Math 563 - What was covered in class

After each class I will add a very brief description of what we covered.

Mon, Aug 23 Durrett 1.1: sigma-fields, measures, properties of probability measures

Wed, Aug 25 Durrett 1.1 (cont): discrete probability measures, sigma-field generated by a collection of sets, Borel sets, Lebesgue measure on real line
Durrett 1.2 : def of random variable (RV), distribution measure of RV,

Fri, Aug 27 Durrett 1.2 (cont) : distribution function of RV and its properties, def of identically distributed, classification of probability measures on the real line - discrete, absolutely continuous, and singular continuous

Mon, Aug 30 Durrett 1.3: def of measurable function, composition of measurable functions is measurable, continuous functions are measurable. The sigma-field generated by a RV. Extended reals, measurability and inf's, sup's, liminf's and limsup's.
Durrett 1.4: Def of integral for simple functions.

Wed, Sept 1 Durrett 1.4 (cont): Def of integral for general function. Meaning of a.e.
Durrett 1.5: Jensen inequality, L^p norm, Holder inequality. Convergence theorems: Fatou's lemma, dominated convergence thm, monotone convergence theorem.

Fri, Sept 3 Durrett 1.5 (cont) : approximation of integrable function by simple functions. Chebyshev's inequality. The change of variables theorem. Def of mean and variance.

Wed, Sept 8 Durrett 1.6: Product measures and Fubini's theorem. Overview of independence of events and conditional probability. Durrett 2.1 : def of independence for events, RV's and sigma-fields. Sufficient conditions for independence. The pi-lambda theorem.

Fri, Sept 10 Durrett 2.1 (cont): More on sufficient conditions for independence.

Mon, Sept 13 Durrett 2.1 (cont): Functions of independent collections of RV''s are independent. Distribution measure for a random vector. Independence of RV's is equivalent to factorization of their joint distribution. Independence and expectation. Variance of sum of independent RV's is sum of their variances.

Wed, Sept 15 Durrett 2.1 (cont): Distribution of sum or two independent RV's is a convolution. Kolmogorov extension theorem and existence of i.i.d. sequences.
Durrett 2.2 : convergence in probability.

Fri, Sept 17 Durrett 2.2 : a.s. convergence implies convergence in probability. L^2 weak LLN. Weak LLN for triangular arrays.

Mon, Sept 20 Durrett 2.2: Weak LLN for iid sequence

Wed, Sept 22 Durrett 2.3: review of "easy" Borel-Cantelli lemma. L^4 strong LLN. Convergence in probability iff every subsequence has a sub-sequence that converges a.s.

Fri, Sept 24 Borel Cantelli Lemma ("hard" direction). E[|X_n|] =infinity implies S_n/n does not converges to a finite limit.

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