Use this tag only if your question is about the modern theoretical footing for probability, for example probability spaces, random variables, law of large numbers, and central limit theorems. Use [tag:probability] instead for specific problems and explicit computations. Use ...

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2
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1answer
28 views

limit of gaussian process

If I have a sequence of gaussian random process $X_{t}^{n}$ which converge in $L^2$ norm to a process $X_t$ for every $t$. can I say that $X_t$ is also gaussian process? thank you
2
votes
1answer
65 views

When do we say independence is the probability of intersection being equal to the product of probabilities? [duplicate]

This is something I never really got in either Elementary Probability Theory or Advanced Probability Theory because my professors mainly discussed independence between 2 objects. Please tell me if my ...
7
votes
0answers
79 views

A matrix with a dense submatrix - application of Chernoff’s Inequality

I am trying to solve an exercise from this book, which I will post here for convenience. I have a bit of a problem understanding how the hint of using Chernoff's bound implies the claim. Specifically ...
-1
votes
1answer
73 views

Proof of Itō's lemma for the Brownian motion

Let $(\Omega,\mathcal A,\operatorname P)$ be a probability space $B=(B_t)_{t\ge 0}$ be a Brownian motion on $(\Omega,\mathcal A,\operatorname P)$ $\mathcal P$ be a sequence of countable subsets ...
2
votes
0answers
33 views

Integration bounds with $P(XY < z)$

In the following problem, why are the bounds of integration what they are? I think $x$ should be integrated between $0$ and $\infty$. If $X$ is greater than $z$ it's OK as long as $y < \frac z x$. ...
2
votes
0answers
24 views

Stochastic order of the minimum of a stopped sequence of random times

Let $S_k$ and $S'_k$ be two sequences of independent random times such that for each $k \in \mathbb{N}$ it holds that $S'_k \leq_{st} S_k$, that is $S'_k$ is stochastically smaller than $S_k$ which ...
1
vote
1answer
39 views

$(X_n)_{n\in\mathbb{N}}$ independent Cauchy-distributed random variables. Convergence of $n^{-\gamma}(X_1+\cdots+X_n)$

I want to solve the following exercise but i am unsure if my ideas are correct or not. Let $(X_n)_{n\in\mathbb{N}}$ be i.i.d. random variables with probability density $$ ...
4
votes
1answer
48 views

Arc Length of largest arc when $n$ points are chosen at random on the circumference of the unit circle.

I am currently doing a little self-study of A Probability Path by Sidney Resnick, and am having trouble with the following problem: Points are chosen at random on the circumference of the unit ...
0
votes
2answers
36 views

Given probabilities of a horse beating each other horse. What is the probability that the horse will finish in a particular position? [duplicate]

I have computed probabilities that horse A beats horse B, horse A beats horse C and so on. I want to find out the probability that the horse will finish in a particular position, say 2nd. Edit: In a ...
1
vote
1answer
45 views

Probability density function of a ratio with summation?

Let $X_i$ be a random variable that is exponentially distributed with parameter $\lambda$ for all $i\in\{1,\dotsc, n\}$ for some $n$. Let $Y_i$ for all $i$ be the random variable defined by: ...
2
votes
0answers
41 views

Unique stationary distribution (or measure?) of a Markov Chain

Let $(X_n)_{n \geq 0}$ be a irreducible, positive recurrent Markov chain. We have a theorem that states that the unique stationary distribution is then given by $$\pi(x)= \frac{1}{E_x[H_x]},$$ where ...
0
votes
1answer
14 views

MCMC( Markov Chain Monte-Carlo Simulation )

Suppose $Q=[q(i,j)]$ is a transitional probability matrix for an irreducible Markov chain. Suppose also $\lbrace X_{n}, n \geq 0 \rbrace$ is Markov Chain such that if $ X_{n}=i$, generate $Y=j$ such ...
0
votes
0answers
29 views

What is the proof of $\mathbb{E}\Phi (X) = \Phi\left(\mu\sqrt{\frac{1}{1+\sigma^2}}\right)$, where $X \sim \mathcal{N}(\mu,\sigma^2)$?

