Modern theory of probability is formulated on the footing of measure theory. Use this tag if your question is about this theoretical footing (for example probability spaces, random variables, law of large numbers, central limit theorems, and the like). Use (probability) for explicit computation of ...

learn more… | top users | synonyms (1)

3
votes
0answers
9 views

Hitting time process of Brownian motion

I am stuck with this problem: Let $(B_t)$ be a standard Brownian motion in $\mathbb{R}$. For $t \geq 0$, let $$ H_t = \inf \{ s \geq 0 : B_s = t \}, \quad S_t = \inf \{ s \geq 0 : B_s > t \}. $$ ...
1
vote
1answer
4 views

Convergence in probability given that covariance matrix goes to $0$

Suppose I have a sequence of random vectors $\{X_n\}$ each of dimension $2\times 1$. Suppose also that I know $$ ...
4
votes
1answer
21 views

Stopped process of Brownian motion

I am baffled about the following problem: Let $(B_t)$ be a standard Brownian motion. Let $$ \tau:= \inf\{ t \geq 0 :B_t = x \} \wedge \inf\{ t \geq 0 :B_t = -y \}$$ be a stopping time, where $x,y ...
2
votes
1answer
23 views

Rates of convergence of an OLS estimator

I have a linear regression model $$ y_t=x_t\beta+e_t,\quad t=1,\ldots,N. $$ Here $x_t$ is non-random and given by $(1,\delta_t t)$ where $\delta_t$ is 1 for odd $t$ and $0$ otherwise. Moreover, ...
2
votes
0answers
18 views

Reverse diffusion of a gas

I have a question about the Ehrenfest model of gas diffusion. Suppose I have two urns, each one with $N = 1,000,000$ balls in it. To start with, the first urn has only black balls, and the second ...
1
vote
0answers
25 views

Strong law of large numbers for square-integrable and uncorrelated random variables with bounded variance

Let $(\Omega,\mathcal{A},P)$ be a probability space and $(X_n)_{n\in\mathbb{N}}$ be a sequence of square-integrable and uncorrelated random variables $\Omega\to [0,\infty]$ with ...
1
vote
0answers
14 views

Distrubution of the maximum of a sequence of random variables.

Let $\{Xn\}_n$ a sequence of independent random variables with the same distribution. Suppose their mean is 0 and their variance is 1. Consider $$\max_{0\le k \le n} X_k(t)-tX_k(1)$$ How can I ...
1
vote
0answers
10 views

Question about upcrossings of Brownian motion

I am very stuck on this problem: Given a Brownian motion $(B_s)$, write $S_t = \sup_{0 \leq s \leq t} B_s$, for each $t>0$. All stopping times and martingales are considered w.r.t the filtration ...
2
votes
1answer
41 views

Convergence to $N(0,1)$ in distribution

Question is as following: $X\sim Po(\lambda)$ $$\frac{X-\lambda}{\sqrt{\lambda}} \,{\buildrel d \over \rightarrow}\, N(0,1)$$ as $\lambda \rightarrow \infty$. Obs. One is asked not ...
1
vote
0answers
12 views

Conditional expectation involving Brownian Bridge

I have no ideas on this problem: Let $(B_t, 0 \leq t \leq 1)$ be a standard Brownian motion in $1$ dimension. Let $Z^y_t = yt+ (B_t -tB_1)$. We call $\{Z^y_t\}_{0 \leq t \leq 1}$ a Brownian Bridge ...
0
votes
1answer
24 views

Does $a^2P(|X |\ge a )\le EX^2 $ hold when $a<0 $?

