By H. Vincent Poor

The matter of detecting abrupt alterations within the habit of an saw sign or time sequence arises in a number of fields, together with weather modeling, finance, snapshot research, and safeguard. fastest detection refers to real-time detection of such adjustments as quick as attainable once they happen. utilizing the framework of optimum preventing thought, this booklet describes the basics underpinning the sector, delivering the historical past essential to layout, examine, and comprehend fastest detection algorithms. For the 1st time the authors compile effects that have been formerly scattered throughout disparate disciplines, and supply a unified therapy of numerous varied ways to the fastest detection challenge. This e-book is key studying for a person who desires to comprehend the elemental statistical systems for switch detection from a basic point of view, and for these attracted to theoretical questions of switch detection. it truly is excellent for graduate scholars and researchers of engineering, information, economics, and finance.

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**Example text**

2 Stopping times Consider a probability space ( , F, P) and a filtration {Fk ; k = 0, 1, . }. A stopping time (or Markov time)8 is an extended random variable T taking values in the set {0, 1, 2, . } ∪ {∞}, with the property that {ω ∈ |T (ω) ≤ k} ∈ Fk , ∀ k ≥ 0. 74) or that {1{T =k} } be an adapted sequence. Thus, a stopping time associated with a filtration is an extended random variable taking values in the time set of the filtration, with the property that it can assume the value k only on events that are measurable with respect to the filtration at k.

4: (the moment-generating function of cumulative sums). d. random variables, each of which has moment-generating function: M(t) = E etYk , t ∈ IR. 66) Let {Fk } be the minimal filtration generated by {Yk }. Consider t ∈ IR such that M(t) < ∞, and define a random sequence {X k } by Xk = where Sk = k =0 Y et Sk , k = 0, 1, . 67) , k = 0, 1, . .. 2 shows that {X k , Fk } is a martingale. Since X k ≥ 0 for each k it follows that {X k } is almost surely convergent. s. likelihood ratio is given by L(y) = et y /M(t).

61) Ak+1 − Ak = E{X k+1 |Fk } − X k , k = 0, 1, . , A0 = 0. 62) and We now consider some examples of martingales and semimartingales. 2: (the likelihood ratio under the null hypothesis). Consider a sequence {Yk ; k = 0, 1, . d. random variables, each of which has pdf p, and let {Fk } be the minimal filtration associated with {Yk }. s. Define a random sequence {X k ; k = 0, 1, . } by k Xk = L(Y ), k = 0, 1, . . 63) =0 Then, since X k = X k−1 L(Yk ), we have E{X k |Fk−1 } = E{X k−1 L(Yk )|Fk−1 } = X k−1 E{L(Yk )|Fk−1 } = X k−1 E{L(Yk )}, where the second equality follows because X k−1 is Fk−1 -measurable, and the third equality follows because L(Yk ) is independent of Fk−1 .