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 [tag:probability-...

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1answer
31 views

Does $X_n \xrightarrow{L_1} X \implies X_n \xrightarrow{\text{qm}} X$?

Let $X_n$ and $X$ be a sequence of random variables. According to All of Statistics (pg. 81), we have that: $$ X_n \xrightarrow{\text{qm}} X \implies X_n \xrightarrow{L_1} X $$ But the book doesn't ...
0
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0answers
29 views

Time-changed Brownian Motion

Let $B_t$ be a standard Brownian motion and let $\tau_{-1}= \inf \{ t \geq 0: B_t(\omega) = -1\}$. By the Continuous Time Stopping Theorem we know that \begin{align} Z_t = B_{t \wedge \tau_{-1}} \...
5
votes
1answer
142 views

Strong Law of Large Numbers for a i.i.d. sequence whose integral does not exist

Prove: Let $X_1 ,X_2 , ... , X_n , ...$ be i.i.d. random variables with $\mathbb{E}[X_1^+]=\mathbb{E}[X_1^-]=+\infty$. If $S_n=\sum_{i=1}^{n}{X_i}$, then $$\limsup_{n\rightarrow\infty}{\frac{...
2
votes
1answer
27 views

Sharpen Doob's Maximal Inequality

Let $B_t$ be a Brownian motion, $B_T^* = \sup_{0\leq t \leq T} B_t$ and $\lambda > 0$. Applying Doob's maximal inequality gives: \begin{align} P(B_T^* \geq \lambda)\leq \frac{\mathbb{E}[B_T^p]}{\...
2
votes
1answer
42 views

Distribution of $\lceil X \rceil - X$ where $X$ has an exponential distribution

Suppose $X$ is a random variable with exponential distribution of parameter $\lambda > 0$. That is, $X$ has density $f(x) = \lambda e^{-\lambda x} \mathcal{1}_{\mathbb{[0,\infty [}}$. The question ...
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0answers
20 views

Book Recommendation for Infinite Dimensional Stochastic Optimization Problem in Discrete Time

Let $X(k)$ be i.i.d. discrete random variables and for all $k=0,1,2,...,N-1$, let $X:=(X(0),X(1)...,X(k))$ and $f := (f(0), f(1),...,f(k))$ with $f(k)$ be the decision function at time $k$, I want to ...
2
votes
1answer
70 views

The jumping times of a càdlàg process are stopping times.

Protter first proves this theorem: Let $X$ be an adapted càdlàg stochastic process, and let $A$ be a closed set. Then the random variable: $T(\omega)=\inf\{t: X_t(\omega)\in A \text{ or } ...
0
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0answers
30 views

In what sense does does linear dependence correspond to random variable dependence?

In linear algebra, there is a theorem that states that $\langle v, w \rangle = 0$ implies that $v$ and $w$ are linearly independent. Now let $V$ be a vector space of real-valued random variables on ...
6
votes
2answers
199 views

Corollaries of the Yoneda Lemma in Analysis?

I am looking for some simple examples of how the Yoneda Lemma can be applied in analysis and probability theory and related fields. A simple candidate example that I can think of and somewhat ...
2
votes
1answer
50 views

Missing crucial step in the derivation of the Stirling's Formula via the Poisson Distribution and CLT

I believe that this is a particular neat example that we've done in class. Unfortunately there is one step I do not quite understand and my Professor had to skip due to the lack of time. I think this ...
1
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1answer
32 views

Show that $\mathbb{E}[h(x)^{2}-2h(x)y+y^{2}]=\mathbb{E}[h(x)^{2}-2h(x)\mathbb{E}[y|x]+\mathbb{E}[y^{2}|x]]$

Let $\mathcal{D}$ be a distribution over $Z=X\times Y$. I am trying to understand the following: Why $\mathbb{E}_{(x,y)\sim\mathcal{D}}[h(x)^{2}-2h(x)y+y^{2}]=\mathbb{E}_{x\sim\mathcal{D}_{x}}[h(x)^{...
1
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0answers
60 views

Surveys in probability? In the current literature sense

I mostly come from an economics background so when I want to find where the current state of knowledge is in specific fields I look for surveys. These are basically primers so that a researcher can be ...
1
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3answers
94 views

Why is the probability of $\emptyset$ equal to 0?