That is, does Chebyshev's inequality hold for when $a $ is negative? I have seen some authors to require that $a $ be positive, but when Reading the proof by Rick Durrett, I cannot see that this is ...
2
votes
0answers
14 views

Bounding $\int_{\infty}^{\infty}|g(s)v^3k(v)|dv$ where $k$ is a second-order kernel

Suppose $k$ is a nonnegative, bounded real-valued function that satisfies $$ \int_{-\infty}^\infty k(v)dv=1,\quad k(v)=k(-v),\quad \int_{-\infty}^\infty ...
1
vote
1answer
20 views

Exit time of Brownian motion from the upper half plane

I am currently reading some books on SLE and struggling on some problems regarding Brownian motion. For a Brownian motion in $\mathbb{R}^2$ starting from $(x,y)$, I don't know how to find the ...
0
votes
0answers
26 views

Three state probability where one state has yet to factor results.

I'm currently trying to explain something to someone else using probability to make it simpler to understand. As I have it now there have been 5 examples that have happened. In one case there are 3 ...
3
votes
0answers
20 views

What is the explicit obstruction to the failure of pointwise convergence in the stochastic integral?

Let $B(\omega,t)$ be a Brownian motion defined on some appropriately filtered probability space $(\Omega,\mathcal{F}_{t},\mathbb{P})$, and let $f(\omega,t)$ be a stochastic process defined on $\Omega$ ...
2
votes
1answer
28 views

Determining if some random variable is a stopping time

I am stuck on this issue: Let $(B_t)$ be a Brownian motion. We know that since $\{0\}$ is a closed set in $\mathbb{R}$ and that $(B_t)$ is a continuous adapted process, $$ \tau:= \inf \{ t\geq 0 : ...
0
votes
1answer
13 views

Multinomial distribution: bounding sum of coordinates' deviations from mean

Okay, given a multinomial random vector $$X=\text{Multinom}(n,\;p_{1},\;\dots,\;p_{k}),$$ so that $$X=(X_{1},\;\dots,\;X_{k})\;\;\;\text{with} \;\;\;\sum_{i=1}^{k}X_{i}=n,$$ I'm looking for a bound ...
1
vote
0answers
28 views

Verifying that a certain process is not a Brownian motion

Let $B$ be a standard Brownian motion in $1$ dimension. Define \begin{equation} \tau = \inf \bigg\{ t \geq 0 : B_t = \max_{0 \leq s \leq 1} B_s \bigg\}. \end{equation} We want to show that $(B_{t+ ...
2
votes
1answer
36 views

Prove that $\int k(w)o(h^2w^2)dw=o(h^2)$ for $\int k(w)dw=1$

Suppose that $k$ is nonnegative real-valued function satisfying $$ \int k(w)dw=1,\quad\int wk(w)dw=0,\quad\int w^2k(w)dw=\kappa_2<\infty.\tag{$\star$} $$ (The limits of the integrals are all ...
2
votes
2answers
32 views

Probability in dice, Feller exercise

I am stuck with exercise 2 of Chapter 4 Feller vol 1 "an introduction to probability theory and its application". Here I report the exercise text: Five dice are thrown. Find the probability that at ...
1
vote
2answers
20 views

probability of a brownian motion being equal to the running maximum

Let $B$ be a standard Brownian motion on $\mathbb{R}$. I would like to show that $$ \mathbb{P} \bigg\{ B_1 = \max_{t \in [0,1]} B_t \bigg\} =0 .$$ I argue that since $\max_{t \in [0,1]} B_t $ has the ...
1
vote
3answers
44 views

Where does this conditional probability law come from?

I was trying to follow a computation done in my class notes, and was having difficulty seeing the inspiration for a part of the manipulation in a question regarding probability. I did some Googling, ...
1
vote
1answer
34 views

Measurability of adapted processes

Let $(\Omega, \mathscr{A}, P)$ be a probability space, $(E, \mathscr{E})$ a measurable space and $X_t : \Omega \to E$, $t \geq 0$ a measurable stochastic process, i.e. the map $X : [0, \infty) \times ...
4
votes
1answer
45 views

probability circle determined by chord determined by two random points is enclosed in bigger circle

Two points $A$ and $B$ are chosen uniformly at random from the interior of a circle $X_1$. Let $X_2$ be the circle whose diameter is the segment $AB$. What is the probability that $X_2$ is contained ...
0
votes
1answer
36 views

Definition of $\sigma$-algebra. Axioms.

