user avatar
user avatar
user avatar
izimath
  • Member for 6 years, 11 months
  • Last seen this week
5 votes
1 answer
1k views

For any unitary matrix $U$, there exists a Hermitian matrix $H$ such that $U=e^{iH}$

4 votes
1 answer
136 views

Prove For $(X_n)_{n \geq1}$ independent RVs, $ X_n \rightarrow X \ \text{ a.s.} \Rightarrow \sum _{n\geq1} P(|X_n -X| \gt \varepsilon) \lt \infty$

4 votes
2 answers
206 views

$rank(M)=rank(M^2)$ whenever $M$ is skew-symmetric

4 votes
1 answer
200 views

Show $\limsup _{t \to \infty} \frac{B_t}{\sqrt{t \ln t}} \leq 1 $ using the fact that $\frac{e^{B_t ^2 / (1+2t)}}{\sqrt{1+2t}}$ is a martingale.

3 votes
1 answer
97 views

$u(t,B_t)$ is a martingale if $u(t,x)$ is polynomial in each variables and satisfies the heat equation

3 votes
1 answer
231 views

Show for i.i.d. r.v., $Y_n /n \rightarrow 0$ a.s. $\Leftrightarrow$ $E|Y_1| < \infty$

3 votes
1 answer
463 views

Show $S^n \setminus \{p,q\} $ is diffeomorphic to $\mathbb R^n \setminus\{0\}$.

3 votes
0 answers
176 views

On the unit sphere $S^2$, show the antipodal map $A:S^2\to S^2$ is orientation reversing using definitions.

3 votes
3 answers
103 views

Let $X,Y$ be independent RVs. If there exists $c \in \mathbb R $ s.t. $P(X+Y=c)=1$, then $X,Y$ are constants

3 votes
0 answers
156 views

Show $(B_t )^2$ i.e. square of a Brownian motion is a Markov process. [duplicate]

3 votes
1 answer
98 views

Construct a continuous function $g$ which vanishes on closed $F$ and "follows" a continuous function $f$ outside $F$.

2 votes
1 answer
104 views

Show $\sup_x \sqrt{n}|F_n(x) -F(x)|=O_p(1)$ where $F_n$ is the empirical distribution function of a iid RVs with a density function.

2 votes
1 answer
62 views

Suppose $X_n \rightarrow X$ a.s. and for each $n$, $X_n \perp \textit F$. Then is it true that $X \perp \textit F$?

2 votes
1 answer
53 views

Show $ \int_ {-\infty} ^{\infty} (\arctan(x+a)) {{1}\over {\sqrt{2 \pi T}}} e^{-x^2 /2T} dx$ can assume any value in $(-\pi/2, \pi/2)$

2 votes
1 answer
44 views

Finding conditional expectation $E[f|g]$ given $f(x):= x^2 /2$ and $g(x):=2(x-1/2)^2$

2 votes
2 answers
66 views

Theory without a concrete example.

2 votes
1 answer
77 views

Finding the distribution of $\int_0 ^T uW_u du$ for a Brownian motion

2 votes
3 answers
124 views

If $\exists f''(0)$, show $\lim _{h \rightarrow 0} \frac{f(h)-2f(0)+f(-h)}{h^2}=f''(0)$

2 votes
1 answer
148 views

Independence of functions of random variables.

2 votes
2 answers
70 views

Convergence of a (barely converging?)series

2 votes
3 answers
671 views

Integration by parts for Riemann integrable fucntion

2 votes
1 answer
250 views

Is the Dirac measure outer regular on a compact Hausdorff space?

2 votes
0 answers
40 views

Show that the compound Poisson process is continuous a.s. at a fixed time.

2 votes
1 answer
124 views

If $f(\cdot, \cdot)$ is measurable in each variable and $g(\cdot)$ is measurable, then $x \mapsto f(x,g(x))$ is measurable?

2 votes
1 answer
27 views

Find a counter-expamle to $\lim_n \limsup_m d(a_n, a_m) =0 \implies (a_n)_n \ \text{is cauchy}.$

2 votes
1 answer
121 views

Show that $\sum_{i=1}^n X_i / \sqrt{n} \Rightarrow W$ implies $EX_1^2 <\infty$ for an i.i.d. sequence $(X_i)$.

2 votes
1 answer
181 views

For a Brownian motion $B(t)$, show $0=\mathbb E[ B(\tau)]$ where $\tau := \max \{\tau_a, \tau_b\}$ is the latter hitting time of the levels $a<0<b$.

2 votes
1 answer
145 views

Question on the proof of Subadditive Ergodic Theorem in Durrett's textbook.

1 vote
1 answer
66 views

Show $ M_1 \# M_2= U_1 \cup U_2 $ for some open subsets $U_i$ of manifolds $M_i$ such that the following properties hold.

1 vote
0 answers
18 views

If $B_t$ is a Brownian motion, do the zeros of the process $M_t -B_t$ behave the same as the zeros of the $|B_t|$?