By H. Jerome Keisler

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**Extra resources for An Infinitesimal Approach to Stochastic Analysis**

**Example text**

T. A) of the equation F (A N ) = ǫ. 16) P0 { ηN (t, δ) > A} ∼ F (A N ). Now we consider the asymptotical estimates. 2. 5−δ . Proof. Let 0 ≤ δ ≤ 1/2. 18) |W 0 (t)| C√ N ∆(a, b)}(1 + o(1)). 18) α(N ) ≤ P{ max 1≤z≤d(1−c)/c(1−d) a 1−a z and taking into account d 1 t ) = 1+t W (t), we obtain W 0 ( t+1 u→ |W (z)| C√ √ N ∆}(1 + o(1)). 20) √ 1 = √ (1 − x−2 ) ln T + 2x−2 + o(x−4 ) x exp(−x2 /2), π which is true as x → ∞. 19), we obtain lim inf N | ln α(N )| ≥ N C σ 2 ∆2 . 5−δ (a(1 − a)) lim sup N . | ln α(N )| C2 1−2δ .

Is computed using the same relationships as the precious item. As a result of the second stage, we get the list of conﬁrmed change-points (LCCP). 38 Change-Point Analysis in Nonstationary Stochastic Models Third stage of the procedure: refining the change-points and calculation of confidence intervals • Take point n1 from the LCCP and consider the subsample around this point. • Find max |YN (n, 0)| = |YN (˜ n1 , 0)| = B1 (the maximum over the subsample). The point n ˜ 1 is called the reﬁned change-point.

18) is trivially fulﬁlled. Suppose θˆN is a certain consistent estimate of the parameter θ constructed by the sample X N = {x1 , . . , xN }. v. λN = λN (x1 , . . , xN ) = I{ θˆN − θ > ǫ}. Let us ﬁx a small enough ǫ > 0. Suppose d > 0 and z ∈ Θ is a ﬁxed vector, such that z = ǫ˜ > ǫ. Then, Pθ { θˆN − θ > ǫ} = Eθ λN ≥ Eθ λN I(f (X N , θ + z)/f (X N , θ) < ed ) ≥ ≥ e−d Eθ+z {λN I(f (X N , θ + z)/f (X N , θ) < ed } ≥ ≥ e−d Pθ+z { θˆN − θ > ǫ} − Pθ+z {f (X N , θ + z)/f (X N , θ) ≥ ed } . 19) Here we used the elementary inequality P(AB) ≥ P(A) − P(Ω\B).