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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
10
accepted Tail $\sigma$-algebra for a sequence of sets
May
10
comment Tail $\sigma$-algebra for a sequence of sets
Thanks! For the tail of a sequence of general $\sigma$-algebras $\{\mathcal{F}_n\}$, (1) can I cay a set $B$ belongs to the tail if and only if $B$ can be built from $\{ \mathcal{F}_n: n \geq N \}$ for any integer $N$? (2) what does "$B$ can be built from $\{ \mathcal{F}_n: n \geq N \}$ " mean? Is it that there exists some mapping $f$, such that $B=f(\{ \mathcal{F}_n: n \geq N \})$ ?
May
10
revised Tail $\sigma$-algebra for a sequence of sets
added 225 characters in body
May
10
revised Tail $\sigma$-algebra for a sequence of sets
added 192 characters in body; added 1 characters in body; deleted 7 characters in body; added 1 characters in body
May
10
revised Tail $\sigma$-algebra for a sequence of sets
edited title
May
10
asked Tail $\sigma$-algebra for a sequence of sets
May
10
comment How to solve forward equation for a continuous-time Markov chain?
@Emre: Thanks! Is eigendecomposition one usual way to solve it? What if the process is given as a linear pure birth problem, i.e. Yule process, with rates $k\lambda, k=1,2,..n$ where values of $n$ and $\lambda$ are unknown? Can the rate matrix with unknown constants still be eigendecomposed?
May
10
revised How to solve forward equation for a continuous-time Markov chain?
added 6 characters in body
May
10
asked How to solve forward equation for a continuous-time Markov chain?
May
9
comment Categories of mathematics
@Arturo: How about set theory, category theory, logic and measure theory? Are they at first level?
May
9
comment Are holding times independent in a continuous-time Markov chain and in a semi-Markov process
Thanks! For any two holding times, (1) if given their beginning states, will they be conditionally independent? (2) if given their beginning and end states, will they be conditionally independent? If yes to both (1) and (2), is it because of Markov property and how to explain that based on that?
May
9
comment Independence of holding time and next state in continuous-time Markov chain
Thanks! (1) I was wondering why given the current state, it corresponds to independent U and V? (2) When U and V are independent exponential r.v.s, why min(U,V) is independent of the event U<V? (3) In the example, at state a, the intervals in the two possibilities are 5 and 10 seconds, while in $P(X_{20} = b | X_{17} = a) $ and $P(X_{20} = b | X_{17} = a)$, the intervals are 3. How do you know the $P(X_{20} = b | X_{17} = a) > 0$ and $P(X_{20} = b | X_{17} = a) > 0$?
May
8
awarded  Nice Question
May
8
comment Indeterminate equation and functional equation
@Qiaochu: In that case, the set of function solutions to the functional equation can still be derived from those to the indeterminate equation.
May
8
comment Questions about algebraic identities
Thanks! Can you say more about differences between formal and functional polynomials? Why in the example they are not equivalent?
May
8
comment Indeterminate equation and functional equation
@Qiaochu: Can I say an indeterminate equation always has a functional equation corresponding to it in the way above, although not vice versa?
May
8
comment Indeterminate equation and functional equation
@Qiaochu: From the beginning, I have already understood the differences in what a solution is to each type of equation. I have been asking about connection between them, like what I did in my last two comments, finding the corresponding equation of the other type for an equation of one type, so that their solutions can be derived from each other in that obvious way.
May
8
comment Indeterminate equation and functional equation
@Qiaochu: The solutions of $x^2+y^2=1$ as an indeterminate equation and of $x^2+f(x)^2=1$ as a functional equation, are similar, although the former is represented as a set of pairs and the latter as a set of functions. For functional equation $f(x)=f(x+1)$, I can't find a corresponding indeterminate equation yet.
May
8
comment Indeterminate equation and functional equation
@Amy: The solutions of indeterminate equation $y=x^2$ and of functional equation $f(x)=x^2$ are essentially the same, except that the former is represented as a set of pairs, and the latter as a function.
May
8
comment Indeterminate equation and functional equation
@Amy: Did you forget the link? I don't think the example is indeterminate, because there are not infinitely many solutions.