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Jul
16
comment Dominated convergence theorem for complex-valued functions?
@ConradoCosta Looks like you missed the negative sign: $\frac{it}{2a_L}\neq -\frac{it}{2a_L}$, so your $A$ doesn't get multiplied by your $C$. It gets multiplied by $\frac{it}{2a_L}-\frac{t^2}{8a_L^2}+O(a_L^{-3})$, which does not equal your $C$. Does that make things clear(er)?
Jul
16
comment Dominated convergence theorem for complex-valued functions?
@ConradoCosta You see, I am having trouble locating that term that you wrote down -- it's not in my calculations anywhere. Perhaps you are confusing $a_-^2$ with $a_S^2$ and $a_+^2$ with $a_L^2$? In that case, note the negative sign in front of $\frac{it}{2a_L}$ in the term on line 2 corresponding to $a_-^2$... You can see that most of the terms in line 3 either combine or cancel out when you do the arithmetic...
Jul
16
comment Dominated convergence theorem for complex-valued functions?
@ConradoCosta I am not sure where in my derivation you found that expression. In the first block of formulas of "specific example" the second equality is from Taylor series expansion of $e^x$; the third equality just plugs in the definitions of $a_-^2$ and $a_+^2$, and expands the squares; the fourth equality is just careful arithmetic -- a lot of terms cancel out; and final equality just evaluates the limit.
Jul
16
comment Dominated convergence theorem for complex-valued functions?
"Example" refers to the "specific example" I gave in my question (sorry that it's long). Your answer allows me to interchange the limit and the integral at the very end of the "specific example." This shows that the probability density function of the random variable $A$ (which you recover from the characteristic function using Fourier transform) approaches Gaussian in the limit (after dividing by $2\pi$, which I omitted for brevity). Convergence in probability described earlier in "specific example" is central-limit-theorem-like, while convergence of densities is "local". Does this explain?
Jul
16
revised Dominated convergence theorem for complex-valued functions?
Fixed a minor typo in the last line by moving the symbol indicating the real part inside the limit.
Jul
16
suggested approved edit on Dominated convergence theorem for complex-valued functions?
Jul
16
accepted Dominated convergence theorem for complex-valued functions?
Jul
16
comment Dominated convergence theorem for complex-valued functions?
Thanks for writing out the answer. I just want to make two notes: 1) it mirrors @DavidC.Ullrich's comment above, and; 2) the "local limit theorem"-like result in the example seems to hold using this flavor of DCT (one could use the same dominating function $g(t)=\exp[-t^2]$ for both real and imaginary parts since sine and cosine are bounded in $[-1,1]$).
Jul
16
comment Dominated convergence theorem for complex-valued functions?
The main problem is that I don't know measure theory all that well (though I never took a formal course in it, I know probability theory pretty well at the graduate engineering student level). In this context, I am not sure how function $g$ would look like. Is $|f_n(x)|$ simply the magnitude of a complex-valued function $f_n(x)$ and $g(x)$ has to be greater than that magnitude for all $x$?
Jul
16
asked Dominated convergence theorem for complex-valued functions?
Jul
1
comment The normal approximation of Poisson distribution
@alexjo There is a tiny typo in the second-to-last line: $\frac{t^3\alpha^{-3/2}}{6}$ should be $\frac{t^3\alpha^{-1/2}}{6}$.
Jun
3
comment DTFT of a triangle function in closed form
This is a very nice answer -- I love the use of the fact that triangle function is a convolution of two rectangular functions. I feel it makes this a very intuitive solution.
May
23
awarded  Popular Question
May
15
asked Hexagon packing in a circle
Apr
24
accepted Distribution of an angle between a random and fixed unit-length $n$-vectors
Apr
24
comment Distribution of an angle between a random and fixed unit-length $n$-vectors
Thanks for a great explanation! One note: the coefficient simplifies to $\frac{\Gamma(\frac{n}{2})}{\sqrt{\pi}\Gamma(\frac{n-1}{2})}$ using the property that defines the Gamma function $\Gamma(t+1)=t\Gamma(t)$.
Apr
24
revised Distribution of an angle between a random and fixed unit-length $n$-vectors
addressed comment
Apr
24
comment Distribution of an angle between a random and fixed unit-length $n$-vectors
Thanks for the explanation -- but how does it apply to $n$ sphere? The volume element looks a lot more complicated -- does one just integrate out the other dimensions? And yes, I overlooked the fact that the angle between vectors cannot exceed $\pi$ (will fix the question).
Apr
22
asked Distribution of an angle between a random and fixed unit-length $n$-vectors
Apr
1
awarded  Popular Question