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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2answers
63 views

Product of i.i.d. random variables uniformly distributed on $(-1,1)$ converges almost surely to $0$

Let $(\Omega,\mathcal{F},P)$ be a probability space and $(X_n)$ i.i.d. random variables with uniform distribution on $(-1,1)$. If $Y_n=X_1\ldots X_n$, prove that $(Y_n)$ converges to $0$ a.s.
2
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1answer
43 views

Lebesgue integral of vector-valued function?

In Bernt Øksendals stochastic differential equations he says that if we have a random variable $X:\Omega\rightarrow\mathbb{R}^d$. He defines the expectation: $E[X]=\int_\Omega ...
1
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1answer
28 views

Congruent measurable sets

I have a question regarding Congruent relations: In Euclidean geometry, two subsets of $\mathbb{R}^{d}$ are said to be congruent if one set can be mapped onto the other by translations and rotations. ...
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0answers
47 views

If the sample space is an Euclidean Space, we can use a different type of PDF

The title resume all the point I'll try to make now. Reading this post, I realize that is possible to have another type of PDF (probability density function). Usually, we have a probability space ...
2
votes
1answer
21 views

Martingale Transform counterexample

I am studying discrete time martingale theory and came across the classical "You can't beat the system" theorem: given a martingale $M$ and a previsible process $C=(C_n)_{ n \ge 1}$ such that $C_n$ is ...
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0answers
13 views

Modelling the ballot theorem as a martingale.

The page 19 in the link http://www.imada.sdu.dk/~jbj/DM839/FL15.pdf provides the explanation of what a ballot theorem is and how we can prove that it is a martingale. It takes a random variable $S_k$ ...
3
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1answer
41 views

Weak law of large numbers works although first moment

I want to prove that the weak-law of large numbers hold if $X_{k}$ has the distribution: $$\mathbb{P}\left(X_k\leq x\right) = \int_{-\infty}^x c\left(1+t^2\right)^{-1}\left(\log(1+t^2)\right)^{-1}dt ...
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1answer
25 views

examples of random variables that the result of their preimage is not in F?

let's assume we have a probability space $(\Omega , F , P)$. and we have a random variable $X$ defined as : $X : \Omega \rightarrow \Bbb{R}$ and we also use a Borel set ($\mathcal{B}$).(making the ...
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1answer
32 views

How do I transform this Probability in weak law of large number form?

Let $X_{1},\dots$ be a sequence of independent random variables. Suppose, for $k=1,2,\dots$ $$P\left(X_{2k-1}=1\right)=P\left(X_{2k-1}=-1\right)=\frac{1}{2}$$ and the probability density function of ...
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votes
2answers
55 views

What probability mass, density or distribution function might corresponds to this moment generating function? [duplicate]

I have somehow come up with a random variable $X$ with moment generating function (assuming it exists) $$M_{X}(t) = -t (1 - e^t)$$ What is the probability mass, density or distribution function? It ...
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1answer
29 views

Weak convergence of measures and compact sets

Suppose that we have a sequence of probability measures $\{ \mathbb{P}_n \}$ converging weakly to a probability measure $\mathbb{P}$. Suppose that $M$ is a metric space with a compact subset $K$. I ...
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1answer
45 views

How to show a sequence of independent random variables do not almost surely converge by definition?

I have a sequence of independent random variables $X_1, X_2, \ldots$ where $$ X_n = \begin{cases} 1 & \quad \text{with probability} \ 1/n \\ 0 & \quad \text{with ...
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1answer
28 views

How to show convergence in probability by just using the definition?

I have a series of random variables $X_1, X_2, \ldots$ where $$ f(X_n) = \begin{cases} 1/n & \quad \text{if} \ X_n = 1 \\ 1-1/n & \quad \text{if} \ X_n = 0 \\ 0 & ...
2
votes
1answer
46 views

What is the intuitive difference between almost sure convergence and convergence in probability? [duplicate]

It is a standard fact in probability that almost sure convergence is stronger than convergence in probability. I can only see the differences in the proof. However, is there a way to view it ...
3
votes
1answer
20 views

Moment generating function and convergent random variables

denote by $X$ and $X_n$, $n\in \mathbb{N}$, random variables and $r\in\mathbb{R}_+$ with $E=\mathbb{E}\left[ e^{rX} \right] < \infty$ and $E_n=\mathbb{E}\left[ e^{rX_n} \right] < \infty$ for all ...
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0answers
33 views

Dealing with Recurrence Relations of Random Variables

Let $(Y_n)_{n\in \mathbb N} $ be some sequence of independent random variables, and $(X_n)_{n\in \mathbb N} $ another sequence, defined recursively as follows: $$X_{n+1} = \alpha X_n + \beta Y_n ...
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0answers
29 views

Probability to get from A to C.

There has been a snowstorm and Bob is trying to drive from A to C. p and q are the probabilities that the two roads are passable. What is the probability that Bob can get from A to C? Note that ...
1
vote
1answer
23 views

Stopping time and the Martingale stopping theorem.

According to the book that I am reading, A nonnegative, integer valued random variable T is a stopping time for the sequence {$Z_{n},n\geqslant0$} if the event T = n depends only on the value of ...
4
votes
1answer
50 views

References for information theoretic statistical tools

Strange statistical concepts like spaces of probability distributions, "metrics" like Fisher information or relative entropy, and convergence with respect to these quantities are necessitated in my ...
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0answers
37 views

Mutual information $I((X,Y,Z);A)$ larger for small pairwise mutual informations $I(X;Y), I(X;Z), I(Y;Z)$?