Let $X \sim \mathcal{N}(\mu,\sigma^2)$. I think it's true that $$\mathbb E \Phi(X) = \Phi \left(\mu\sqrt{\frac{1}{1+\sigma^2}}\right)$$ where $\Phi$ is the cdf of standard normal. This holds up ...
1
vote
1answer
34 views

$E[|X(t)|]\leq K\implies E[|X(\tau)|]\leq K $?

Let $X(t)$ be a stochastic process. Assume that, for every $t\leq M$, it holds $$E[|X(t)|]\leq K, $$ for some constant $K$. Let now $\tau\in[0,M]$ be random (stopping time). Is it true that also ...
2
votes
3answers
49 views

The limit of the integral when the set is decreasing in probability to zero.

This is an exercise problem(#2 in section 3.2) from 'A course in probability theory'. If $E(\vert X \vert ) < \infty$ and $\lim_{n \to \infty} P(A_n) = 0,$ then $\lim_{n \to \infty} \int_{A_n} X \ ...
0
votes
1answer
22 views

How does the probability change when changing the decission?

Let's say you play roulette and you set 2 coins of 2 of the 3 columns or dozens. Then your chance is 2/3rd, so about 66% to win one coin. Let's assume that you loose because the other dozen or column ...
1
vote
1answer
29 views

Necessary and sufficient condition for $X_n/n\rightarrow0$ almost surely when $(X_n)$ is i.i.d.

Let $X_n$ be a i.i.d. sequence of random variables. My question is how to find a necessary and sufficient condition for $X_n/n\rightarrow0$ a.s. First, I assume that $EX_i^p<\infty$, $p>1$. ...
1
vote
2answers
37 views

Sum of Two Continuous Random Variables

Let $X$ and $Y$ be two random variables with common PDF. $f_X(x)=f_Y(y)=e^{-x}$, for all $x, y > 0$ I have set up the integral according to the formula for convolution for two continuous random ...
6
votes
0answers
95 views

Upper and Lower Bounds on $Var(Var(X\mid Y))$

Are there any particular properties that \begin{align*} Var(Var(X\mid Y)) \end{align*} satisfies so that we can derive any upper and lower bounds on it. For example, if we replace $Var$ with ...
3
votes
1answer
21 views

Equivalent definitions of reflecting brownian motion

In what follows, $D$ is a precompact subset of $\mathbb{R}^n$ with say $C^3$ boundary, $\eta(a)$ is the inward unit normal vector to the boundary at $a$.I'm trying to prove the equivalence of the ...
1
vote
0answers
18 views

L1 and L2 regularization and L1 and L2 space

I am looking to characterize the difference of the function obtained in the solution process of $L^1$ and $L^2$ regularization. It is known that $L^1$ regularization gives sparse solutions. In $L^2$ ...
1
vote
1answer
16 views

Bayes' Rule for Parameter Estimation - Parameters are Random Variables?

Let $(\Omega, \mathcal{F}, P)$ be a probability space and let $\mathbf{X}: \Omega \to \mathbb{R}^n$, $\mathbf{Y}: \Omega \to \mathbb{R}^m$ be jointly continuous random vectors. That is, there exists ...
0
votes
1answer
20 views

A question involving Markov processes

Let $(S, \mathcal{B}, m)$ be a measurable space and $X_p := L^p(S, \mathcal{B}, m)$. Let $T_t \in \mathcal{L}(X_p, X_p)$ be a bounded linear operator defined by $$(T_t f)(x) = \int\limits_S P(t, x, ...
1
vote
1answer
22 views

A question about Chapman-Kolmogorov equation

I'm reading ''Functional Analysis'' - K. Yosida and at page 379 there is the following claim "The hypothesis that the particle has no memory of the past implies that the transition probability P ...
0
votes
0answers
19 views

