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Math BSc 5th Sem > Probabiliy Theory > Flashcards

Probabiliy Theory Flashcards

1
Q

Define weak convergence and convergence in distr. 31.01.19 (“Saying measure isnt enough here,…”) 21.01.20

A

Definition 2.1 pg 48-49 (“Saying measure isnt enough here, you really need to say probability measures on B(R)”) 21.01.19

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2
Q

What are equivalent properties to weak convergence? (Exam 21.01.20+22.1.2019)

A

Prop 2.7 pg 54 Show “2)=>3)” (Me: maybe mention mu is image of P through Y, that we use Prop 1.13), “3)=>1)”

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3
Q

State the upcrossing inequality. Show the upcrossings in a graph (“writing N1,N2,..) Prove it. (Exam 21.01.20, 22.01.19)

A

Proposition 3.25 pg 98 (“Draw sth like Fig 3.3. on blackboard)

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4
Q

Define (sub/super-)martingale (Exam 21.01.20, 28.01.19)

A

Definition 3.10 pg 88

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5
Q

Define conditional expectation (Exam 21.01.20, 28.01.19, 31.01.19) What is E[X|F] if F is A or trivial? (Me)

A

Theorem 3.2 pg 83

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6
Q

Show E[X|F]=Sum^N_i ( E[X|Ai]*1_Ai ) if (Ai) s.s.o. A (sig-Alg) is a (disj) partition of Omega. (Exam 21.01.20: Example 3.3. for N=2. (I.e. conditional expectation for partition into two parts.))

A

Example 3.3.1) pg 84

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7
Q

Show E[X|F]=E[X] for F={ {}, Omega } (Me)

A

Example 3.3.2) pg 85

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8
Q

What is E[X|F] if X is F-measurable? Prove it.

(Exam 31.01.19)

A

Theorem 3.5 (3++ for prove of theorem 2 only special case)

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9
Q

Define independence of X and F (sub sig-alg). What follows? (Me)

A

and conclude E[X|F]=E[X] P-a.s. follows Example 3.3.3) pg 85;

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10
Q

Show E[E[X|F]]=E[X] (Me)

A

pg 85

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11
Q

State dominated/monotone convergence theorem and Fatou’s lemma

A

pg 17, Dominated convergence: theorem 1.11, Fatou: lemma 1.9

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12
Q

State Jensen’s inequality (Me) (For 4+ prove)

A

pg. 85

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13
Q

Define characteristic function and state and prove its properties (23.01.19, 05.02.18)

A

Definition 2.13. , Remark 2.14. pg. 60 Check: pg 47 “He wanted to know the definition and “some” properties. So, I gave the basic properties, but Sznitman was not happy with that. So, I also had to give the uniqueness property and the continuity theorem…..” (there’s more) 07.02.18

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14
Q

“What can you tell me about the conditional expectation if X is in L2? Prove it.” (Exam 21.01.20)

A

“Theorem 3.6” pg 87, check if there is more and update this answer!

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15
Q

“Martingale convergence: State the Theorem. Give an example of a Martingale that converges P-a.s. but not in L1.” (show) + “He wanted me to show why Sn converges P-a.s. to infinity.” (Exam 2x21.01.20)

A

Martingale convergence: theorem 3.27, pg 99 “… example of a Martingale that converges P-a.s. but not in L1. I stated example 3.11.3.). He wanted me to show why Sn converges P-a.s. to infinity.” I think he means Example 3.11. 3) on page 89 (asymmetric random walk)

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16
Q

“State Doob’s Decomposition” (Exam 05.02.20, 05.02.18)

A

Proposition 3.19, page 94

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17
Q
  1. “What conditions characterize the conditional expectation?” 2. “What if the sigma algebra is finite?” (Me: I suppose 2. refers to 1.) (Exam 05.02.20)
A
  1. Me: Maybe he means Example 3.3. 1), pg. 84?? 2. If 1. correct, then I guess we can always partition Omega into disjoint sets into finest sets possible.
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18
Q

“Proof Doob’s Decomposition” (Exam 05.02.20, 05.02.18)

A

Proposition 3.19, pg. 94

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19
Q

State the Continuity Theorem. Prove it (23.01.19) (Exam 05.02.20, 23.01.19)

A

Theorem 2.18, pg 66

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20
Q

“Prove the first part of the second statement of the continuity Thm: mu_n is tight” (probably prove all, its short) (Exam 05.02.20)

A

Theorem 2.18, pg 66

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21
Q

“What are applications of the Continuity theorem?” (Exam 05.02.20)

A
  1. symmetric stable distributions (pg. 67)
  2. Central limit theorem (at the end of the proof)
    (3. when over 80% check out exercises, maybe..)
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22
Q

Check if “some example not in notes” converges in distribution (Update this question and answer) (Exam 05.02.20)

A

Check exercises and update here!!

