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The expression above is only approximate (and therefore the symbol ) because it does not take low-frequency phenomena into account; we have in fact omitted the integral on {| | < 1} in (654) This minor inaccuracy is not detrimental to the heuristic reasoning to come, which seeks to explain that the regularity of the GMBM depends in fact on h(t) = lim inf n + hn (t) Still, to eliminate the technical problems, let us suppose that functions of the de nition of W (d ) cancel k j j+1 themselves outside [2 , 2 ] if we have = 2j (this means neglecting constants 2 3 and 8 which appear when describing the dependence of function (l) on (613)) 3 Under these assumptions, formula (655) is simpli ed because integrals are zero k except if the scale index j of = 2j is equal to n 1; we obtain:

Y(h) (t)

(656)

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This clari es the natural correspondence between hj+1 and the scale j In particular, we understand why the regularity of the GMBM depends on the behavior of hn (t) when n becomes large This result is Theorem 2 of [AYA 99], the precise statement of which we now recapitulate THEOREM 66 Let: Y (h) (t) = sup , lim

.

Y (h)(t + ) Y (h)(t)

be the pointwise H lder exponent for all t t ,

of a GMBM Y (h) Then: (as)

Y (h) (t) = h(t)

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is the generalized multifractional function of the GMBM To conclude this section on the GMBM, it should be added that, under additional assumptions bearing on the multifractional sequence of a GMBM, the generalized multifractional function of a GMBM has been identi ed in [AYA 04a, AYA 04b] 65 Estimate of regularity In this section, we will estimate the regularity of the processes by means of quadratic variations It is of course not the only method: see 2 and 9 for alternative approaches 651 General method: generalized quadratic variation First, we will x the general framework of identi cation methods In particular, in this section, we shall identify processes indexed by The hypothesis that the processes are Gaussian authorizes us to proceed to the identi cation starting from a unique trajectory of the process, which we suppose to have been observed in discrete p times: we will assume X( N ) as known for 0 p N The various estimators which p we use are built from the generalized quadratic variation of the observation X( N ) which we can write:

case of RDP messages, intermediate nodes remove the signature and certi cate of the previous intermediate node Eventually node A receives the REP; it veri es the authenticity of the response, and can start communicating with node X The authors in [51] do not discuss whether it is possible for intermediate nodes to respond to RDP messages if they have a fresh route to the destination It seems possible to extend the scheme in a manner similar to the double signature approach of SAODV discussed earlier in this section This could increase the ef ciency of the protocol, but may also increase the security risk and complexity ARAN also supports a route maintenance scheme similar to the approach used in AODV If, for example, node B discovers that the link to node C is broken, it sends an error message (ERR) towards the source of the route The ERR message has the following format: ERR, IPA , IPX , certB , NB , t PrKB Since it is in general dif cult to distinguish malicious ERR message from correct ERR messages, especially in very volatile networks, it may be useful to maintain a count of the number of ERR messages that each node generates If a node generates an abnormally high number of ERR messages (compared with other nodes), it is likely that this node is malicious (since ERR are signed and it can be veri ed that such a node actually generated those messages) Hence such a node must be avoided during routing ARAN also supports key revocation, presumably for nodes that have been veri ed to be behaving maliciously Node T (the trusted certi cate server) is responsible for generating such messages The revocation messages have the following format: revoke, certR PrKT Each node receiving this message broadcasts the message to its neighbors Nodes keep track of the certi cates that have been revoked until those certi cates expire When nodes receive the revocation message they also check to see whether any routes go through the untrusted node If they do discover such routes, they initiate the route discovery process for those routes Thus, we see that the approach in ARAN is to make use of digital signatures 423 Security-Aware Ad Hoc Routing.

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