Probabiliy Theory Flashcards
Define weak convergence and convergence in distr. 31.01.19 (“Saying measure isnt enough here,…”) 21.01.20
Definition 2.1 pg 48-49 (“Saying measure isnt enough here, you really need to say probability measures on B(R)”) 21.01.19
What are equivalent properties to weak convergence? (Exam 21.01.20+22.1.2019)
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)”
State the upcrossing inequality. Show the upcrossings in a graph (“writing N1,N2,..) Prove it. (Exam 21.01.20, 22.01.19)
Proposition 3.25 pg 98 (“Draw sth like Fig 3.3. on blackboard)
Define (sub/super-)martingale (Exam 21.01.20, 28.01.19)
Definition 3.10 pg 88
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)
Theorem 3.2 pg 83
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.))
Example 3.3.1) pg 84
Show E[X|F]=E[X] for F={ {}, Omega } (Me)
Example 3.3.2) pg 85
What is E[X|F] if X is F-measurable? Prove it.
(Exam 31.01.19)
Theorem 3.5 (3++ for prove of theorem 2 only special case)
Define independence of X and F (sub sig-alg). What follows? (Me)
and conclude E[X|F]=E[X] P-a.s. follows Example 3.3.3) pg 85;
Show E[E[X|F]]=E[X] (Me)
pg 85
State dominated/monotone convergence theorem and Fatou’s lemma
pg 17, Dominated convergence: theorem 1.11, Fatou: lemma 1.9
State Jensen’s inequality (Me) (For 4+ prove)
pg. 85
Define characteristic function and state and prove its properties (23.01.19, 05.02.18)
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
“What can you tell me about the conditional expectation if X is in L2? Prove it.” (Exam 21.01.20)
“Theorem 3.6” pg 87, check if there is more and update this answer!
“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)
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)
“State Doob’s Decomposition” (Exam 05.02.20, 05.02.18)
Proposition 3.19, page 94
- “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)
- 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.
“Proof Doob’s Decomposition” (Exam 05.02.20, 05.02.18)
Proposition 3.19, pg. 94
State the Continuity Theorem. Prove it (23.01.19) (Exam 05.02.20, 23.01.19)
Theorem 2.18, pg 66
“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)
Theorem 2.18, pg 66
“What are applications of the Continuity theorem?” (Exam 05.02.20)
- symmetric stable distributions (pg. 67)
- Central limit theorem (at the end of the proof)
(3. when over 80% check out exercises, maybe..)
Check if “some example not in notes” converges in distribution (Update this question and answer) (Exam 05.02.20)
Check exercises and update here!!
State Helly’s Thm
Theorem 2.9 pg 57
After def of weak convergence: “example” (31.01.19) “why we want convergence only at point of continuity?” 22.1.19
“(Dirac measure on 1/n)” 22.1.19, think about this, research and update
Prove the martingale convergence theorem (31.01.19)
theorem 3.27, pg 99
“Proof of Martingale convergence theorem using the upcrossing inequality (with exactly saying why E(U_inf)
Me: State Hölder’s inequality, Cauchy-Schwarz inequality, Chebyshev
page 17
State the Central Limit Theorem, outline proof (Apparantly never came)
Theorem 2.20, pg 68
Define symmetric stable distribution and “How to get the corresponding random variables -> Calculation of characteristic function of compound Poisson distribution.” (Exam 21.01.20)
I think page 67??
“3-Theories Thm proof+example” 23.01.19
theorem 1.37 Kolmogorov’s 3-series thm, pg 37
# 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
pg 38
“State the theorem we use in the 3-series theorem proof and prove it” 23.01.19
Thm 1.34, pg 1.34
(Lemma 1.24. (First lemma of Borel Cantelli), pg 26)
State and prove the lemmata of Borel Cantelli (in order) (“Idea of pf” 07.02.18)
Lemma 1.26., pg 26 update for second
“Martingales and Markov chains (last prop in lecture notes)” 28.01.19
Proposition 4.34, pg 150
Calculate the characteristic function for the compound Poisson distribution 28.01.19
update
SSLN: state and prove 28.01.19
pg. 40
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
definition 3.38, pg 114
# Define tightness. (Exam 05.02.20, 31.01.19) (Me: + prove the prop)
Proposition 2.11 pg 59
Does P-as convergence imply L1 convergence?
Say no and state which additional conditions are required. Then state and prove the whole proposition: Proposition 3.41 pg 115
State how conditional expectation can be viewed as an orthogonal projection (in the sense of functional analysis) 28.01.19 Prove it.
Theorem 3.6, pg 87 “He wanted to know in detail why E[X(Z-Z’)]=E[Z(Z-Z’)]” 08.02.18
Define markov chain
pg 140
Doob’s inequality with proof (05.02.18)
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
Kolmogorov’s inequality & (2nd) proof
pg 110 (pg 33)
State and prove Kolmogorov’s 0-1-law 07.02.18
Theorem 1.30, pg 32
State Lebesgue-Stieltjes
distribution and distribition function of r.v. pg. 12 Lebesgue-Stieltjes pg 13
Give an example with no weakly convergent subsequence (31.01.19, 31.01.19)
update
State and prove the tower properties (ME)
Proposition 3.8, pg. 88
State Dynkin’s lemma
Lemma 1.18, pg 21
“Give two applications of the continuity theorem we had in the lecture” 05.02.18
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
“Convergence of stochastic series” 07.02.18
update
“Theorem about L1-convergence of martingales (just stating the theorem)” 07.02.18
update
Martingale convergence theorems
update
State only characterisations of martingale convergence in L^p (Me)
Theorem 3.35, Theorem 3.42
Convergence of Xn:=E[X|Fn] (Me)
Proposition 3.36, pg 112, Proposition 3.43 pg 117 (difference and prove easiest ones)
Def of stochastic kernel and construction of canonical Markov chains
def 140, pg 148
State and prove optional stopping theorem
Corollary 3.24
