0
votes
1answer
32 views

How to prove the inequality using Jensen's inequlaity?

How to prove the above inequality? I am learning probability by myself and it has been confusing me for days. Thanks!
1
vote
3answers
21 views

How to prove the above expectation inequality?

If $\mathbb{E}[|X|^k]<\infty$ then for $0<j<k$, $\mathbb{E}[|X|^j]<\infty$, and furthermore $\mathbb{E}[|X|^j]\leq(\mathbb{E}[|X|^k])^{j/k}.$ How to prove the above expectation ...
1
vote
1answer
11 views

Order between probability measures: sets full below

Consider a product space $X = \{0,1\}^\mathbb{Z}$ and the space of probability measures on $X$, $\mathcal{M}(X)$. We say that for any two $a, b \in X$, $$a \prec b \iff a_x \leq b_x \, \, \, \, \, ...
0
votes
1answer
16 views

How to do you compute the probability a record occurring in a sequence of independent experiments?

Consider a sequence of independent experiments, each of which produces a random integer in N with the probability mass function ${p_k}$. The pmf is the same for all the experiments and also strictly ...
2
votes
4answers
419 views

What does it really mean when we say that the probability of something is zero? [duplicate]

Conventionally, people will say a probability of zero is equivalent as saying that the event is impossible. But when we look at the probability from a mathematics perspective, probability is defined ...
0
votes
1answer
15 views

Soft: Interpretation of a periodic event on circle group

Recently I've been exploring probability measures on topological groups, derived from the (essentially) unique Haar measure defined thereon. I had begun to focus on the example of the circle group ...
0
votes
1answer
19 views

Def. of total probability on wiki

As following: I am confused about "Bn is measurable". Bn is measurable means Bn is a sigmal-algebra. According to the definition of sigma-algebra: Bn must contain whole sample space, empty space; ...
1
vote
0answers
12 views

Can the theorem on method of a single probability measure be extended to general metric space?

I don't know whether or not there is a name for the following theorem, so I just write it down: Let $(E,\mathscr{E},\rho)$ be separable metric space, and random elements $X$,$X_n$, $n \geq 1$ ...
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 ...
3
votes
0answers
31 views

Joint distribution expressed with conditional distribution.

Let $(\Omega,\mathfrak{A},\mathbb{P})$ be a probability space, $(\Omega',\mathfrak{A}')$, $(\Omega'',\mathfrak{A}'')$ two measure spaces and $$X\colon(\Omega,\mathfrak{A},\mathbb{P})\rightarrow ...
1
vote
2answers
69 views

More general definition of expected value

Let $X$ be a random variable with pdf $f$. I would like to know why: $$\operatorname{E} [X] = \int_\Omega X \, \mathrm{d}P = \int_\Omega X(\omega) P(\mathrm{d}\omega)= \int_{-\infty}^\infty x f(x)\, ...
1
vote
1answer
42 views

Problem about $\sigma$-algebra

Space $\Omega$, $\mathcal C$ is a algebra,$\mathcal F=\sigma(\mathcal C)$ is a $\sigma$-algebra. define:$\mathcal F_\omega=\{B\in\mathcal F|\omega\in B\}$,$\mathcal C_\omega=\{B\in\mathcal ...
1
vote
1answer
33 views

An inequality about signed measure.

Suppose $\mu$ is signed measure,then: $$|\mu(A)|\le\epsilon\Rightarrow|\mu|(A)\le2\epsilon$$ I tried to use the Jordan composition of $\mu$: $$\mu^+(C)=\mu(C\cap D),\mu^-(C)=-\mu(C\cap D^c)$$ so ...
2
votes
1answer
98 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
66 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
24 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 $\mathcal{F}:=(\mathcal{F}_t)_{t\in [0,1]}$ ...
3
votes
1answer
74 views

What does $\mathbb{P}(d\omega)=dw$ actually mean?

