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**Extra info for An introduction to stochastic modeling**

**Example text**

J > 0, |z|k+1 , k (k + 1) |z| ≤ 1 − . 7) or log(1 − z) = − k j=1 zj + O (|z|k+1 ), j where we write O to indicate that the constant in the error term depends on . 1 Show that there are exactly 2d − 1 additive subgroups of Zd of index 2. Describe them and show that they can all arise from some p ∈ P. 2 Show that if p ∈ Pd , n is a positive integer, and x ∈ Zd , then p2n (0) ≥ p2n (x). 3 Show that if p ∈ Pd∗ , then there exists a ﬁnite set {x1 , . . , xk } such that: • p(xj ) > 0, j = 1, .

Therefore, for |θ| ≤ n1/8 , (n − 1) √ √ λ(θ/ n) φ(θ/ n) √ √ 2+ φ(θ/ n) φ(θ/ n) = | θ|2 − tr( ) + q˜ 2+α (θ ) + qˆ α (θ ) |θ |α + |θ |α+2 + o nα/2 nα/2 , which establishes that for α = 1, 2 |Fˆ n (θ )| = O 1 + |θ |2+α nα/2 , |θ| ≤ n1/16 , and for α = 0, for each r < ∞, lim sup |Fˆ n (θ )| = 0. 9. 34) while for α = 0, we choose r independent of n and then let r → ∞. 1 Exponential moments The estimation of probabilities for atypical values can be done more accurately for random walks whose increment distribution has an exponential moment.

Part (d) follows from (b), (c), and Taylor’s theorem with remainder. Part (e) is immediate from the product rule for expectations of independent random variables. 2 Characteristic functions and LCLT 27 We will write Pm (θ ) for the mth order Taylor series approximation of φ about the origin. Then the last proposition implies that if E[|X |m ] < ∞, then φ(θ ) = Pm (θ ) + o(|θ|m ), θ → 0. 10) Note that if E[X ] = 0 and E[|X |2 ] < ∞, then 1 P2 (θ ) = 1 − 2 Here, d d E[X j X k ] θ j θ k = 1 − j=1 k=1 E[(X · θ)2 ] θ· θ =1− .