User izimath

1

4

votes

1

answer

105

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.

Nov 11 '19 at 8:51 izimath 984

3

4

votes

1

answer

420

views

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

linear-algebra matrices matrix-exponential unitary-matrices

Jun 30 '19 at 23:28 Rodrigo de Azevedo 17k

1

4

votes

1

answer

79

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

Apr 25 '18 at 16:05 izimath 984

3

votes

1

answer

38

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

Nov 21 '19 at 0:45 izimath 984

1

3

votes

2

answers

121

views

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

linear-algebra matrices matrix-rank

Aug 29 '19 at 6:09 Rodrigo de Azevedo 17k

3

votes

1

answer

80

views

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

general-topology continuity

Mar 14 '19 at 7:57 quarague 4,050

3

votes

0

answers

80

views

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

brownian-motion markov-process conditional-probability

Dec 12 '18 at 15:53 izimath 984

1

3

votes

1

answer

175

views

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

random-variables independence

May 1 '18 at 4:08 izimath 984

1

2

votes

1

answer

26

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

Jul 3 at 8:17 Tipping Octopus 1,335

2

votes

1

answer

104

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

Oct 28 '19 at 17:00 Jonathan Hole 2,484

2

votes

0

answers

26

views

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

stochastic-processes poisson-process

Jul 20 '19 at 2:54 izimath 984

2

votes

1

answer

252

views

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

differential-geometry smooth-manifolds diffeomorphism

Jun 27 '19 at 16:33 Si Kucing 5,524

1

2

votes

0

answers

92

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

Apr 10 '19 at 15:51 izimath 984

1

2

votes

1

answer

44

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

Dec 1 '18 at 6:13 Community♦ 1

1

2

votes

1

answer

49

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

Nov 23 '18 at 14:13 izimath 984

2

votes

3

answers

114

views

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

calculus analysis

Jun 9 '18 at 10:11 Paramanand Singh 71.2k

2

votes

1

answer

82

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

Jun 6 '18 at 13:29 Galton 106

2

votes

1

answer

43

views

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

stochastic-processes brownian-motion

May 8 '18 at 3:04 hypernova 5,482

1

2

votes

3

answers

83

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

May 1 '18 at 14:48 BCLC 1

2

votes

2

answers

66

views

Theory without a concrete example.

soft-question

Apr 18 '18 at 13:08 Mikhail Katz 34.5k

2

votes

1

answer

33

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

Apr 17 '18 at 8:29 Davide Giraudo 148k

2

votes

3

answers

401

views

Integration by parts for Riemann integrable fucntion

calculus real-analysis integration

Oct 5 '17 at 16:27 copper.hat 152k

1

2

votes

2

answers

62

views

Convergence of a (barely converging?)series

sequences-and-series

Apr 21 '17 at 17:04 Robert Israel 400k

1

vote

1

answer

30

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

Nov 29 at 11:46 izimath 984

1

vote

1

answer

127

views

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

probability-theory ergodic-theory stationary-processes

Nov 23 at 6:27 WhoKnowsWho 2,492

1

vote

1

answer

60

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

Oct 27 at 19:32 saz 106k

1

vote

0

answers

12

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

Oct 15 at 2:49 izimath 984

1

vote

1

answer

24

views

Supermartingale with bdd increment and positive variance before $\tau$ satisfies $P_k(\tau >u) \leq \frac{4k}{\sigma \sqrt{u}}$ for large $u$.

probability markov-chains martingales stopping-times

Sep 24 at 6:46 Fei Cao 408

1

vote

1

answer

91

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

Sep 2 at 21:47 izimath 984

1

vote

0

answers

38

views

Is $\sum_{n=1}^{\infty} \frac{a_n}{(\sum_{k=1}^n a_k)^2}$ convergent whenever $\sum_{n=1}^{\infty} a_n =\infty$ for unbounded $a_n >0$?

sequences-and-series convergence-divergence

Jun 30 at 7:12 izimath 984

Stack Exchange Network

current community

your communities

more stack exchange communities

Next privilege

Next badge

70 Questions

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.

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$

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

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

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

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

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

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

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

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

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.

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

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

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

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.

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

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

Theory without a concrete example.

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

Integration by parts for Riemann integrable fucntion

Convergence of a (barely converging?)series

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.

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

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 $a<0<b$.

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 $|B_t|$?

Supermartingale with bdd increment and positive variance before $\tau$ satisfies $P_k(\tau >u) \leq \frac{4k}{\sigma \sqrt{u}}$ for large $u$.

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

Is $\sum_{n=1}^{\infty} \frac{a_n}{(\sum_{k=1}^n a_k)^2}$ convergent whenever $\sum_{n=1}^{\infty} a_n =\infty$ for unbounded $a_n >0$?