All Questions
Tagged with probability-theory solution-verification
843
questions
1
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1
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45
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Deviation with respect of the Mode instead from the Mean Value
Deviation with respect of the Mode instead from the Mean Value
I am trying to figure out if the following calculations make sense. If I try to make a deviation measure from the Mode value "$\nu$&...
- 1,586
0
votes
0
answers
65
views
How to prove null recurrence when there are infinitely many states?
I have a random walk on a chessboard, where each square is a state/vertex.
I understand that the location of pieces on the board describes a Markov chain, and one way to look at this is through a ...
- 235
0
votes
0
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29
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Proof that $\mathcal{A}_1 \cap \mathcal{A}_2$ forms a valid $\sigma$-algebra over $\Omega$ using Boolean algebra
I'm trying to proof that $\mathcal{A}_1 \cap \mathcal{A}_2$ forms a valid $\sigma$-algebra over $\Omega$, if $\mathcal{A}_1$ and $\mathcal{A}_2$ are both each a valid $\sigma$-algebra over $\Omega$, ...
- 145
1
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1
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303
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Let $X_i$ be a sequence of independent random variables, then $P( \sup_n X_n < \infty) \in \{0,1\}$ Help justifying a step.
Let $X_i$ be a sequence of independent random variables, then $P( \sup_n X_n < \infty) \in \{0,1\}$. (Here the random variables take values on the reals, not on the extended reals).
So my solution ...
- 520
2
votes
1
answer
103
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Show that $\mathbb{E}(\mathbf{1}_{\{X>t\}}|\mathcal{G})$ is $(\mathcal{B}(\mathbb{R})\otimes\mathcal{G},\mathcal{B}(\mathbb{R}))$-measurable.
Let $X$ be a non-negative integral random vairable on $(\Omega,\mathcal{F},\mathbb{P})$ and let $\mathcal{G}\subseteq\mathcal{F}$ be a sub-$\sigma$-algebra. I want to show that $$\mathbb{E}(X|\mathcal{...
- 5,930
1
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0
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44
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From discrete time stopping theorem to continuous
I would like to generalize by dyadic discretization some theorem I have seen in finite time setting.
Here is the theorem
Let $(X_t)_t$ be a continuous martingale and $\tau$ a bounded stopping time ($\...
- 1,381
2
votes
1
answer
91
views
Stochastic process and reaching time to an interval
I have the following exercice to do and I would like to know if what I did is correct please.
Consider $X_t$ a continuous stochastic process and $\tau$ the reaching time to the interval $[a,b]\subset[...
- 1,381
0
votes
2
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182
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Sufficient condition for $L^1$ convergence using uniformly integrabllity
I would like to prove the following result : let $(Xn)_n$ be a sequence in $L^{1}(\Omega,\mathcal{F}_t, \mathbb{P})$ that converges almost surely to $X\in L^1$. Then if $(X_n)_n$ is uniformly ...
- 1,381
0
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0
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29
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Sub-Gaussian Norm Basic Property: Infimum is Minimum?
This is the definition of a sub-gassian norm of a random variable $X$:
$$
\| X \|_{\Psi_2} = \inf\{ t > 0: \mathbb{E}\exp(X^2/t^2) \leq 2 \}.
$$
It is claimed that we have:
$$
\mathbb{E}\exp(X^2/\| ...
- 593
0
votes
0
answers
19
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Proof of a maximal inequality in a finite interval
I consider the set $ I = \left\{ 0,…, N\right\}$ and $X$ a sub martingale on this set. I would like to prove the following inequality for $\lambda>0$
$$
\lambda\mathbb{P}(max(X_0,… X_N)\geq\lambda)\...
- 1,381
1
vote
1
answer
131
views
Brownian motion is not of bounded variation
I would like to prove that Brownian motion, denoted $B_t$, is not of bounded variation using the fact that its quadratic variation is finite.
Here is my attempt:
Consider $[t,s]\subset[0,+\infty)$ and ...
- 1,381
-1
votes
1
answer
67
views
Prove that random variable is a stopping time
After a question I asked concerning my failure to prove that a random variable is a stopping time I come to propose another proof that I hope is better.
We consider $X$ a continuous process with its ...
- 1,381
0
votes
1
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341
views
How to prove that a Markov chain is transient?
I have a Markov chain $\{Y_n: n\geqslant 0\}$ where the $Y_n$ are integer-valued.
The probability of going from any state $i$ to its right (i.e., from state $i$ to state $i+1$) is $p$,
and the ...
- 235
1
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0
answers
128
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What is the expected length of an interval on an arc of a circle that can be constructed using exponential variates?
Consider a circle $S$ of length $\theta$. Now suppose, we delete an interval $I$ of length $|I|$ (I'll drop the $|\cdot|$ notation for length and directly write $I$) from it.
Now on $S-I$, I choose a ...
- 1,285
1
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0
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53
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Prove that a continuous and $\mathcal{F}_t$ adapted process is progressively measurable
I would like to show that if $X_t$ is a continuous and adapted stochastic process (real valued) then it is progressively measurable.
Here is my attempt : consider $B\in\mathcal{B}(\mathbb{R})$. Then ...
- 1,381