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52 2. Modèles phénoménologiques de cours
D. Sornette, Y. Malevergne / Physica A 299 (2001) 40–59 49
The condition (21) imposes some stringent constraints on admissible matrices At.
Indeed, while At are not diagonal in general, their average must be diagonal. This
implies that the o -diagonal terms of the matrices At must take negative values, sufciently
often so that their averages vanish. The o -diagonal coe cients quantify the
in uence of other bubbles on a given one. The condition (21) thus means that the
average e ect of other bubbles on any given one must vanish. It is straightforward to
check that, in this linear framework, this implies an absence of correlation (but not an
absence of dependence) between the di erent bubble components E[X (k) X (‘) ]=0 for
any k = ‘.
In contrast, the diagonal elements of At must be positive in majority in order for
EP[Aii]= (i)−1,
for all i’s, to hold true. In fact, on economic grounds, we can exclude
the cases where the diagonal elements take negative values. Indeed, a negative value
of Aii at a given time t would imply that X (i)
t abruptly change sign between t − 1 and
t, what does not seem to be a reasonable nancial process.
3.4. Renewal theory for products of random matrices
In the following, we will consider that the random d × d matrices At are invertible
matrices with real entries. We use Theorem 2:7 of Davis et al. [38], which synthesized
Kesten’s Theorems 3 and 4 in Ref. [25], to the case of real valued matrices. The proof
of this theorem is given in Ref. [39]. We stress that the conditions listed below do
not require the matrices (An) to be non-negative. Actually, we have seen that, in order
for the rational expectation condition not to lead to trivial results, the o -diagonal
coe cients of (An) have to be negative with su ciently large probability such that
their means vanish.
Theorem 1. Let (An) be an i.i.d. sequence of matrices in GLd(R) satisfying the following
set of conditions that we state in a heuristic manner (see Ref. [32] for technical
details). Provided that the following conditions hold;
H1: stationarity condition;
H2: ergodicity;
H3: intermittent ampli cation of the random matrices;
H4: the fattailness of the distribution is not controlled by that of the additive
part Bt;
then;
• there exists a unique solution ∈ (0; 0] to the equation
1
lim
n→∞ n ln EP[A1 ···An ]=0; (22)
• If (Xn) is the stationary solution to the stochastic recurrence equation in (18)
then X is regularly varying with index . In other words; the tail of the marginal