Convergence in laws versus convergence in distribution

For rvs $X_{n}$ in metric space S, convergence in laws means $\int f(X_{n})dP\to \int f(X)dP$ for $f\in C_{b}(S)$ whereas in distribution means $F_{X_{n}}\to F_{X}$. In Dudley page 296, for example ...
0
votes
0answers
25 views

The Itō integral $\sum_{i=1}^nH_{t_{i-1}}\left(B_{t_i}-B_{t_{i-1}}\right)$ of an simple process $H$ is independent of the choice of $(t_0,\ldots,t_n)$

Let $(\Omega,\mathcal A,\operatorname P)$ be a probability space $\mathbb F=(\mathcal F)_{t\ge 0}$ be a filtration on $(\Omega,\mathcal A)$ $B=(B_t)_{t\ge 0}$ be an $\mathbb F$-adapted Brownian ...
1
vote
1answer
26 views

is subset of probability measures with finite second moment a Borel set?

We know that the space of the probability measures on $\mathbb{R}^n$ endowed with the topology of weak convergence is a Borel space. My question is that the subset of those that have finite second ...
1
vote
2answers
36 views

Intuition of joint density of min(X,Y) and max(X,Y)

The problem is to find the joint density of $U = min(X,Y)$ and $V=max(X,Y)$ when both are exponential random variables. The solution to it is: I can finish it after the first step but I don't ...
0
votes
1answer
61 views

Sum of Two Continuous Random Variables

Consider two independent random variables $X$ and $Y$. Let $$f_X(x) = \begin{cases} 1 − x/2, & \text{if $0\le x\le 2$} \\ 0, & \text{otherwise} \end{cases}$$.Let $$f_Y(y) = \begin{cases} ...
7
votes
2answers
89 views

When is the union of $\sigma$-algebras atomless?

Suppose that we are given a probability space $(\Omega, \mathcal{F}, \mathsf P)$ and an increasing sequence of $$\mathcal{F}_1\subset \ldots \subset\mathcal{F}_n\subset \mathcal{F}_{n+1} \subset ...
3
votes
0answers
27 views

Computing an Ito Integral using the Definition

Let $B_t$ be a brownian motion adapted to $\mathcal F_t$. For general $\mathcal F_t$-adapted processes $X_t$ the Ito-integral could be defined as $$ \int_0^t X_s dB_s = \lim_{n\to \infty} \int_0^t ...
0
votes
1answer
34 views

Quadratic variation of the Brownian motion and Itō's lemma

Let $(\Omega,\mathcal A,\operatorname P)$ be a probability space $B=(B_t)_{t\ge 0}$ be a Brownian motion on $(\Omega,\mathcal A,\operatorname P)$ $\mathcal P$ be a sequence of countable subsets ...
0
votes
1answer
25 views

binomial (undefined function)

I’m trying to answer this question using binomial pmf. 1-i have a shelf contains 4 ( 500 pages book) what is the probability that when i select 1 book i get a book consists at least of 200 pages ? ...
0
votes
1answer
49 views

How to calculate sampling error?

Given a reservoir of size $S$ with each element taking a value of error or not an error, we attempt to estimate the number of errors inside the reservoir through the following We poll the reservoir ...
0
votes
2answers
31 views

Is random vector a vector of observations or list of random variables.

I am stuck on a very basic concept in probabilities. The question goes like this. Is a vector of observations of a random variable (for example a series of coin toss states) the same as a random ...
1
vote
0answers
25 views

Mutual information as a fraction of entropy?

Suppose I have two (discrete) random variables $(X,Y)$ with some joint distribution $P$. The mutual information $I(X;Y)$ is informally defined as the reduction is the remaining entropy in $X$ once the ...
7
votes
1answer
106 views

need reference for fact about products of measures

Need a reference (textbook or paper) for the following (probably well known) fact: Suppose $(X,M)$ is a measurable space and $\lambda$, $\nu$ are two different probability measures on $(X,M)$. Write ...
1
vote
3answers
28 views

Independence of intersections

Let $(\Omega,\mathfrak{A},P)$ be a probability space and $A,B,C\in\mathfrak{A}$ some events where $A$ and $B$ are independent. I'm a bit confused now as I intuitively think that this also implies ...
3
votes
1answer
36 views

Stochastic process $\exp(W_t - t/2)$ approaches zero for large $t$, but it is a martingale?

