Tagged Questions
-1
votes
0answers
23 views
How to convert additive measure v to characteristic function?
1
how to convert characteristic function to additive measure v?
2.
and convert back to characteristic function?
characteristics function are from density function, pdf by fourier transform
1
vote
1answer
32 views
Upper bound on a probability generating function with a finite first moment
If $X$ is discrete random variable taking values in non-negative integers $\{0,1,\ldots\}$, its probability generating function is defined as follows:
$$G(z)=\mathbb{E}(z^X)=\sum_{x=0}^\infty ...
2
votes
1answer
36 views
Proving a property for some general transformation $\mathbf{f}: \mathbb{R}^s \to (0,1)^{k}$
The author of the prob/stat textbook I am using casually threw out a statement that I've been driving myself crazy trying to prove. It goes like this:
Let $s < k-1$ for some positive integers $s$ ...
0
votes
0answers
31 views
Combining convergence in probability and the means of the positive sequence of r.v. implies convergence in L 1
Let $\{X_n\}$ be a collection of positive random variable with $X_n \rightarrow X$ in probability. Prove that if $E(X_n) \rightarrow E(X)$, then $X_n \rightarrow X$ in $L^1$.
My partial answer:
Let ...
4
votes
1answer
35 views
$L^{p}$ functions from Rudin Exercises 3.5
I am attempting a question from Rudin's "Real and Complex Analysis" Chapter 3 question 5. I shall summarise the question as below: Suppose that $f$ is a complex measurable function on $X$, $\mu$ a ...
2
votes
2answers
109 views
Interpretation of dP in Radon-Nikodym Theorem
[Radon-Nikodym Theorem]
Let $(\Omega, \Sigma, P)$ be a probability space. Suppose that $(\Omega, \Sigma, \mu)$ be a measure space with $\mu(A)=0$ implies $P(A)=0$, then there exist a function $f:X ...
1
vote
0answers
15 views
Analytic random function
Let $t\in[0,1]\mapsto X_t\in[0,1]$ an analytic random function.
I'd like to say that the deterministic function $t\mapsto \mathbb{E}[X_t]$ is still analytic.
What are the minimal conditions needed? ...
2
votes
0answers
48 views
Bernoulli shift on $S^\mathbb{Z}$
Why is the Bernoulli shift ergodic? I know the proof for $S^\mathbb{N}$. The best would be if you could argue from this fact to the integer case, so that way I can see how one usually goes from ...
1
vote
0answers
68 views
Borel-Cantelli lemma proof $\text{P}\left(\lim_{n\to\infty}{ M_n\over{\log n}}=\frac{1}{\lambda}\right)=1$
Show $\displaystyle \text{P}\left(\lim_{n\to\infty} {M_n\over{\log n}}=\frac{1}{\lambda}\right)=1$where $M_n=\max_{1\leq k\leq n}X_k$ and $X_k \text{~ Exp}(\lambda) $
! All I have so far is (which ...
1
vote
1answer
44 views
Independence of $n$ random variables
Let $A_1,A_2,\ldots,A_n$ be independent subsets of probability space $(\Omega, \Sigma, P)$ (For every $I\subseteq \{1,2,\ldots,n\}$, $P(\bigcap_{j\in J}A_j)=\prod_{j\in J}P(A_j) )$. Prove that ...
1
vote
1answer
31 views
$X_n \overset{a.s.}{\longrightarrow} X$ and $X_n \overset{L^1}{\longrightarrow} Y$ implies $X = Y$ a.s.?
If I have a sequence of random variables $\{X_n\}_{n \geq 0}$ such that
$$X_n \overset{a.s.}{\longrightarrow} X \quad\textrm{and}\quad X_n \overset{L^1}{\longrightarrow} Y$$
then is it always true ...
2
votes
1answer
71 views
integer Random Walk with step size governed by a distribution.
This problem is for a final exam I am taking in a graduate probability class. Collaboration has been explicitly allowed, but with the remark that the professor felt he couldn't stop us even if he ...
1
vote
1answer
48 views
Brownian Motion and the Functional CLT
Suppose we have a time series $(x_t\mid t\in \mathbb{Z})$ for which the partial sum process $X_T$ defined on the unit interval by $$ X_T(\xi)=\omega_T^{-1}\sum_{t=1}^{[T\xi]} ...
