Normal distribution & Central limit theorem

L¨ohr/Winter Winter term 2015/16

Exercises to the lecture Probability Theory II

Exercise sheet 6

Normal distribution & Central limit theorem

Exercise 6.1. (4 Points)

Let X be standard normally distrubuted, and Xn Poisson distributed with parametern.

Xn−n

√n

n→∞

=⇒ X.

Hint: Use the central limit theorem, or alternatively Exercise 1.1

Exercise 6.2(Stability characterisation of the normal distribution). (4 Points) Let X, Y be independent, identically distributed, square integrable, R-valued random vari- ables with

L

X+Y

√2

= L(X).

(a) Show that X is centered, i.e. E(X) = 0.

(b) Show that X is normally distributed (with arbitrary variance).

Hint: Use the central limit theorem.

Exercise 6.3. (4 Points)

Let (Xⁿ)ⁿ∈Nbe a sequence ofR-valued, independent, identically distributed random variables withE(X₁) = 0 and Var(X₁) = 1. Show forSⁿ:=Pⁿ

k=1Xk: lim sup

n→∞

Sⁿ

√n = ∞ a.s.

Hint: Use the CLT and Kolmogorow’s 0-1-law.

Exercise 6.4 (CLT for non-identically distributed random variables). (4 Points) Let (Xn)n∈N be a sequence of independent random variables with

P {Xⁿ= 1}

= P {Xⁿ=−1}

= ₂¹_n, P {Xⁿ= 0}

= 1−n¹. Find a sequence (aⁿ)ⁿ∈N of real numbers such that

Sn^∗ := X₁+· · ·+Xn

aⁿ

converges in distribution to a standard normally distributed random variable.

Please turn

Due Wed, 02.12. at the beginning of the exercise session

Probability Seminar:

24.11.: Stefan H¨afner (University of Duisburg-Essen)

Higher oder variance reduction for discretised diffusions via regression 01.12.: Alexey Muravlev (Steklov Mathematical Institute, Moscow) .

Tue, 16:15 – 17:15in WSC-S-U-3.03