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Just a layman and slow learner. Thank you for your enlightenment and patience, and for giving me the best memory in my life.


May
7
asked Subjects studied in number theory
May
7
comment What kind of topological vector space is $\mathbb{C}^n$?
@Theo: Thanks! I am "flattered" for being recognized as "an avid reader of Wikipedia".
May
7
comment What kind of topological vector space is $\mathbb{C}^n$?
@Asaf: I know that. I just wonder if $\mathbb{C}^n$ is studied as a Banach space only, or further as a Hilbert space, in complex analysis. How are its norm and inner product defined?
May
7
asked What kind of topological vector space is $\mathbb{C}^n$?
May
7
comment Elementary and measure definitions of conditional expectation and probability
@Didier: (4) I know P(A|X=x) and E(Y|X=x) do not exist in general, but I consider cases when they exist. However ill-posed my questions are, my intention is to find connection between, when exist, conditional probabilities/expectations defined in non-measure theoretical courses/books and those in measure-theoretical courses/books.
May
7
revised Elementary and measure definitions of conditional expectation and probability
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May
7
comment Elementary and measure definitions of conditional expectation and probability
@Didier: Thanks for following up! (1) Yes, I have tried some books. (2) f and h are functions defined on the codomain of r.v. X. So f(X) and h(X) are r.v.s. I made a typo by saying its domain is R. Now I corrected it. (3) Some more clarification: when I said P(A|X=x) (when it exists) is a p.m., I wanted to say it is when x is fixed and A is varying; when I said P(A|X=x) (when it exists) is a r.v., I wanted to say P(A|X=) is a function of x when A is fixed, so P(A|X) is a r.v..
May
7
revised Elementary and measure definitions of conditional expectation and probability
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May
7
comment Questions about independence between random variable and $\sigma$-algebra
Thanks! For 2, I would like to use the following, but not sure if it is correct. A random variable can be approximated pointwise by a sequence of simple functions, and each simple functions can be represented as a sum of finitely many step functions. For each random variable, is the subset corresponding to each such step functions in the sigma algebra generated by the random variable?
May
7
accepted Questions about independence between random variable and $\sigma$-algebra
May
6
accepted Different versions of Riesz Theorems
May
6
comment Continuous a.s. process
@Shai: Thanks! Why is distribution of increment being normal not required in the definition in your last edit? How is it implied from that definition? Any reference about that definition?
May
6
comment How to solve this PDE?
Thanks! I was wondering in "since you can choose c freely, you can combine these into arbitrary functions of $(x−1)e^{-ay}$", why does $(.)^{c/a}$, when $c$ is arbitrary, become an arbitary function?
May
6
comment How to solve this PDE?
Really nice! Thanks! I was wondering what types of PDE can be solved by method of characteristics, by separation of variables and by other methods respectively? In other words, what are some guidelines for one to know what method to solve his PDE?
May
6
accepted How to solve this PDE?
May
6
comment How to solve this PDE?
Thanks! Do you also know what page(s) on that book are relevant to solving my PDE?
May
6
revised How to solve this PDE?
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May
6
accepted Continuous a.s. process
May
6
comment Continuous a.s. process
(4) Does "2.No" mean you agree with that the three quoted definitions not having continuity a.s. are correct?
May
6
comment Continuous a.s. process
@Shai: Thanks! I was wondering: (1) what parts in the three definitions respectively imply continuity a.s.? (2) how "any constant function corresponds to a Gaussian process" can be a counterexample, isn't constant function continuous a.s.? (3) in Wikipedia en.wikipedia.org/wiki/…, the definition of Wiener process explicitly requires continuity a.s. besides those said in the first definition quoted in my post. So is it redundant in Wikipedia's definition?