1
vote
1answer
38 views
Non-centered Gaussian moments
I would like to find a (closed nice) expression for the non-centered Gaussian moments with mean $\mu$ and variance $\sigma$. In found something in wikipedia:
...
1
vote
1answer
55 views
Product of a sequence tending to 0 and a sequence of random variables converging in distribution
Let $X_n$ be a sequence of random variables and suppose $X_n \rightarrow X$ in distribution. Let $a_n$ be a sequence of constants with $a_n \rightarrow 0$. Must $a_n X_n \rightarrow 0$ almost surely? ...
0
votes
1answer
40 views
Construction of Lebesgue-Stieltjes measure on $\mathbb{R}^d$ $d=1,2$
I need some help o understand the construction of measure induce by a function.
For $d=1$, given right-continous function $F:\mathbb{R}\rightarrow [0,1]$ that increasingly monotonic such that ...
0
votes
0answers
31 views
Lebesgue-Stiletjes measure on $\mathbb{R}^2$
Let $F$ be a two variable continous function that satisfy
\begin{equation}
F(x_1,y) \leq F(x_2,y) \text{ for } x_1\leq x_2,
\end{equation}
\begin{equation}
F(x,y_1) \leq F(x,y_2) \text{ for } y_1\leq ...
1
vote
1answer
55 views
Prove Convergence in Probability is closed under multiplication
This seems like a pretty plain question, but I can't figure it out.
Let $X_n \to X$ in probability, and $Y_n \to Y$ in probability. Show that $X_N Y_N \to XY$ in probability.
So far I can only show ...
1
vote
1answer
57 views
Writing the sigma-algebra generated by $X_1, \ldots, X_n$
This is somewhat of a stupid question, but I am stuck on coming up with an airtight justification. Let $X_1, \ldots, X_n$ be defined on a probability space $(\Omega, \mathcal{F}, P)$. There are ...
3
votes
2answers
82 views
Every uncountable Polish Space has a copy of $\{0, 1\}^\mathbb{N}$
I am having trouble verifying corollary 7.8 on p. 6 in this document
http://www.math.ucla.edu/~biskup/275b.1.13w/PDFs/Standard-Borel-Spaces.pdf
My troubles are with the definition of the "tree" ...
1
vote
1answer
31 views
Limit involving probability
Let $\mu$ be any probability measure on the interval $]0,\infty[$. I think the following limit holds, but I don't manage to prove it:
$$\frac{1}{\alpha}\log\biggl(\int_0^\infty\! x^\alpha ...
2
votes
0answers
50 views
Inequality change in $\mathbb{E}[ \max |\cdot|] $ due to $\max$
Let $m$ be a probability measure on $W \subseteq \mathbb{R}^p$, so that $m(W)=1$.
Find $m$, a locally-bounded function $f:\mathbb{R}^n \times \mathbb{R}^m \times \mathbb{R}^p \rightarrow ...
3
votes
0answers
101 views
An absolutely continuous cumulative distribution function that fails to have a Riemann-integrable pdf.
We know that if a real-valued random variable $ X $ on a probability space has an absolutely continuous cumulative distribution function (cdf) $ F $, then $ X $ possesses a probability density ...
1
vote
1answer
31 views
Probability space defined by function from X to [0,1]
Let $X$ be a non-empty countable set. If there is a function $f:X\rightarrow [0,1]$ such that $p(S)=\sum_{x\in S} f(x)$ for all $S \in 2^X$, then prove that $(X,2^X, p)$ is a probability space.
My ...
1
vote
1answer
73 views
How compute $\lim_{p\rightarrow 0} \|f\|_p$ in a probability space?
I not solve the follow limit
$$\lim_{p\rightarrow 0} \bigg[\int_{\Omega} |f|^p d\mu \bigg]^{1/p} = \exp\bigg[ \int_{\Omega} \log|f|d\mu \bigg],$$
where $(\Omega, \mathcal{F}, \mu)$ is a probability ...
3
votes
2answers
68 views
How is this done by dominated convergence theorem?