I'm wondering why is the probability of the empty set, $\emptyset$, equal to 0? \begin{equation} P(\emptyset) = 0. \end{equation} Isn't the empty set always included when you take the all the ...
3
votes
1answer
32 views

Show that a cadlag adapted martingale is a local martingale (help using DCT to show uniform integrability)

EDIT 2: With the correct definition, I think I have a proof. Want to show $\lim_{M\to\infty} \sup_t E[|X_{t\wedge n}|; |X_{t\wedge n}|\ge M]=0$. Fix $n$. Note that $\sup_t E[|X_{t\wedge n}; |X_{t\...
2
votes
1answer
41 views

Probabilistic constraint implying deterministic constraint?

Suppose $X$ is an $N$-dimensional random variable $X := [X_1 \; X_2 \; \cdots \; X_N]$ such that all entries can either be 0 or 1 while satisfying the following: (i) $\mathbb{P}(X_i = 1) = p_i \; \; ,...
6
votes
2answers
71 views

Why does Slutsky's Theorem Fail to Generalize? [closed]

What is a counterexample to the claim that $X_n \rightsquigarrow X$, $Y_n \rightsquigarrow Y$ implies that $X_n + Y_n \rightsquigarrow X + Y$? I know that Slutsky's Theorem guarantees the case that $...
2
votes
0answers
73 views

Radon-Nikodym with respect to Stochastic Measure?

Question This question is now concerning stochastic processes. Let $(X_t)_{t\geq0}$ be defined on the probability space $(\Omega,\mathcal{F},P)$ with $\mathcal{F}_t=\sigma(X_s:s\leq t)$. Assume that ...
0
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0answers
38 views

Bayesian Estimation: calculating an integral

I am reading a book on Bayesian filtering and I have a question regarding calculating transition density $p(X_t|X_{t-1})$. My question is how the term $p(X_t|X_{t-1}, V_{t}=v)$ is converted to the ...
2
votes
1answer
61 views

Uniqueness of the uniform spherical distribution

Suppose that $X,Y$ are random vectors on some (possibly different) probability spaces mapping to $\mathbb R^n$ for some $n\in\mathbb N$. Suppose furthermore that $\|X\|=r>0$ for all realizations ...
1
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1answer
30 views

Which functions arise from a probability measure in this way?

Given a probability measure $\mathbf{P}$ on the interval $I=[0,1]$, we get a corresponding function $f:(\mathbb{R}_{>0})^2 \rightarrow \mathbb{R}_{>0}$ as follows: $$f(x,y) = \int_\mathbf{P} x^q ...
4
votes
1answer
71 views

Convergence of the integral $\int_0^\infty f(x)\frac{xf'(x/(1-1/N))}{f(x/(1-1/N))}\ \mathsf dx$ as $N\to\infty$

How can calculate this integral $$\lim_{N \to \infty} \int_{x=0}^{\infty}f(x) \frac{x f'\left(\frac{x}{1-1/N}\right)}{f\left(\frac{x}{1-1/N}\right)} dx$$ where $f(x)$ is a probability density function?...
1
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2answers
34 views

How to interpret probability of a nonrepeating event?

I'm wondering about the meaning of ascribing probabilities to the outcomes of nonrepeating events. As a concrete example, here in the UK we're pollsters are currently predicting the result of the ...
2
votes
2answers
75 views

Probability: Balls in baskets

I'm self learning and I stumbled upon the following exercise but I'm not sure if I solved it correct as I'm very new to this. Problem: 7 balls fall independently into 7 baskets. Let $X_i$ = number of ...
2
votes
2answers
60 views

Are real numbers generated uniformly at random guaranteed to be unique?

Suppose I can generate numbers uniformly at random from an infinite set, such as: $$r \in \mathbb{R} : 0 < r < 1$$ Each number has an infinitely small probability of being generated. Does ...
0
votes
0answers
40 views

If $(Y_n)$ is independent and $X_n=\prod\limits_{k=0}^nY_k$, what is $\bigcap\limits_n \sigma(X_k\,;\,k\geqslant n)$?

