0
votes
0answers
16 views

Find inequality for gaussian density

Let $C>0$ be a fixed constant. Is it true that $$Cx^2 e^{-x^2}\leq e^{-\frac{x^2}{C}}?$$ More generally, if we have a power $x^p$ in front of the exponential, do we have that $$(C^{1/2}x)^p\leq ...
0
votes
0answers
13 views

Hermite polynomials with non standard variance (not equal to one)

It is known for probabilists that if $p(x)=\frac{1}{\sqrt{2\pi}} e^{-\frac{x^2}{2}}$ then the derivatives of $p$ can be expressed in terms of the Hermite polynomials as follows: $$\frac{d^n}{dx^n} ...
1
vote
1answer
54 views

$X_n$ are r.v.s, is it true that $E[\sum_{n=1}^{\infty} X_n] = \sum_{n=1}^{\infty} E[X_n] $? [duplicate]

$X_n$ are r.v.s, is it true that $E[\sum_{n=1}^{\infty} X_n] = \sum_{n=1}^{\infty} E[X_n] $? My feeling is that this is not necessarily true. But cannot come up with an example. Can someone provide ...
0
votes
1answer
18 views

Random variables with all moments. Is this statement true?

Let $X$ be a random variable such thta $X\neq 0$, $P$-a.s. Then $$X\in \bigcap_{p\geq 1} L^p(\Omega) \iff \frac{1}{X} \bigcap_{p\geq 1} L^p(\Omega).$$ In other words, is the space $\bigcap_{p\geq 1} ...
0
votes
1answer
22 views

Integrated Brownian motion: independent stationary increments?

Let $B_t$, $t\in [0,T]$ be a $d$-dimensional standard Brownian motion. Let $\sigma:[0,T] \rightarrow \mathbb R^{d\times d}$ be a deterministic function such that $$\sigma(u) = diag( \sigma_1(u), \dots ...
0
votes
1answer
20 views

Show that convergence in probability implies convergence almost surely

Let {$X_n : n ≥ 1$} and X be random variables on a probability space (Ω,F,P). Suppose $X_n ≤ X_{n+1}$ for every $n ≥ 1$ and $Xn → X$ in probability. Show that $ X_n → X $a.s.. Can someone help me ...
1
vote
0answers
32 views

Fix point theorem for measures? metric on space of measures?

I have the following problem: I consider a probability space $(\Omega, \mathcal{F}, \mu)$ where $\Omega= C_0([0,1])$ (space of continuous functions on $[0,1]$ starting from 0), $\mathcal{F}$ is a ...
0
votes
1answer
20 views

relationship between $L^p$ convergence and a.s. convergence.

$X_n$ are r.v. which convergent to $X$ a.s.(1) and $\sup_n\mathbb E[|X_n|]<\infty$ (2) there is a counterexample that $X_n$ do not convergent to $X$ in $L^1$:$n\chi_{[0,\frac{1}{n}]}$ But when ...
1
vote
0answers
16 views

Uniform Convergence and exponential inequality to demonstrate Stirling's formula

If $0<R\leq\sqrt n$ and $Z_{k} \sim exp(1)$ are independent and identically distributed as prove that $$P\left[\left|\dfrac{Z_{1}+Z_{2}+\cdots +Z_{n} -n}{\sqrt n }\right|\leq R\right] \geq 1- ...
0
votes
0answers
22 views

Integrability in conditional expectation.

Suppose $X$ is a random variable in $(\Omega,\mathcal F)$.$\mathbb E\left(|X|\right)<\infty$. $Y=\mathbb E[X|\mathcal F_0]$ ,here $\mathcal F_0\subset\mathcal F$. Then I want to show $Y$ is ...
3
votes
2answers
151 views

If you have two envelopes, and …

Suppose you're given two envelopes. Both envelopes have money in them, and you're told that one envelope has twice as much money as the other. Suppose you pick one of the envelopes. Should you switch ...
0
votes
0answers
23 views