Terry Tao wrote in his blog
Fix ${k \geq 1}$. If ${X}$ has finite ${k^{th}}$ moment, say ${{\bf E}|X|^k \leq C}$, then from Markov’s inequality (14) one has
$$
\displaystyle {\bf P}(|X| \geq ...
3
votes
1answer
62 views
Convergence of a positive series with an application to sums of Poisson random variables
Main question: Let $a_n \geq 0$ and $b_n = \sum_{k=1}^n a_k \uparrow \infty$. Is it true that
$$
\sum_{n=1}^\infty \frac{a_n}{b_n^2} < \infty \,?
$$
I strongly suspect there is either a short, ...
0
votes
2answers
47 views
Proving $X_n\overset{\text{a.s.}}{\to}X \Rightarrow X_n\overset{L^1}{\to}X$ when $X_n$ are dominated by a r.v. with a finite mean
Let $X_n\overset{\text{a.s.}}{\to}X$, $|X_n|<Y$, and $E(Y)<\infty$. Then $X_n\overset{L^1}{\to}X$. Wikipedia says the statement without a proof. Could someone prove it or give a reference?
2
votes
1answer
71 views
An inequality $\frac{(1-\lambda)^{2}}{4}\leq m\{x\in [0,1]: |f(x)|\geq \frac{\lambda}{2}\}$
In this question $f$ is a Lebesgue measurable function on $[0,1]$ with the property that $\|f\|_{2}=1,\|f\|_{1}=1/2$. I am trying to prove that
$$
\frac{(1-\lambda)^{2}}{4}\leq m\left\{x\in [0,1]: ...
2
votes
0answers
56 views
Identify the smallest $c$ such that $P(|X_n| \ge c \sqrt{ln n} \text{ i.o.}) = 0$ for normally distributed $X_n$
The problem is to show that $P(|X_n| \ge c \sqrt{ln n}\text{ i.o.}) = 0$ for standard normal $X_n$ that are not necessarily independent. Also, identify the smallest such $c$.
I am thinking that the ...
1
vote
1answer
67 views
Discontinuous point of a distribution function on ${\Bbb R}^n$
Can the points of discontinuity of a distribution function on $\mathbb R^n$, ($n\gt1$) be uncountable？ What about $\left(\{-\infty\}\cup\Bbb R\cup\{+\infty\}\right)^n$?
1
vote
1answer
103 views
Where are algebras and $\sigma$ algebras studied?
I am taking a course on real analysis (mainly about Lebesgue measure
etc') and a few lectures back the lecture introduced to concept of
algebra and $\sigma$-algebra.
It feels a bit strange to see it ...
1
vote
2answers
93 views
Examples that are not Lebesgue integrable for any $p$
I've been trying to think up different examples of functions such that $EZ^p = \infty$ (with $Z>0$) for all $p$, but each time it becomes rather messy. Can anyone suggest some interesting but ...
3
votes
3answers
59 views
Simple question: Why does $E(|X|) < \infty$ imply $E(|X|I_{|X|>a} )$ tends to $0$ as $a$ tends to infinity
Simple question: Why does $E(|X|) < \infty$ imply $E(|X|I_{|X|>a} )$ tends to $0$ as $a$ tends to infinity?
I've seen it in a few proofs and I can't see why this is the case, I've tried a proof ...
1
vote
1answer
71 views
$\sum_{n=1}^{\infty} E|X_n - X| < \infty$ imples $X_n$ converges to $X$ almost surely
$\sum_{n=1}^{\infty} E|X_n - X| < \infty$ imples $X_n$ converges to $X$ almost surely
I'm sure this is an obvious one line proof and I'm being stupid here, but I cannot see how to show the above. ...
5
votes
1answer
413 views
Convergence in probability, not almost surely
This is a classic example of convergence in probability, but not almost surely, but I am trying to rigorously prove it as opposed to "arguing against" the almost sure convergence.
...
2
votes
1answer
40 views
Compact subclasses of $R^\mathbb{N}$
I am following this source: http://www.hss.caltech.edu/~kcb/Notes/Kolmogorov.pdf and agree with everything done in sections 1-3.
In section 4, I cannot fill in the detail for for Lemma 4, because I ...
