Residues and fractional parts via characteristic functions

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Problem 1.12.9 from Жуковский М.Е., Родионов И.В., Шабанов Д.А. "Основы теории вероятностей" [ЖРШ15]More problems in Chapter 1.12 "Characteristic Functions"

A random variable ξ\xi takes only integer values, and φ(t)\varphi (t) is the characteristic function of ξ\xi. Prove that for any natural kk we have

P(ξ≡0( mod k))=1k∑j=0k−1φ(2πjk) \mathbb {P}(\xi \equiv 0(\bmod k))=\frac{1}{k} \sum _{j=0}^{k-1} \varphi \left(\frac{2 \pi j}{k}\right)
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Problem 4.45 from Прохоров А.В., Ушаков В.Г., Ушаков Н. "Задачи по теории вероятностей" [ПУУ86]More problems in Chapter 4 "Analytic Methods of Probability Theory"

Let ξ1,ξ2,…\xi_{1}, \xi_{2}, \ldots be independent identically distributed integer-valued random variables, and let ηn=ξ1+…+ξn\eta_{n} = \xi_{1}+\ldots +\xi_{n}. Prove that if 0<P(ξ1 is divisible by two)<10 < \mathbb {P}\left(\xi_{1} \text{ is divisible by two}\right) < 1, then

lim⁡n→∞P(ηn is divisible by two)=1/2. \lim _{n \rightarrow \infty } \mathbb {P}\left(\eta _{n} \text{ is divisible by two}\right) = 1 / 2.
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Problem 6.132 from Прохоров А.В., Ушаков В.Г., Ушаков Н. "Задачи по теории вероятностей" [ПУУ86]More problems in Chapter 6 "Limit theorems of probability theory"

Let the random variables ξn,n=1,2,…\xi_{n}, n = 1,2, \ldots, have distributions concentrated on the interval [0,1)[0,1). Prove that if for every integer k≠0k \neq 0

E[e2πikξn]→0 \mathbb {E}\left[e^{2 \pi i k \xi _{n}}\right] \rightarrow 0

as n→∞n \rightarrow \infty, then for any α\alpha and β,0⩽α<β<1\beta , 0 \leqslant \alpha < \beta < 1,

P(α⩽ξn<β)→β−α. \mathbb {P}\left(\alpha \leqslant \xi _{n} < \beta \right) \rightarrow \beta -\alpha .
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Problem 6.133 from Прохоров А.В., Ушаков В.Г., Ушаков Н. "Задачи по теории вероятностей" [ПУУ86]More problems in Chapter 6 "Limit theorems of probability theory"

Let the random variables ξ1,ξ2,…,ξn\xi_{1}, \xi_{2}, \ldots , \xi_{n} be mutually independent and identically distributed, where P(0⩽ξn<1)=1\mathbb {P}\left(0 \leqslant \xi_{n} < 1\right) = 1. Let ηn={ξ1+…+ξn}\eta_{n} = \left\{ \xi_{1}+\ldots +\xi_{n}\right\} be the fractional part of the sum ξ1+…+ξn\xi_{1}+\ldots +\xi_{n}. If for k≠0k \neq 0

E[e2πikηn]→0,n→∞, \mathbb {E}\left[e^{2 \pi i k \eta _{n}}\right] \rightarrow 0, \quad n \rightarrow \infty ,

then for any α\alpha and β,0⩽α<β<1\beta , 0 \leqslant \alpha < \beta < 1,

P(α⩽ηn<β)→β−α. \mathbb {P}\left(\alpha \leqslant \eta _{n} < \beta \right) \rightarrow \beta -\alpha .

(This assertion serves as an analogue of the central limit theorem when random variables are added modulo 1.)

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References

[ЖРШ15] Жуковский М.Е., Родионов И.В., Шабанов Д.А. Основы теории вероятностей. МФТИ, 2015.
[ПУУ86] Прохоров А.В., Ушаков В.Г., Ушаков Н. Задачи по теории вероятностей. Наука. Главная редакция физико-математической литературы, 1986.