Let $Y_0, Y_1, Y_2, \dots$ be independent random variables. For $n \in \mathbb{N}$, define $$X_n:=Y_0Y_1\cdots Y_n.$$ Is it true that $\bigcap\limits_n \sigma(X_{n+1},X_{n+2},\dots)=\bigcap\...
1
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2answers
55 views

Proofs related to chi-squared distribution for k degrees of freedom

I was reading a proof related to chi-squared distribution for k degrees of freedom from wiki. https://en.wikipedia.org/wiki/Proofs_related_to_chi-squared_distribution I think I might understand the ...
1
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1answer
47 views

Probability with coins

I'm self learning and I stumbled upon the following task, but I struggle to find the solution: Two players flip coins. The first player flips 3 coins, the second player flips 2 coins. The player that ...
2
votes
2answers
71 views

Probability: breaking keyboard

I'm trying to self-learn theory of probability, I came across the following basic problem that I think I solved but I'm not sure as I'm very new to this. Problem: A keyboard manufacturer states that ...
0
votes
0answers
96 views

Formula for $E(\max(X-x,0))$ in terms of $E(X)$ and an integral of the CDF of $X$

I am reading this paper and I am having trouble with the algebra used there for equation 1, which reads: $$E[\max(0,p_T-k_C)|H_T]= \bar{p_T}-k_C +\int^{k_C}_0Q_{p_T}(x)dx$$ In the paper it is ...
0
votes
1answer
36 views

How can probability and expected value be same in the limit?

I have seen similar arguments at other places but couldn't convince myself so far about it. I am reading some literature related to graph theory but let me post the analogous problem which avoids the ...
6
votes
0answers
74 views

Rigorous justification for “complex” change of variable in integration

Suppose that I have $X_1,\ldots,X_n$ i.i.d. $\sim $ $X$ and $Y_1,\ldots,Y_n$ i.i.d. $\sim$ $Y$ for some continuous $X$ and $Y$. Consider the r.v.'s $\bar{X}=\frac{1}{n}\sum_jX_j$ and $\bar{Y}=\frac{1}{...
2
votes
3answers
26 views

Characteristic Function of a Conditioned Random Varaible

Let $(X_j)$ be iid random variables and $N \sim \mathrm{Poisson}(\lambda)$, independent of $X_j\, \forall j$. Define $S_n:=\sum_j^n\,X_j$ and consider $S_N$. Find the characteristic function of $S_N$....
-1
votes
1answer
67 views

Let $X \sim\operatorname{unif} (1,2)$. Find the distribution of $ Y=X+2/X $

If $ X $ follows the uniform distribution in $ (1,2) $ what is the distribution of $ Y= X + \frac{2}{X} $ ? I thought that $ P( X + 2/X <=y )$ => $ P(X^2-2xy +2 <=0)$ , where y is at $(2\sqrt{...
0
votes
1answer
22 views

Can we prove that the “speed” of convergence implied by LNN is proportional to $\sqrt n$?

Let $\overline X_n$ be the average of $n$ i.i.d. instances of the random variable $X$. Let's say $E(x)=0$ for simplicity. The CLT roughly says that $\overline X_n$ is distributed normally around its ...
1
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0answers
35 views

If a process is previsible, is the stopped process previsible? [closed]

Assume we have a filtered probability space $(\Omega, \mathscr F, \{\mathscr F_n\}_{n \in \mathbb N}, \mathbb P)$ where $A = \{A_n\}_{n \in \mathbb N}$ is an $\{\mathscr F_n\}_{n \in \mathbb N}$-...
0
votes
0answers
29 views

Probability distribution of $\omega'(n)$. [duplicate]

$\omega(n)$ is the number of distinct prime factors of $n$ and $\omega'(n)$ is the number of distinct prime factors of $n$ with multiplicity. For example if $p,q$ are prime numbers then $\omega(p^2q)=...
0
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0answers
13 views

Where did I make a mistake in this transformation of random variable?

