the spread should not be even smaller at $T_6$ that you said is later than $T_5\,.$ Oh my goodness! Commented Mar 4 at 14:22