""Def. A family $\mathcal F$ of subsets of $\Omega$ is said to be a $\sigma$-algebra on $\Omega$ if: (A.1) $\Omega\in\mathcal F$ (A.2) $\ A\in\mathcal F\implies\ A^c\in\mathcal F$ (A.3) $\ ...
9
votes
2answers
84 views

true story about probability? [duplicate]

A women's organization was contemplating suing a famous American university when it learned that the percentage of women who received tenure in the university was smaller than the percentage of men. ...
-2
votes
0answers
23 views

a totally bounded subset of $\mathbb{L}_1(\mathbb{R}^{d})$ and Kolmogorov-Riesz compactness theorem

Let $\lambda$ be the Lebesgue measure on $\mathbb{R}^{d}$ , $\mathcal{F}$ a set of all probability densities $f$ such that $\mathcal{F}$ is a totally bounded subset of $\mathbb{L}_1(\mathbb{R}^{d})$ ...
1
vote
1answer
37 views

What does it mean to say the smallest σ-algebra?

I am just starting out on measure theory. What does it mean to say the smallest σ-algebra?
1
vote
2answers
23 views

Prove that lim sup of a function belongs to a certain sigma algebra

I am so baffled with this problem: Let $B$ be a standard Brownian motion, $\{ \mathcal{F}_t \}$ be the filtration generated by the Brownian motion. I would like to show that for any $k>0$, ...
2
votes
2answers
44 views

A question about sum of n random variables

Let $X_1, \ldots, X_n$ be random variables. We know that $X_1, \ldots, X_n$ are $\sigma(X_1, \ldots, X_n)$ - measurable. But how about $X_1 + \cdots + X_n$? Is it $\sigma(X_1, \ldots, X_n)$ - ...
2
votes
2answers
51 views

What is a non-decreasing sequence of sets?

What is a non-decreasing sequence of sets and how come it can have a limit? It appear in a probability theory book
1
vote
0answers
32 views

Partial Correlation Coefficient

I have the following questions on computing the correlation coefficient. Let us say we have two discrete random variables $X_1$ and $X_2$, where $X_1$ has $n_1$ outcomes and $X_2$ has $n_2$ ...
0
votes
0answers
10 views

Iso-density locus of Gaussian mixture distribution

I would like to known what is the equation of the iso-density locus of a Gaussian mixture distribution. Is such an iso-density locus a union of ellispoids? Let's say that this Gaussian mixture is in ...
1
vote
0answers
17 views

How to calculate convolution of function defining a measure

Given the function $F(t)=2-2e^{-t}$ defining a measure on $(\mathbb{R}_+,\mathfrak{B}(\mathbb{R}_+))$ and I want to calculate the convolution of this function with itself. I tried to do that by using ...
0
votes
2answers
40 views

Does really convergence in distribution or in law implies convergence in PMF or PDF?

Ref :Introduction to Mathematical Statistics-Prentice Hall (1994) by Robert V. Hogg, Allen Craig. Now , in the above problem it has been shown that a sequence converges to a random variable X in ...
1
vote
2answers
46 views

Interchanging the order of a double infinite sum

I'm stuck at a proof of Wald's first equation about interchanging the order of a double infinite sum: Suppose $X_n \ge 0$ and $1_{\{\cdot \}}$ be indicator function. $$ \sum_{n=1}^\infty ...
0
votes
2answers
51 views

What is my probability space and measurable space?