The stochastic process $$ S_t = \exp\left( W_t - \frac{1}{2} t \right) $$ is a martingale (for example this could be seen by noting that it solves the SDE $dS_t = S_t dB_t$, which has no drift). But ...
3
votes
1answer
40 views

Why is the solution of a stochastic differential equation wrt the Brownian motion suitable for a model of a disturbed time continuous process

Let $(\Omega,\mathcal A,\operatorname P)$ be a probability space $\mathbb F=(\mathcal F_t)_{t\ge 0}$ be a filtration on $(\Omega,\mathcal A)$ $B=(B_t)_{t\ge 0}$ be a Brownian motion on ...
1
vote
1answer
44 views

example computing expectation

I am trying to understand the following example: A fair die is rolled, and whichever number comes up, a fair coin is then flipped that many times. Let $N$ be the outcome of the die roll, and $X$ the ...
1
vote
0answers
14 views

Convergence in “the” Skorohod topology for monotone functions

Let $x_n$, $x$ be nondecreasing cadlag functions on $[0,T]$. To get $x_n\to x$ in Skorokhods $M_1$-topology, we only have to prove convergence on a dense subset of $[0,T]$ including 0. (see Whitt: ...
2
votes
2answers
43 views

Expectation of the maximum as the number of random variables goes to infinity?

Suppose $v_1,v_2,...,v_n$ are $n$ i.i.d. continuous random variables with the range $[\underline v,\bar v]$, does the expectation of the maximum of the random variables $E[\max v_i, i=1,2,\dots,n]$ go ...
0
votes
2answers
41 views

Conditional Probability $P(B| A\cap B) = 1$?

$P(B|A\cap B) = 1$? I'm a little confused about the probability of $B$ occurring given that $B\cap A$ occurred. If $B\cap A$ happens, does this guarantee $B$'s occurrence or must we consider the ...
1
vote
1answer
53 views

Itô integral with respect to a diffusion

Let $(\Omega,\mathcal A,\operatorname P)$ be a probability space $\mathbb F=(\mathcal F)_{t\ge 0}$ be a filtration on $(\Omega,\mathcal A)$ $B=(B_t)_{t\ge 0}$ be an $\mathbb F$-adapted Brownian ...
1
vote
1answer
60 views

How does one define the Fourier transform of a probability distribution?

Say $p_X$ and $p_Y$ are two probability distributions on a $m$ element set. Then I see an equality written as, $$\sqrt{m} \vert \vert p_X - p_Y \vert \vert _2 = \sqrt{ \sum_{k=0}^{m-1} \vert ...
0
votes
1answer
25 views

Equivalent definitions for expected value of random variable

Let $(\Omega, P)$ be a probability space. One definition for the expected value of a random variable $X$ is $$E(X)=\sum_{x\in \mathbb{R}} xP(X=x).$$ The notes I am reading say that this definition ...
1
vote
1answer
62 views

Proof that random variable is almost surely constant

If a random variable $X : \Omega \to \mathbb R$ is $\{ \emptyset, \Omega \}$-measurable, then it is constant. I want to generalise this result: Now if $\mathcal G$ is a $\sigma$-algebra such that ...
-2
votes
3answers
45 views

Book to learn Mathematical Probability theory? [closed]

What are some good references to , good book to learn Mathematical Probability theory ? Please help .
4
votes
1answer
27 views

Itô symmetry for elementary predictable stochastic processes

Let $(\Omega,\mathcal A,\operatorname P)$ be a probability space $\mathbb F=(\mathcal F)_{t\ge 0}$ be a filtration on $(\Omega,\mathcal A)$ $B=(B_t)_{t\ge 0}$ be an $\mathbb F$-adapted Brownian ...