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izimath

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70 Questions

Votes Activity Newest Views
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.

probability stochastic-processes brownian-motion martingales
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
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
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
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
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
1
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