4
votes
1answer
79 views
$L_p$ complete for $p<1$
It is rather straight forward to show that $L_p$ is complete for $p\geqslant 1$, but I am having trouble showing the same thing when $p<1$. For the former case I have shown that every absolutely ...
1
vote
1answer
96 views
Composition of uniformly integrable function (or random variable)
If I have a sequence of uniformly integrable functions (or random variables) $X_n$ and I compose these with a function $f$ what conditions on $f$ make $f(X_n)$ uniformly integrable? Further, my ...
2
votes
1answer
114 views
Conditional expectation and martingales
I have a few questions concerning martingales. Let $Y\in \mathcal{L}^1(\Omega,\mathcal{F},\mathbb{P})$ be given, and $(\mathcal{F}_n)$ a filtration, and define $X_n:=\mathbb{E}[Y|\mathcal{F}_n]$.
We ...
5
votes
1answer
220 views
Why are continuous functions not dense in $L^\infty$?
Why are the continuous functions not dense in $L^\infty$?
I mean both concretely (i.e. a counter example) and intuitively why is this the case.
1
vote
2answers
77 views
A probability question on sum
Let $X_{1}$, $X_{2}$, $X_{3}$, $X_{4}$, $X_{5}$ and $X_{6}$ be real-valued random variables
that have the same probability distribution with finite moments, and they are independent. Does anyone know ...
1
vote
1answer
113 views
certain stochastic process as martingale
Let $(\Omega, \mathcal{F}, (\mathcal{F}_n)_{n\in\{0,1,2,\ldots,N\}}, P)$ be a stochastic basis, carrying an adapted and integrable stochastic process $X=X_n$. Show that X is a martingale iff ...
2
votes
1answer
227 views
Conditional Moment Generating Function With A Twist
Let $X$, $X'$ be identically distributed (not necessarily iid) random variables with compact support, on the same probability space. Define
$G_t(x):=\mathbb{E}[e^{t(X'-X)} | X=x]$
In other words a ...
4
votes
2answers
369 views
Prove the time inversion formula is brownian motion
Let $B=(B_t)_{t\geq 0}$ be a brownian motion. Show the time inversion formula $\hat{B}=(B_t)_t\geq0$ is a brownian motion, where for $t \geq 0$ we set $\hat{B}=0$ for $t=0$ and $\hat{B}=tB_{1/t}$ for ...
1
vote
0answers
54 views
Equation Involving Bilateral Laplace Transform
Assume that $f(x,y)$ is a compactly supported, joint probability density function on $\mathbb{R}^2$ and nice enough for the following function to make sense:
$$P_t(y):=e^{ty}-\int_{-\infty}^\infty ...
0
votes
0answers
58 views
A different Uniform Bound on a Sequence of Uniform Bounded Integrals
Let $m$ be a probability measure on $W = \mathbb{R}^m$, so that $m(W)=1$.
Consider a sequence $\{X_k\}_{k=1}^{\infty}$ of compact sets $X_k \subset X = \mathbb{R}^n$ such that $X_k \rightarrow X$.
...
2
votes
1answer
74 views
When does non-negativity of the integral of a function imply that the function itself is non-negative?
Let $(\Omega,\Sigma)$ be a measurable space and $(\omega_k)_{k\in\mathbb{N}}$ a sequence of elements of $\Omega$. Let
$$
\mathcal{M}:=\left\{\sum_{k=1}^\infty a_k\cdot\delta_{\omega_k}: ...
1
vote
1answer
127 views
Absolute continuity between measures, one induced from the other by a measurable mapping?
$(\Omega, \mathcal{F},\mu)$ is a measure space, and $f: \Omega \to \Omega$ is a measurable mapping. Let $\nu$ be the measure on the same measurable space induced from $\mu$ by $f$ .
I wonder if there ...
2
votes
1answer
200 views
Sigma-field of a sequence of Random Variables
Problem:
Suppose $\tilde{X}=(X_1,X_2,\dots)$ is a sequence of RVs on $(\Omega,\mathcal{B})$. Prove that $\sigma(\tilde{X})$ is generated by events of the form:
$$\bigcap_{i=1}^m \{X_i\leq ...