I am currently reading S. Shreve's book Stochastic Calculus II, and I have a question regarding Example 1.6.4 (p.35-36) which describes a change of measure, but I am puzzled by the notation. ...
2
votes
1answer
19 views

Kernel density estimation in the limit of infinity many samples

Let ($x_1, ..., x_n$) be i.i.d. samples drawn from some distribution $P$ with an unknown probability density function $f$. Its kernel density estimator is \begin{align} \hat{f}_h(x) = ...
1
vote
0answers
10 views

Ito integrals and joint distribution with copulas

Let $X_{t}$ and $Y_{t}$ be two brownian motions and let their joint distribution be given by $F$. So in regularly correlated BM's where $dX_{t}dY_{t}=\rho dt$, we have a bivariate normal distribution ...
1
vote
0answers
33 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
37 views

Meaning of $P(Y|X=x)$

Suppose that $X$ and $Y$ are two random variables on $(\Omega, \mathcal H, P)$ with values in $(\mathbb R,\mathcal B_{\mathbb R})$. I want to understand what is "formally" the expression $P(Y|X=x)$ ...
1
vote
1answer
39 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
28 views

About a $\sigma$-finite measure

Consider a probability space $(\Omega,\mathcal H,P)$ and a real random variable $X$ such that $E(X)$ is well defined (also infinite values are allowed). Is it true that the measure ...
0
votes
1answer
33 views

Is the set $\limsup\limits_n\{\frac{X_n}{\log(n)}>\frac{1}{\lambda}\}$ equal to $\{\limsup\limits_n\frac{X_n}{\log(n)}>\frac{1}{\lambda}\}$?

Difference between $\limsup\limits_n\{\frac{X_n}{\log(n)}>\frac{1}{\lambda}\}$ and $\{\limsup\limits_n\frac{X_n}{\log(n)}>\frac{1}{\lambda}\}$ are the sets equal ? I think they would ...
2
votes
3answers
124 views

Probability of events in an infinite, independent coin-toss space

I am studying Steven E. Shreve's Stochastic Calculus book. Example 1.1.4 (p.4-6) constructs a probability measure on the space of infinely many coin tosses $\Omega_\infty$. In the example the ...
2
votes
1answer
47 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
20 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 ...
0
votes
2answers
28 views

A proposition about positive random variables and expected values

I have problems to give a proof for the following proposition: Consider a random variable $X$ with values in $[0,+\infty]$. If $P(X=+\infty)>0$, then $E(X)=+\infty$ (notation: $E(X)=\int X ...
4
votes
2answers
146 views

A counter example of Brownian Motion

Here is an example in my textbook to illustrate why we need the continuous sample path in the definition of Brownian motion. Let $(B_t)$ be a Brownian motion and $U$ be a uniform random variable on ...
2
votes
1answer
23 views

Filtration from a Brownian Motion

The textbook I am reading defines the filtration induced from a Brownian Motion as follows. Let $\{B(t): t \geq 0\}$ be a Brownian Motion defined on some probability space, then we can define a ...
1
vote
1answer
15 views

Limit distribution is invariant

Consider a homogeneous Markov chain $\{X_n\}_{n\in\mathbb N}$ with a countable (but not necessarily finite) state space $S$. Suppose that there exists a limit distribution $\pi$, namely: ...
2
votes
1answer
60 views

Relation between uniformly distributed random variable and i.i.d Bernoulli sequence (Cantor space)

(Uniform RV <==> i.i.d Bernoulli sequence) (1) Let $(X_n)_n$ be a sequence of i.i.d. Bernoulli random variables($P(X_n=0)=P(X_n=1)=\frac 12$) on a probability space. Then show that $\xi:= \sum_n ...
0
votes
0answers
41 views

Sum of independent discrete random variable

Here is my attempt of deriving the sum of independent random variable in the discrete case : $\underline{\textbf{Sum of independent random variables}}$ Let $\mathcal{C_1}, \mathcal{C_2}$ be ...
2
votes
1answer
70 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 ...
1
vote
1answer
107 views

Real valued random variables and cumulative distribution functions (c.d.f.)

