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1d
comment Independent random variables and integrability
What does $n|\varepsilon$ mean? Is it just supposed to be multiplication $n\varepsilon$? It looks like a printing error.
1d
comment Probability distribution of $\int_0^t \frac{W_s}{s} \,ds$
Yes, I was referring to the interchange of the $du\,dv$ integral with expectation (which after all is another integral). The obvious theorem to cite is Fubini's, but one needs to verify that the hypothesis of Fubini's theorem is satisfied, by showing that $\int_0^t \int_0^t \frac{E[|W_u W_v|]}{uv}\,du\,dv < \infty$ (note the absolute value inside the expectation). That is what I meant by "justify the interchange of integrals".
2d
answered What does $\sim$ in $X\sim \mathcal{N}(\mu,\sigma^{2})$ really mean?
2d
comment How to smoothly approximate a sign function
It would help if you explain why you want something other than the $\tanh$ function. What is it about $\tanh$ that doesn't work for your purposes? Otherwise, people may waste a lot of time coming up with other functions that also won't work for you.
2d
answered If $X$ is standard Normal then find $\lim_{x\to0}P(X>x+\frac{a}{x}|X>x)$
2d
comment What is the distribution of a stochastic process?
Then the distribution of $X$ is the pushforward measure on $C(I,E)$. This has the advantage that it gives you a finer $\sigma$-algebra and hence more measurable sets. For cadlag processes, you might replace $E^I$ with the Skorohod path space.
2d
comment What is the distribution of a stochastic process?
This is certainly one way to do it. However, more commonly one deals with processes that have some regularity, and uses smaller path spaces. For instance, if $X$ is a continuous process (i.e. the map $t \mapsto X_t(\omega)$ is continuous for every, or almost every, $\omega$), then we may instead view $X$ as a mapping from $\Omega$ into the space $C(I,E)$ of continuous functions from $I$ to $E$, equipped with something like the Borel $\sigma$-algebra coming from the topology of uniform convergence on compact sets.
2d
answered Stopped brownian motion
2d
revised If $p$ is a positive multivariate polynomial, does $1/p$ have polynomial growth?
correct my calculus
May
2
comment Checking that $(C[0,1], \|\cdot\|_1)$ is not Banach.
Rather than actually going to the trouble of computing that integral and then looking for bounds on the answer, it is simpler just to note that $0 \le |f_n(x) - f_m(x)| \le 1$ for $0 \le x \le 1/n$, and $|f_n(x) - f_m(x)| = 0$ otherwise, and therefore $\|f_n - f_m\| \le 1/n$.
May
1
comment Probability distribution of $\int_0^t \frac{W_s}{s} \,ds$
@Richard: One can also see that $I$ is Gaussian by writing the Riemann integral as a limit of Riemann sums, and then writing those sums in terms of the independent increments of Brownian motion. Using a similar argument, you can show that $\int_0^t W_t \mu(dt)$ is Gaussian for any finite Borel measure $\mu$ on $[0,t]$.
May
1
comment Probability distribution of $\int_0^t \frac{W_s}{s} \,ds$
I think this needs some explanation as to why the interchange of integrals is justified when computing $E[I^2]$.
May
1
answered Components of $\sigma$ algebras
Apr
30
comment Components of $\sigma$ algebras
What do you mean by "restricted to its $i$th coordinate"? Do you mean the projection?
Apr
30
comment Components of $\sigma$ algebras
I see, one inclusion is true, but the misconception you had was "every Borel set in $\mathbb{R}^n$ is a product of $n$ Borel sets from $\mathbb{R}$"?
Apr
30
revised If $p$ is a positive multivariate polynomial, does $1/p$ have polynomial growth?
left out reciprocal
Apr
30
asked If $p$ is a positive multivariate polynomial, does $1/p$ have polynomial growth?
Apr
30
asked Does the inverse of a polynomial matrix have polynomial growth?
Apr
28
comment Selfadjoint Operators: Sesquilinear Form (II)
Let us continue this discussion in chat.
Apr
28
comment Selfadjoint Operators: Sesquilinear Form (II)
I have a little trouble following your purported proof because it is not clear what spaces various elements lie in, nor how exactly you are defining $\hat{\mathcal{H}}_s$.