All Questions
Tagged with probability-theory solution-verification
843
questions
4
votes
1
answer
959
views
Hitting time of Brownian motion against a square root curve
$B$ denotes Brownian motion and the hitting time I am interested in is
$$\tau = \inf\{t \geq 0: B_t = b\sqrt{a+t}\}$$
where $a,b >0$. I first want to show that $\tau < \infty$ almost surely. I ...
CommunityBot
- 1
2
votes
0
answers
38
views
A proof of the Glivenko-Cantelli theorem
I would like to have the following proof of the Glivenko-Cantelli theorem checked, which states that if ${X_1,X_2,\dots}$ are iid copies of a real random variable ${X}$, then almost surely, one has
$\...
- 1,011
1
vote
1
answer
107
views
Conditional distribution given order statistics
Let $X_1,\dots,X_n$ be i.i.d, with $X_1$ having a continuous cumulative distribution function. Let $T = (X_{(1)},\dots,X_{(n)})$ be the order statistics. Show given $T$, the conditional distribution ...
- 47.4k
1
vote
0
answers
27
views
Centered Subgaussian Variables have better Properties
I am trying to understand the following proof:
Main Confusion: In particular, I am having a very hard time understanding the chain of inequalities in the proof for (3)': I think the first equality is ...
- 593
0
votes
1
answer
17
views
Understanding the proof for Properties of Subgaussian Variables
Here are the definitions, statements and the proof that I am stuck on:
I am stuck on the last part of the proof where the author claims that setting $C = e$ automatically guaranties that (1) holds ...
- 39.2k
2
votes
2
answers
71
views
A convergence property for iid sequence of Cauchy random variables
A real random variable ${X}$ is said to have a standard Cauchy distribution if it has the probability density function $\displaystyle {x \mapsto \frac{1}{\pi} \frac{1}{1+x^2}}$. If ${X_1,X_2,\dots}$ ...
- 1,011
0
votes
0
answers
15
views
Proving statement about cumulative distribution function
I want to prove the following statement:
Let F fulfill the properties of a cumulative distribution function. Define $$X^-(\omega) = \inf \{z \in \mathbb{R}: F(z) \geq \omega\} \quad X^+(\omega) =\inf ...
- 11
1
vote
1
answer
58
views
Law of large number for non-integrable random variables
Let ${X_1,X_2,\dots}$ be iid copies of an unsigned random variable ${X}$ with infinite mean, and write ${S_n := X_1 + \dots + X_n}$. Show that ${S_n/n}$ diverges to infinity in probability, in the ...
- 3,527
-1
votes
1
answer
36
views
Sigma-algebra generated by $\sin(2\omega)$
If $\Omega = (0, 2\pi)$ and $X(\omega) = \sin(2\omega)$, is $\sigma(X) = \{S: x \in S \cap [0, \pi / 4] \Leftrightarrow \pi / 2 - x \in S \cap [\pi / 4, \pi / 2] \Leftrightarrow \pi + x \in S \cap [\...
- 39.7k
0
votes
1
answer
51
views
Microstate interpretation of Shannon entropy
Let ${A}$ be a finite non-empty set of some cardinality ${|A|}$, and let ${X}$ be a random variable taking values in ${A}$. Define the Shannon entropy ${{\bf H}(X)}$ to be the quantity $\displaystyle ...
- 9,298
4
votes
2
answers
394
views
Show that $\frac{1}{n}X_n\to 0$ a.s.
Show that for any sequence $(X_n)_{n\in\mathbb{N}}\in (L_{\mathbb{P}}^2)^{\mathbb{N}}$ of identically distributed random variables it is $\frac{1}{n}X_n\to 0\text{ a.s.}$.
The professor suggested ...
- 425
4
votes
4
answers
892
views
A step in the proof of Fubini theorem (Theorem 2.36, Folland)
This is a first case of the proof of the Fubini-Tonelli theorem, given in Folland's Real Analysis. I'm confused with the line underlined in blue at the end (namely, 'the preceding argument applies to' ...
- 40.8k
0
votes
0
answers
47
views
Estimate the number of Cramer model primes
This is a sub-post of the problem Probabilistic Riemann hypothesis.
Consider a sequence of independent Bernoulli random variables $(X_n)_{n \geq 3}$ of parameters $1 / \log n$ (so $X_n = 1$ with ...
- 1,011
0
votes
1
answer
29
views
If $X = g(Y)$, then Is the sigma-algebra generated by $X$ a subset of the sigma-algebra generated by $Y$?
Let $g$ be a continuous function. Let random variables $X$ and $Y$ satisfy $X = g(Y)$. Do we always have $\sigma(X) \subset \sigma(Y)$?
I want to disprove this. My thought is that there must be a set $...
- 2,269
2
votes
1
answer
46
views
Sufficient Condition on Almost Surely Convergence
Let $f_n \in [0, 1]$ and suppose if we want to show
$$
\lim_{n \to \infty} f_n = 1
$$
almost surely, is it enough to show
$$
\lim_{n \to \infty} \mathbb{P}\{ f_n = 1 \} = 1?
$$
If not, what if we add ...
- 13.2k