Convergence in probability (in measure) of a weighted sum of random functions

Consider a mesh of points $\pi_{n} = (t_{1n},\ldots,t_{K_{n}{n}})$ with $0 < t_{1n} < \ldots < t_{K_{n}n} < 1$ and weights $(w_{1n},\ldots,w_{K_{n}n})$ such that ...
4
votes
1answer
46 views

Slowly varying function without limit at infinity

A function $f:\mathbb R \to \mathbb R$ is slowly varying at infinity if for any $t>0$ $$ \lim_{x\to +\infty}\frac{f(xt)}{f(x)}=1. $$ Is there a bounded function slowly varying at infinity whose ...
2
votes
1answer
81 views

About strong law of large numbers

I came across a problem: Let $X_1,X_2,...$ be independent random variables with uniform(0,1) distribution from probability space $(\Omega,\cal F,P)$. Prove that: ...
0
votes
1answer
35 views

the conditions for a measurable function to be the uniform limit of simple functions

In our homework we are asked to prove that, on a measurable space $(\Omega,\mathcal{F})$, every function $f:\Omega \rightarrow R, f\geq 0$ can be written as the uniform limit of an increasing limit of ...
1
vote
1answer
41 views

Normal Distribution and Iterated Logarithm

Let $X_n$ be independent $N(0, \sigma^2)$-distributed random variables with partial sum $S_n := \sum_{k=1}^n X_k$, $n \geq 1$. Then I read the following results. $$ \sum_{k = 1}^n \mathbb P (S_n > ...
0
votes
0answers
27 views

cantor staircase function uniform distribution on cantor set

suppose Cantor staircase function $F$ is extended to have $F(a)=0$ for $a<0$ and $F(a)=1$ for $a>1$. Then how can one show that $F$ is the cumulative distribution function of the uniform ...
3
votes
1answer
45 views

Construct independent $X_n$ such that $\sum_{n=1}^\infty Var(X_n)=\infty$

How to construct an independent random variable $(X_n)_{n\in\mathbb{N}}$ such that $\sum_{n=1}^\infty X_n$ converges and $\mathrm{Var}(X_n)$ is uniformly bounded by some constant C, but ...
1
vote
1answer
63 views

asymptotics from Laplace transform

Suppose I know that a non-negative random variable with density $f$ has the following Laplace transform: $$\hat{f}(s)=\int_0^{\infty}e^{-st}f(t)dt=\frac{1}{\cosh(\sqrt{2s}x)}$$ where $s>0$ and ...
1
vote
1answer
37 views

Convergence of the expectation of a non-continuous function

Suppose that $F_{n}$ converges to $F$ weakly, where $F$ is a continuous distribution function. Also, suppose that $g$ is a bounded, continuous function and $\{x_{n}\}$ is a real-valued sequence of ...
1
vote
1answer
34 views

inverse Laplace transform by integral

I've seen this formula for the inverse Laplace transform in several books: $$f(t)=\mathcal{L}^{-1}\{F\}(t)=\frac{1}{2\pi i}\lim_{T\to\infty}\int_{\alpha -iT}^{\alpha +iT}e^{st}F(s)ds$$ where $f$ is ...
2
votes
1answer
37 views

How could I recreate the proof of the Dominated Convergence Theorem?

I saw a proof of the Dominated Convergence Theorem that goes like this: If $X_n \to X$, $|X_n| \le Y $, and $E[Y] < \infty$, prove that $E[X_n] \to E[X]$. First, define $Z_n = X_n + Y$. Then, ...
1
vote
2answers
37 views

Need help with some equivalent statements of measurability [duplicate]

I want to know why the above statements are true. Thank you!
1
vote
1answer
26 views

Every measure of natural numbers and the power of natural numbers as their sigma algebra looks like this…

Let X= $ \mathbb{N} $ ans S= P($ \mathbb{N} $) . Prove that every measure $\mu $ in $(X,\mathcal S)$ can be obtained by an unique non-negative extended sequence of real numbers $(a_{n})$ as follows ...
0
votes
0answers
45 views

Dependent Expectation in Random Numbers Illustrated by Prime Repetition in Pi

When approximating Pi, appending each numerical digit as you refine, what is the first repetition of a four-digit prime number? For instance the first repetition of any one-digit number in the ...
0
votes
0answers
31 views

Problems with convergence in mean

I hope you can help me with the following problem Let $\{ e_i : i\in \mathbb{Z}\}$ be and independent U.I. sequence of scalar random variables with zero mean. Let $\{ A_j : j \geq 0\}$ be a sequence ...
1
vote
0answers
22 views

Cesaro means converges in first mean to 0.