Let $X$ be a random variable with values in $\mathbb R$ (we fix the Lebesgue measure on $\mathbb R$), then is well defined a c.d.f. $F_X$ such that $$F_X(x)=X_\ast P(]-\infty,x])=P(X\in]-\infty,x])$$ ...
0
votes
1answer
51 views

An example of stochastic process

I use the following definition for a stochastic process. Let $(\Omega, \mathcal F, P)$ be a probability space, $(E, \mathcal E)$ be a measurable space, and $T$ be a non-empty set. A collection ...
0
votes
1answer
45 views

A field being a sigma field if and only if it's a monotone class

The exercise is as follows: "The limit of an increasing (or decreasing) sequence An of sets is defined as its union ∪nAn (or the intersection ∩nAn). A monotone class is defined as a class ...
0
votes
0answers
24 views

Distribution of an upper limit of a sum of random variables

Let $\{X_{n,j},n\geqslant1,j\geqslant1\}$ be independent and identically distributed random variables. Denote $S_{n,k}=\sum_{j=1}^kX_{n,j}$. Let $\{Z_n,n\geqslant1\}$ be a sequence of random variables ...
3
votes
1answer
41 views

Help with conditional expectation

I need help finding a conditional expectation: Let $X$ be a $(0,1)$ uniform random variable i.e. $\mathbb{P}(X \in A)=\lambda((0,1)\cap A)$ where $\lambda$ is the Lebuesgue measure. We define the ...
-1
votes
2answers
39 views

Convergence of running maximum of uniform random variables [closed]

Let $X_1, X_2, ... X_n$ be an IID sequence of IID random variables that have a uniform distribution $(0,1)$. Let Max$(n) =$ max$(X_k:1\le k \le n)$, where $n\in \mathbb N$. How do I show that ...
1
vote
1answer
45 views

Intersection of countable many sets of measure $1$

Consider a probability space $(X,\mathscr M,\mu)$ and a collection of measurable sets $\{A_n\}_{n\in\mathbb N}$ such that $\mu (A_n)=1$ for every $n$. Then I don't unterstand the following result: ...
3
votes
1answer
41 views

countably additive function P

This problem comes from exercise 1.3.5(b) of 'A First Look at Rigorous Probability Theory'. It asks to give an example of a countably additive function $P$, defined on all subsets of $[0,1]$, which ...
0
votes
1answer
58 views

Measure extension theorem(unique) [closed]

Please give an example of two probability measures $\mu \not = \nu$ on $\cal{F} $= all subsets of {1, 2, 3, 4} that agree on a collection of sets C with $\sigma(C)=\cal{F}$ . thanks in advance.
1
vote
1answer
31 views

Showing that a set is in terminal $\sigma$-Algebra

I am reading a probability theory book (from Bauer) and I found the following statement in the book that I cant understand: Given a sequence of independent random variables $(X_i)_{i\in\mathbb{N}}$ ...
2
votes
1answer
66 views

Intuition in probability theory

Good afternoon. Could you please suggest me some books or may be articles where I can read about the intuition of Kolmogorov's axiomatics. I know it, I can solve university problems but I can't feel ...
0
votes
1answer
36 views

Measurability and knowledge

there seems to be a subtle relationship between knowledge and measurability. If I have a stochastic process $(X_n)_n$, then for example a stopping time ( other examples would be martingales, ...
2
votes
1answer
30 views

Convergence in total variation

There are the very basic convergence types in probability theory: almost sure, in $L^p$-norm, in measure and in distribution. Besides that there is the concept of convergence in total variation norm. ...
0
votes
1answer
72 views

How to make sense out of this: Ergodic theorem for Markov chains

We had the ergodic theorem for Markov chains, stating that: For a state space $S \subset \mathbb{N}$ and all functions $f \in L^1$ (meaning that $\sum_{s \in S} |f(s)|\pi(s) < \infty$) and an ...
2
votes
1answer
38 views

law of iterated logarithm

Wikipedia claims see this link that the law of the iterated logarithm marks exactly the point, where convergence in probability and convergence almost sure become different. It is apparent from the ...
0
votes
1answer
46 views

Prove that a Modified Cantor Distribution is Atomic.

Consider a measurable space $\{\mathcal{I},\mathcal{B}\}$, where $\mathcal{I} = [0,1]$ and $\mathcal{B}$ are the Borel sets on $\mathcal{I}$. And also, denote $\mathcal{C}$ as the cantor set on ...