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Aug
10
awarded  Nice Question
May
19
comment Simulating a SDE
$P=S?~~~~~~~~~$
May
18
revised Supporting hyperplane of a convex set
added 161 characters in body
May
18
revised Supporting hyperplane of a convex set
added 161 characters in body
May
18
answered Supporting hyperplane of a convex set
Apr
13
reviewed Approve Transitive Relations
Mar
27
comment To show a given function is not the viscosity solution.
@smiley06 A function $\phi$ with behaviour near 1 as per my comment and fast decay elsewhere has the properties you need.
Mar
25
comment Two-sided hitting time of Brownian motion
@Math-fun That's the third event in my comment.
Mar
25
comment Two-sided hitting time of Brownian motion
@Math-fun An event is a set of outcomes to which a probability can be assigned. In particular, $\{|W(t)|>a\}$, $\{T_a\le t\}$ and $\{|W(t)|>a\text{ and } T_a\le t\}$ are events, but $\{|W(t)|>a| T_a\le t\}$ is not, since it doesn't refer to a particular set of outcomes.
Mar
25
awarded  Popular Question
Mar
24
comment Two-sided hitting time of Brownian motion
Interesting. $\{|W(t)|>a|T_a<t\}$ is not an event.
Mar
23
comment Two-sided hitting time of Brownian motion
This is equivalent to solving the heat equation on the space interval $[-a,a]$ with boundary conditions $P(x, 0)=1$ and $P(-a, t) = P(a, t) = 0$. I believe this gives a Fourier series solution with no nice closed form. Notes on the heat equation.
Mar
23
comment If $X_n = Y_n + Z_n$ in distribution, and $X_n$ and $Y_n$ converge in distribution, does $Z_n$?
@Frank Using $\mathbf{1}$ as, perhaps non-standard, shorthand for the indicator function, I mean that $Z_n$ equals $2U$ when $n$ is odd and zero when $n$ is even.
Mar
23
comment If $X_n = Y_n + Z_n$ in distribution, and $X_n$ and $Y_n$ converge in distribution, does $Z_n$?
@Frank In your comment, you say "Taking characteristic functions, we see $f_z=f_X\bar{f_Y}$". You can't do this without an assumption of independence! [You end up with a conclusion which holds for independent random variables.]
Mar
23
comment If $X_n = Y_n + Z_n$ in distribution, and $X_n$ and $Y_n$ converge in distribution, does $Z_n$?
@thomas You state: "An important assumption... is that all $X_n$ and $Y_n$ are defined on the same probability space." You mean $Y_n$ and $Z_n$, surely.
Mar
23
comment If $X_n = Y_n + Z_n$ in distribution, and $X_n$ and $Y_n$ converge in distribution, does $Z_n$?
@Frank Buried in thomas's answer is a counterexample to your claim: let $U\sim N(0,1)$, $X_n=U$, $Y_n=(-1)^nU$ and $Z_n=2U\mathbf{1}(\text{n odd})$. You just need to verify that in this case the hypothesis for your claim holds, but the conclusion does not.
Mar
23
awarded  Custodian
Mar
23
reviewed Edit Is the geometric series of a set of $n$ RVs a martingale?
Mar
23
revised Is the geometric series of a set of $n$ RVs a martingale?
wrong index
Mar
23
comment Is the geometric series of a set of $n$ RVs a martingale?
Please show what you have tried.