I would appreciate any suggestions to prove the following statement If $\{ X_i: i=1,..\}$ is a sequence of independent uniformly integrable (U.I) random variables ...
3
votes
1answer
54 views

My understanding of “$\sigma-$algebra represents information”.

In stochastic process $\{X_t\}_{t\ge0}$ adapted to $\{\mathcal F_t\}_{t\ge0}$ where $\mathcal F_s\subset\mathcal F_t,\forall s<t$. Many textbook say that $\{\mathcal F_t\}_{t\ge0}$ represents a ...
0
votes
1answer
19 views

Expected Value of a Minimum Function using a Beta Distribution

Let $X$ be a IID random variable with support in $[0,1]$ and CDF given by a Beta distribution, i.e. $X \sim Beta(\alpha,1)$. Suppose we have a function of the form: $$ Z_t = \phi(X_t,y_{t-1}) = ...
2
votes
1answer
96 views

Convergence in probability and convergence of $\sup_{n\geq m}E(X_n\mid\mathcal{F_m}).$

Suppose $X_n$ is adapted to the filtration $(\mathcal{F}_n)_{n=1,2,\ldots}$, $X_n$ is positive and bounded above by some fixed $M$. If $X_n$ converges to a constant $x$ in probability, does this imply ...
0
votes
1answer
24 views

Divide a space into disjoint sets which has a small measure.

triple $(\Omega,\mathcal F,\mathbb P)$ ,$\mathbb P$ is a finite measure. I have seen a statement in a textbook : "$\forall \epsilon<\mathbb P(\Omega)$,we can divide $\Omega$ into finite number of ...
0
votes
0answers
68 views

Finding test of critical region for sum/variance of normal distributions

Let $Y_1,....,Y_n$ denote independent, identically distributed random variables such that $Y_1$ has a normal distribution with mean $\theta$ and standard deviations $\theta$, where $\theta$ > 0. ...
2
votes
1answer
106 views

Equality about limsup.

Suppose $\sum_{n=1}^\infty \mathbb P(A_n)=\infty$,then: $$\limsup_{n\to\infty}\frac{(\sum_{k=1}^n \mathbb P(A_k))^2}{\sum_{i,k=1}^n\mathbb P(A_i\cap A_k)}=\limsup_{n\to\infty}\frac{\sum_{1\le ...
2
votes
1answer
67 views

How to understand the exchangeable $\sigma$-algebra?

Suppose there are $(\Omega,\mathcal F,\mathbb P)$ and r.v. $\xi_i$(i$\ge$1) $\xi_i:(\Omega,\mathcal F,\mathbb P)\to(\mathbb R,\mathcal B,\mu)$ $A\in$ the exchangeable $\sigma$-algebra $\mathcal E ...
1
vote
1answer
26 views

Adapted and backward adapted?

I understand the following: Consider a probability space $(\Omega, \mathcal{A},P)$ and a Brownian motion $B=\{B_t, t\in [0,1]\}$ on this space and denote $\mathcal{F}:=(\mathcal{F}_t)_{t\in [0,1]}$ ...
2
votes
1answer
56 views

Understanding an application of Fubini's theorem

I'm going over some lecture notes for a course in statistical theory. There is a "proof" that the density for a $k$-dimensional multivariate normal random variable (with non-singular covariance) is, ...
1
vote
1answer
44 views

AN application of Schwarz inequality.

In the proof of Chung-Erd$\ddot{o}$s inequality: Let $X_k=1_{A_k}$,then: $$(\mathbb E(\sum_{i=1}^nX_k))^2\le\mathbb P(\sum_{i=1}^nX_k\gt0)\mathbb E[(\sum_{i=1}^nX_k)^2]$$ The textbook said this ...
2
votes
1answer
20 views

Existence of an exponential double integral (for the probabilists: Are the $L^p$-norms of Brownian local time integrable in the space variable?)

