4.5 Asymptotic developments
In this part we propose to derive asymptotic developments for pattern p-values from their recursive expressions. For under- (resp. over-) represented patterns, the main result is given in theorem 9 (resp. 12). In both cases, theses results are also presented in a simpler form (where only main terms are taken into account) in the following corollaries.
Proposition 7. For any x = (x(a-1),...,x0)' and all β ≥ 0 xβ = Rβx is given by  = Pβ and
Proof. As  =  for all j ≤ 0 it is trivial to get the expression of . If we suppose now that the relation (28) is true for some i and β then, thanks to the relation (27) we have
and so the proposition is proved through the principle of recurrence.     □
Lemma 8. For all i ≥ 0 and a ≤ b ∈ ℕ and r > 0 we define
If r ≠ 1 we have for all i ≥ 0 we have
and (case r = 1) for all i ≥ 0 we have
Proof. Easily derived from the following relation
Theorem 9. If P is primitive and admits a diagonal form we denote by λ > ν the largest two eigenvalues magnitude of P by P∞ = limi→+∞ Pi/λi (a positive matrix) and we get for all α ≥ 1 and i ≥ 0
uniformly in β and where  is a polynomial of degree i which is defined by  and for all i ≥ 1 by the following recurrence relation:
Proof. See appendix B.     □
Corollary 10. With the same assumptions than in the theorem 9, for all α ≥ 1 and β ≥ (i+1)α we have
Proof. Equation (37) and the lemma 8 gives
and the result is then proved by a simple recurrence.     □
Proposition 11. For any x = (x(a-1),...,x0)' and all β ≥ 0,  is given by  and
Proof. Using equation (28) we get
which gives the proposition.     □
From this result (very similar to proposition 7) it is possible to get a new theorem
Theorem 12. If P is primitive and admits a diagonal form we denote by λ > ν the largest two eigenvalues magnitude of P by P∞ = limi→+∞ Pi/λi (a positive matrix) and we get for all α ≥ 1 and i ≥ 0
uniformly in β and where  is a constant term defined by
and for all i ≥ 0 by the following recurrence relation
and  is a polynomial of degree i which is defined by
and for all i ≥ 1 by the following recurrence relation:
Proof. Easy to derive from the proof of theorem 9.     □
Corollary 13. We have the same assumptions than in the the theorem 12, for all α ≥ 1 and β ≥ (i + 1)α we have
Proof. Easy to derive from the proof of corollary 10.     □