All Questions
Tagged with probability-theory solution-verification
843
questions
2
votes
0
answers
80
views
$0$-$1$ law for brownian motion
I would like to prove the following theorem : Let $B$ be a brownian motion and $\mathcal{F_t}$ its natural filtration. Then for all $A\in\mathcal{F}_{0^{+}}$ we have $\mathbb{P}(A)\in\left\{0,1\right\}...
- 1,381
0
votes
0
answers
283
views
Relationship between Markov chains and i.i.d. random variables
I am studying Markov chains. I understand that a sequence of i.i.d. random variables is a special type of Markov chain. However, I am trying to prove that a finite-valued Markov chain is a sequence ...
- 235
0
votes
0
answers
66
views
Modification of a stochastic process and complete filtration
I consider $B_t$ $t\in[0,T]$ a (real valued) stochastic process adapted for the filtration $\mathcal{F}_t$ and $\bar{B}_t$ à modification of $B_t$. I would like to show that $\bar{B}_t$ is adapted ...
- 1,381
2
votes
0
answers
74
views
Chernoff bound and Polynomial Markov
I'm currently struggling the following statement;
Show that upper bound of polynomial Markov is better than the Chernoff upper bound; that is, for given $\delta > 0,$
$$\inf_{k = 0,1,2,...} \frac{\...
- 769
3
votes
0
answers
48
views
Is my proof of Markov Property for Reflected BM correct?
I want to show that $|B_{t}|$ is a Markov Process where, $B_{t}$ is a Standard Brownian Motion. I have seen the proof here and here. But I don't understand why the method below might fail (or if it's ...
- 1,285
2
votes
0
answers
68
views
Distribution function of $X-Y$, where $X,Y$ are uniformly distributed
I want to compute the distribution function of $X-Y$, denoted by $F_{X-Y}$, where $X,Y$ are indepedently uniformly distributed on $[-1,1]$.
\begin{align*}
&F_{X-Y}(z)=\begin{cases}
0,&z\leq -2\...
- 4,564
2
votes
2
answers
143
views
Find the density of $\cos(X)$ when $X$ is an exponential.
I want to find the density of $\cos(X)$ where $X$ is an exponential with density given by
$$
f_{X}(x) = re^{-rx}\mathbb{1}_{[0,+\infty}(x),\quad r>0
$$
My attempt is the following :
First we notice,...
- 1,381
3
votes
1
answer
68
views
Solution verification on a Borel Cantelli Exercise
I have a question regarding the following exercise that I just solved.
Let $(X_n)$ be any sequence of random variables, show that there exists a sequence of positive constants $(l_n)$ such that one ...
0
votes
0
answers
89
views
Prove that the first return time is finite a.s. for a Markov chain
Suppose $X$ is an irreducible Markov Chain on a discrete state space $E$. I would like to prove that
$$
P_x[\tau_x^1 < \infty]=1
$$
where $\tau_x^1=\inf\{n>0: X_n=x\}$.
Is it necessary to know ...
- 563
0
votes
1
answer
95
views
Probability that player B wins [duplicate]
We have two players, A and B, who play a game in several rounds. The game stops when one of them wins two more rounds than the other. What is the probability that B wins if in each round the ...
- 55
1
vote
2
answers
44
views
$Gx$ is distributed uniformly in the set $\Bbb Z_2 ^n$.
I read the probabilistic proof of the Gilbert Varshamov bound in Coding Theory, and I came across the following argument, which I would like to discuss:
Suppose that $G$ is a random matrix in $M_
{n×k}...
- 2,802
0
votes
0
answers
41
views
$X_n$ s.t $E[\sup _n |X _{n+1}-X_n|]< \infty$ and $X_0 =0$. Prove that a.s $\limsup X _n=-\liminf X _n =\infty$ or $\lim X _n$ exsits and finite
$X_n$ s.t $E[\sup _n |X _{n+1}-X_n|]< \infty$ and $X_0 =0$
Prove that a.s $\limsup X _n=-\liminf X _n =\infty$ or $\lim X _n$ exsits and finite.
hint: you can prove ${Y _n}$ martingale then $\sup ...
- 617
1
vote
0
answers
66
views
How can I show that $\Bbb{E}\left(\exp(-\mu T)\right)=\exp\left(-a\sqrt{2\mu}\right)$?
Let $B$ be a Brownian motion and for any $a$ define $T:=\inf\{t>0: W_t\geq a\}$.
I want to show that $\Bbb{E}(\exp(-\mu T))=\exp(-a\sqrt{2\mu})$.
My idea was to use the optional stopping theorem.
...
- 299
0
votes
1
answer
51
views
Random walk and inequality involving the Green function on N step
I would like to check if my arguments are the correct ones in proving
$$
G_N(x,y):=\sum_{k=0}^N P_x[X_k=y]\leq G_N(y,y)
$$
where $X$ is a random walk on $\mathcal{Z}$ with $E_0[|X_1|]<\infty$ and $...
- 563
2
votes
0
answers
58
views
Show biasedness of estimator of discrete uniform distribution
Let be $X:\Omega\to[0,1,2,3,\dots, \theta]$ a discrete random variable which obeys a uniform distribution. We don't know $\theta$ and estimate it by the maximum of the sample $(x_1,x_2,\dots,x_n)$, ...
- 4,564