I have encountered the following integral and, with a lot of handwaving and some identities for Gaussian integrals (see for example ...
1
vote
0answers
37 views

Bernoulli measure

Does anyone know an elementary proof (or somewhere I can find it) of the construction of Bernoulli measure on the set of infinite binary sequences? I am having trouble to show that the measure defined ...
1
vote
1answer
40 views

Tails sets are Borel

I am trying to proof a particular case of Kolmogorov's law in the set E of infinite binary sequences. Eventually, I'm supposed to prove that a certain type of subsets of this set is in the Borel sigma ...
0
votes
1answer
60 views

A Borel-Cantelli lemma exercise.

Suppose ${A_n}$ is a sequence of events. If $P(A_n)\to 1$ as $n\to\infty$,prove there exists a subsequence ${n_k}$ tending to infinity such that $$P(\cap_kA_{n_k})>0$$ The textbook gives a hint ...
2
votes
1answer
32 views

Can we find two measures $\nu$, $\mu$ which $\nu\ll\mu$ and $\mu$ is $\sigma$-finite while $\nu$ is not $\sigma$-finite?

Can we find two measures $\nu$, $\mu$ which $\nu\ll\mu$ and $\mu$ is $\sigma$-finite while $\nu$ is not $\sigma$-finite? I want to justify the Radon-Nikodym theorem but couldn't find an example.
2
votes
1answer
48 views

A problem on verify conditional expectation

Suppose X and Y are independent.Let $\varphi $ be a function with $E(|\varphi(X,Y)|)< \infty$ and let $g(x)=E(\varphi(x,Y))$.The conclusion is $E(\varphi(X,Y)|X)=g(X)$ So the first step is to ...
0
votes
1answer
26 views

A problem about indefinite integral in measure theory

tirple$(\Omega,\mathcal{A},P)$ Suppose $\xi$ is a random variable.Indefinite integral$$\varphi(B)=\int_B\xi\mathbb{d}P \quad\forall B\in\mathcal{A}$$ I saw in a textbook: If $E(\xi)$ exists(not ...
2
votes
1answer
81 views

A possible incorrect application of Law of Large numbers

A friend left this teaser for me. He asked me to first compute: $$ \lim_{n \to \infty} \frac{\binom{2n}{n}}{2^{2n}}$$ Using Stirling's approximation (and another method), I got the answer as $0$. ...
0
votes
1answer
61 views

A problem about Borel-Cantelli lemma

I am doing exercise in a textbook and I got confused with these two problems: Fisrt, in problem 21,I noticed that "there is a subsequence $\{n_k\} $ tending to infinity s.t. ...
0
votes
2answers
47 views

If probability density functions converge a.e., then cumulative density functions converge

I have read a conclusion in a textbook: Suppose $f_n,f$ are density functions of some r.v. also $f_n\to f$ a.e., then $$\int f_n \mathrm{d}x \to\int f \mathrm{d}x $$ Fisrt I want to use ...
1
vote
1answer
39 views

Bolzano–Weierstrass theorem for random variables?

I am wondering if there is something similar to the Bolzano–Weierstrass theorem for random sequences. Namely, let $\{x_n\}$ be a bounded random sequence. Is it true that, under some reasonable ...
2
votes
1answer
71 views

Conditional probability explained?

Let $F_A$ be the CDF to the random variable $A$ ( and $B$ another independet rv), how do we get that $P(A+B \le s) = \int_{\mathbb{R}} P(A+B \le s\mid A=x ) \, dF_A(x)$ (This is probably a ...
2
votes
1answer
40 views

Prove $X_n \xrightarrow P 0$ as $n \rightarrow \infty$ iff $\lim_{n \to \infty} E(\frac{|X_n|}{|X_n|+1} )= 0$

Let $X_1, X_2, ...$ be a sequence of real-valued random variables. Prove $X_n \xrightarrow P 0$ as $n \rightarrow \infty$ iff $\lim_{n \to \infty} E(\frac{|X_n|}{|X_n|+1} )= 0$ Attempt: Suppose ...