The arctangent of a standard Cauchy random variable $Z\sim\text{Cauchy}(0,1)$ is uniformly distributed in $[-\frac{\pi}{2},\frac{\pi}{2}]$. The proof is straightforward: $$P(\arctan(Z)\leq t)=P(Z\...
1
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1answer
18 views

How to calculate the central moment giving function of a distribution

Is there a function which gives the central moments instead of just moments of a distribution and if so how to calculate this function for a distribution e.g. the normal distribution.
4
votes
1answer
39 views

Probability in $S_{15}$

We consider the set of permutations of the first fifteen natural numbers. What is the probability that $1$ and $2$ aren't contiguous? My attempt: Denote by $C_{12}=$ "The numbers $1,2$ are ...
1
vote
1answer
29 views

'Bounds' on the Covariance Matrix

We define covariance of random vector ${\bf X}$ as \begin{align} Cov({\bf X})=E \left[ \left( {\bf X}-E[{\bf X}] \right) \left( {\bf X}-E[{\bf X}] \right)^T \right]. \end{align} In the scalar case ...
0
votes
0answers
11 views

statistical test for comparison of time series of rare events

I have two time series of a binary variable which assumes value 1 if an event happens and 0 otherwise. Both series report the occurrence of the same event and are not necessarily equal because the ...
1
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0answers
17 views

Probability of at least X same choices by N people choosing out of a set of Y options

I am currently facing the following problem: There is a set of integers from 1 to Y, and N people choosing a random number of this set. What is then the probability that there is at least one number ...
1
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0answers
39 views

Conditional expectation and partition theorem

Let $X,Y$ be two random variables and consider $E[g(Y)\mid X]$. Suppose $\cup_{i=1}^N A_i= \mathbb{R}$, then do we have some thing like $$E[g(Y)\mid X]=\sum_i E[g(Y)\mid 1_{Y\in A_i}, X]P[\{Y\in A_i\}...
0
votes
0answers
30 views

Application Strong Markov Property

I am considering a random walk $S_n$ on a state space $\mathbb{Z}^d$. I want to show that $E_x\left[\sum_{n=0}^{\tau_A-1}{1_{\{S_n=y\}}}\right]=\frac{1}{P_x[\tau_A<\tau_y]}$, where $\tau_A=\inf\{n\...
1
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0answers
53 views

Extending a probability measure to the sigma field obtained by adjoining a new set

Suppose that $P$ is a probability measure on a sigma field $\mathcal{B}$ and suppose $A\not\in\mathcal{B}$. Let $$\mathcal{B}_{1}=\sigma(\mathcal{B},A)$$ be the sigma algebra generated by $\mathcal{B}...
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votes
1answer
16 views

Convergence of time translated iid process

Asume we have a sequence of i.i.d. rv $(Y_n)_{n\geq 0}$, with finite expectation. If $\sqrt{n}^{-1}Y_n\rightarrow 0$ almost sureley, can one conclude, that $\sqrt{n}^{-1}Y_{n+m}\rightarrow 0$ almost ...
-2
votes
0answers
25 views

Calculate conditional expectation using formal definition

Let $\xi$ and $\eta$ be two random variables uniformly distributed on $[0, 1]$. What is conditional expectation $\Bbb{E}[\xi|\xi+\eta=1]$? I know that i can get it using symmetry: $$\Bbb{E}[\xi|\xi+\...
1
vote
0answers
33 views

Independent random variable in the limit (need extension?)

Given a sequence of rv $(X_n)$ on $(\Omega,\mathcal{F},P)$ with values on $(E,\mathcal{E})$ where $\mathcal{F}=\bigcup \mathcal{F}_n$ and $\mathcal{F}_n=\sigma(X_s:s\leq n)$. Lets say there is a ...
0
votes
0answers
9 views

Bounds for transition density and its derivative

Suppose the process $X_t$ has a transition density $p(t,x,y)$, which is continuously differentiable w.r.t $y$. In my proof, I use the following properties of $p$ and $p_y$: There exist functions $\...
0
votes
1answer
108 views

Average number of terms required for a sum of exponential variables to reach a specific limit

I have a sum $\sum_{i=1}^\infty Y_i$ where $Y_i=AX_i+a$ if $X_i>X_{lower}$ and $Y_i=BX_i+a$ if $X_i<X_{lower}$. Here $X_{lower}, A, a, B$ are positive constants and all $X_i$'s are i.i.d ...