I have the following difference equation $$ \tilde{u}_k = \begin{cases} u_k & \text{if $\gamma_k = 1$, no signal lost} \\ \tilde{u}_{k-1} & \text{if $\gamma_k = 0$, signal lost} \end{cases} ...
0
votes
0answers
21 views

AI Bayes Network Question? [duplicate]

A) Given this Bayes Net Answer and explain: 1) True or False 2) True or False B) Given this Bayes Net: Answer and explain: 3) True or False 4) True or False
1
vote
0answers
40 views

Semigroup of operators: weak continuity at 0+ implies weak continuity at any t > 0

Let ($E$, $d$) be a metric space. Consider the semigroup $\{P(t)\}_{t\geq 0}$ of bounded linear operators on the Banach space $\hat{C}(E)$ of continuous real functions on ($E$, $d$) vanishing at ...
1
vote
3answers
46 views

The probability of Breakeven On a Coin Toss Game

I was walking the other day around my work office in NYC and thought of this interesting scenario in a game of coin flips. You have $500 in your pocket. This is your entire life savings. You play a ...
1
vote
2answers
36 views

Four dice, probability that difference of some outcomes is equal to others

I roll four dice which gives me outcomes $x_1, ..., x_4$. I want to determine the probability $$P\left((x_2-x_1) = (x_4-x_3)\right)$$ I have already calculated other probabilities in this setting and ...
1
vote
0answers
13 views

Estimating the discrete laplacian to prove recurrence of simple random walk for d=2

Given a function $f : \mathbb Z\times \mathbb Z \rightarrow \mathbb{R}$ we define the discrete laplacian of $f$, $\triangle_df$, by the following rule $\triangle_df(x,y)= \dfrac{f(x + 1, y)+f(x, y + ...
1
vote
1answer
27 views

distinguishing probability measure, function, distribution

I have a bit trouble distinguishing the following concepts: probability measure probability function (with special cases probability mass function and probability density function) probability ...
0
votes
2answers
42 views

Why is $E[X1_A]=0$ if $P(A)=0$?

I know this is trivial and intuitive, but I'm not able to convince myself rigorously. If $P(A)=0$, why is it true that $E(X1_A)=0$? Every book discards it out as an obvious fact. I tried to prove it ...
0
votes
1answer
33 views

Constructing a joint distribution given $P(X\in A \mid Y)_\omega$

For random variables $X,Y,Z$, I am given for any measurable set $A$ $$P(X\in A \mid Y)=P(Z\in A\mid Y) \text{ a.s. }\iff (X,Y)\overset{d}{=} (Z,Y).$$ The direction $\Leftarrow$ doesn't seem too ...
0
votes
3answers
25 views

probability of the empty set for arbitrary probability measures

I have a probability space $(\Omega, \mathcal{P}(\Omega), P)$. I want to know the probability of the empty set $\{\}$. Intuitively, I would say this probability is zero. It certainly is for the ...
0
votes
0answers
9 views

Density of this quantity for a Geometric Brownian Motion?

If we define $X_T = X_t e^{(\mu-\frac{1}{2}\sigma^2 ) (T-t) + \sigma W_{T-t}}$ where $W_{T-t}$ is a classical weiner process. How would we go about deriving the density and expectation for $X_{max} - ...
0
votes
0answers
18 views

CDF of RVs taking infinite values

How can we define the CDF of a RV that takes positive infinite values with a tagged probability? Thanks in advance
4
votes
0answers
35 views

Convergence of $n^{-\gamma}T$ where $T$ a hitting time for uniform rvs, can I use CLT?

Let $X_1,X_2,\dots$ be iid uniform on $\{1,\dots,n\}$ and define $T=\inf\{k:X_k=X_r \text{ for some }r<k\}$. The objective is to figure out when $n^{-\gamma} T$ converges weakly to some ...
4
votes
1answer
38 views

Almost surely vs expectation

Let $X_1, X_2, X_3 \dots$ be a sequence of random variables. In the limit as $i \rightarrow \infty$ we have $$ X_i \rightarrow 0 \text{ almost surely} $$ Does it follow that In the limit ...