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23
Q

State Helly’s Thm

A

Theorem 2.9 pg 57

24
Q

After def of weak convergence: “example” (31.01.19) “why we want convergence only at point of continuity?” 22.1.19

A

“(Dirac measure on 1/n)” 22.1.19, think about this, research and update

25
Q

Prove the martingale convergence theorem (31.01.19)

A

theorem 3.27, pg 99
“Proof of Martingale convergence theorem using the upcrossing inequality (with exactly saying why E(U_inf)

26
Q

Me: State Hölder’s inequality, Cauchy-Schwarz inequality, Chebyshev

A

page 17

27
Q

State the Central Limit Theorem, outline proof (Apparantly never came)

A

Theorem 2.20, pg 68

28
Q

Define symmetric stable distribution and “How to get the corresponding random variables -> Calculation of characteristic function of compound Poisson distribution.” (Exam 21.01.20)

A

I think page 67??

29
Q

“3-Theories Thm proof+example” 23.01.19

A

theorem 1.37 Kolmogorov’s 3-series thm, pg 37

30
Q 
# Define convergence in probability.
Show relationship between convergence P-a.s., in probability and in L^p.

State and prove the Weak Law of Large Numbers

A

pg 38

31
Q

“State the theorem we use in the 3-series theorem proof and prove it” 23.01.19

A

Thm 1.34, pg 1.34

(Lemma 1.24. (First lemma of Borel Cantelli), pg 26)

32
Q

State and prove the lemmata of Borel Cantelli (in order) (“Idea of pf” 07.02.18)

A

Lemma 1.26., pg 26 update for second

33
Q

“Martingales and Markov chains (last prop in lecture notes)” 28.01.19

A

Proposition 4.34, pg 150

34
Q

Calculate the characteristic function for the compound Poisson distribution 28.01.19

A

update

35
Q

SSLN: state and prove 28.01.19

A

pg. 40

36
Q

Define uniform integrability 28.01.19, 07.02.18 “Give for uniform integrability: firstly, one which uses the conditional expectation and the sub-sig-algebras, second example from the lecture” “He gave me an example and I had to show that it is indeed uniformly integrable” 07.02.18

A

definition 3.38, pg 114

37
Q 
# Define tightness. (Exam 05.02.20, 31.01.19)
(Me: + prove the prop)
A

Proposition 2.11 pg 59

38
Q

Does P-as convergence imply L1 convergence?

A

Say no and state which additional conditions are required. Then state and prove the whole proposition: Proposition 3.41 pg 115

39
Q

State how conditional expectation can be viewed as an orthogonal projection (in the sense of functional analysis) 28.01.19 Prove it.

A

Theorem 3.6, pg 87 “He wanted to know in detail why E[X(Z-Z’)]=E[Z(Z-Z’)]” 08.02.18

40
Q

Define markov chain

A

pg 140

41
Q

Doob’s inequality with proof (05.02.18)

A

3.6 Doob’s inequality, convergence in Lp, pg 110 “Attention: he is really picky about all the indices” 07.02.18 Explain “properly” why (H*X)_n=X_n-X_T^n, n>=0

42
Q

Kolmogorov’s inequality & (2nd) proof

A

pg 110 (pg 33)

43
Q

State and prove Kolmogorov’s 0-1-law 07.02.18

A

Theorem 1.30, pg 32

44
Q

State Lebesgue-Stieltjes

A

distribution and distribition function of r.v. pg. 12 Lebesgue-Stieltjes pg 13

45
Q

Give an example with no weakly convergent subsequence (31.01.19, 31.01.19)

A

update

46
Q

State and prove the tower properties (ME)

A

Proposition 3.8, pg. 88

47
Q

State Dynkin’s lemma

A

Lemma 1.18, pg 21

48
Q

“Give two applications of the continuity theorem we had in the lecture” 05.02.18

A

update “As i Then mentioned the symmetric stable distribution with its characteristic function he wanted to know how we derived this characteristic function” 05.02.18

49
Q

“Convergence of stochastic series” 07.02.18

A

update

50
Q

“Theorem about L1-convergence of martingales (just stating the theorem)” 07.02.18

A

update

51
Q

Martingale convergence theorems

A

update

52
Q

State only characterisations of martingale convergence in L^p (Me)

A

Theorem 3.35, Theorem 3.42

53
Q

Convergence of Xn:=E[X|Fn] (Me)

A

Proposition 3.36, pg 112, Proposition 3.43 pg 117 (difference and prove easiest ones)

54
Q

Def of stochastic kernel and construction of canonical Markov chains

A

def 140, pg 148

55
Q

State and prove optional stopping theorem

A

Corollary 3.24

Math BSc 5th Sem (2 decks)

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