User izimath

5 votes

1 answer

1k views

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

asked Jul 12, 2015 at 16:49

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$

probability borel-cantelli-lemmas

asked Apr 25, 2018 at 15:47

4 votes

2 answers

206 views

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

linear-algebra matrices matrix-rank

asked Aug 28, 2019 at 8:58

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.

probability stochastic-processes brownian-motion martingales

asked Nov 7, 2019 at 9:31

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

brownian-motion martingales heat-equation

asked Nov 12, 2019 at 6:15

3 votes

1 answer

231 views

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

random-variables independence

asked Apr 24, 2018 at 13:36

3 votes

1 answer

463 views

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

differential-geometry smooth-manifolds diffeomorphism

asked Jun 13, 2019 at 7:09

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.

differential-geometry orientation

asked Apr 10, 2019 at 15:51

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

probability probability-theory random-variables independence variance

asked Apr 26, 2018 at 0:54

3 votes

0 answers

156 views

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

brownian-motion markov-process conditional-probability

asked Dec 12, 2018 at 15:53

3 votes

1 answer

98 views

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

general-topology continuity

asked Mar 13, 2019 at 13:10

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.

probability central-limit-theorem

asked Jun 6, 2018 at 12:18

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 $X \perp \textit F$?

probability random-variables independence probability-limit-theorems

asked Nov 23, 2018 at 13:28

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 $(-\pi/2, \pi/2)$

calculus integration improper-integrals convolution

asked Dec 1, 2018 at 5:59

2 votes

1 answer

44 views

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

probability-theory random-variables conditional-expectation

asked Apr 16, 2018 at 10:39

2 votes

2 answers

66 views

Theory without a concrete example.

soft-question

asked Apr 17, 2018 at 14:50

2 votes

1 answer

77 views

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

stochastic-processes brownian-motion

asked May 8, 2018 at 1:55

2 votes

3 answers

124 views

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

calculus analysis

asked Jun 1, 2018 at 12:45

2 votes

1 answer

148 views

Independence of functions of random variables.

random-variables independence

asked Jan 16, 2017 at 12:44

2 votes

2 answers

70 views

Convergence of a (barely converging?)series

sequences-and-series

asked Apr 21, 2017 at 15:06

2 votes

3 answers

671 views

Integration by parts for Riemann integrable fucntion

calculus real-analysis integration

asked Oct 5, 2017 at 4:16

2 votes

1 answer

250 views

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

general-topology measure-theory

asked May 23, 2019 at 14:50

2 votes

0 answers

40 views

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

stochastic-processes poisson-process

asked Jul 20, 2019 at 2:54

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?

real-analysis measurable-functions

asked Sep 5, 2019 at 9:03

2 votes

1 answer

27 views

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

metric-spaces examples-counterexamples cauchy-sequences

asked Jul 3, 2020 at 7:19

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 $(X_i)$.

probability-theory central-limit-theorem probability-limit-theorems

asked Sep 1, 2020 at 19:54

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 $a<0<b$.

brownian-motion martingales stopping-times

asked Oct 27, 2020 at 2:47

2 votes

1 answer

145 views

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

probability-theory ergodic-theory stationary-processes

asked Nov 19, 2020 at 23:17

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.

general-topology manifolds differential-topology problem-solving

asked Nov 29, 2020 at 6:43

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 $|B_t|$?

stochastic-processes brownian-motion

asked Oct 15, 2020 at 2:49

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70 Questions

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

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$

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

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.

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

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

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

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

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

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

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

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.

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

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

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

Theory without a concrete example.

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

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

Independence of functions of random variables.

Convergence of a (barely converging?)series

Integration by parts for Riemann integrable fucntion

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

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

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

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

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

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 $a<0<b$.

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

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.

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 $|B_t|$?