Is the mutual information $I((X,Y,Z);A)$ larger for small pairwise mutual informations $I(X;Y), I(X;Z), I(Y;Z)$? In particular, in the extreme case that the pairwise mutual informations are ...
2
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0answers
30 views

Does the quadratic covariation process define a measure?

In the context of stochastic integration (when we define the space $L^2(M)$), we define the (possibly infinite) measure $$P_M = P \otimes [M]$$ by $$E_M[Y] = E\left[\int_0^\infty Y_s(\omega) ...
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0answers
30 views

Calculate the discrete density of the variables of a Markov chain

$X$ and $Y$ are independent random variables of Bernoulli with parameter $\frac{2}{3}$. $Z=X+Y$ $\{X_n\}_{n \in \mathbb{N}}$ with values in {0,1,2} having $Z$ such as initial law and the transition ...
2
votes
1answer
50 views

Weak convergence of probability measures and uniform convergence of functions

I am stuck on Problem 4.12 of Karatzas and Shreve's book Stochastic Calculus and Brownian Motion: Suppose that $\{ \mathbb{P}_n \}$ is a sequence of probability measures on $(C[0, \infty), ...
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1answer
28 views

Find the joint probability density function of Max and Min

This is the problem 1.2.13 of Karlin's book An introduction to stochastic modeling: Let X and Y be independent random variables each with the uniform probability density function ...
2
votes
0answers
11 views

(joint) Functional CLT for partial sums and counting process

Assume you are given a sequence of random variables $(X_i)_{i\geq1}$. Assume moreover that they are sufficiently smooth, say $\mathbb E[X^2]<+\infty$. Define the diffusion-scaled partial sum as ...
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0answers
28 views

Question on a proof involving tightness and almost sure convergence of a sequence

I'm having a hard time understanding the proof of Lemma 17 in this article. Essentially, the assertion of the lemma boils down to replacing a constant in a sequence of random variables that satisfies ...
1
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0answers
26 views

Range of a standard brownian motion, using reflection principle

With a standard brownian motion $B_t$, I'm trying to find the distribution of the "range": $$R_{t} = \sup_{0 \leq s \leq t} B_s - \inf_{0 \leq s \leq t} B_s = \overline{M_t}-\underline{M_t}$$ The ...
1
vote
1answer
44 views

Joint convergence of stochastic processes

Suppose I have processes $X_n(t)\overset{d}{\longrightarrow} X(t)$, and $Y_n(t)\overset{p}{\longrightarrow} ct$ for some constant $c$. Then, can I conclude like in Slutsky's theorem that ...
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0answers
32 views

What is the probability of unions of intersections? [on hold]

Suppose we have two unions of (possibly overlapping) events. Let me denote the unions as: $$A = IE_A^1 \cup \dots \cup IE_A^{k_A}$$ $$B = IE_B^1 \cup \dots \cup IE_B^{k_B}$$ Each $IE_X^y$ is a ...
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0answers
33 views

Reflection principle for simple random walk

Let $(X_n)$ be a sequence of independent random variables, such that $P(X_i=1) = P(X_i=-1) = 1/2$. Then, the reflection principle states that for all $a > 0$, $$P(\max_{1\leq k\leq n} S_k \geq a) ...
1
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1answer
20 views

What is the distribution of $\min\limits_{1\le i\le N}\frac{X_i}{Y_i + C}$ as $N \rightarrow \infty$?

Let $Z_i = \frac{X_i}{Y_i + C}$ with $i = 1, 2, \dotsc, N$ denote the sequence of the random variables, where $X_i$ and $Y_i$ are exponentially distributed independent random variables with different ...
2
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0answers
23 views

Conditional expectation of another expectation expression.

What is the intuition and the proof behind the given below expression where $U,V,W$ are random variables: $E[V | W]$ = $E[E[V | U,W] | W]$ I know that $E[V | W]$ can be treated as a random variable ...
-1
votes
1answer
23 views

proof for the probability$P(A^{c}\cup C)$.

Let $A,B,C$ be pair-wise independent (like A and B are independent) such that$P(A^{c}\cap B)=0.1$ and $P(B\cap C)=0.2$ Show that $P(A^{c}\cup C)\ge \frac{7}{8}$. $P(A^{c}\cup C)=1-P(C^{c}\cap ...
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0answers
15 views

independence of chi square distributions

We already knew that if two independent chi-squared random variables, then their sum is also chi-squared with the degree of freedoms is the sum of theirs. How about the converse? If $X\sim\chi^2(n)$ ...
0
votes
1answer
63 views

Distribution of a function of a random variable

Suppose we have continuous random variable $X$ with distribution $f_X$. That is $$ P\left(a \le X \le b \right) = \int_{a}^{b} f_X(x) \ dx $$ Now suppose I have a function $\phi: \Bbb{R} ...
1
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1answer
23 views

Finding out the constant term in cumulative distribution function.

Let$$F(x)=\begin{cases} 0,\ if\ x<0\\ \frac{x^2}{10},\ if\ 0\le x<1\\\frac{x+2}{8},\ if\ 1\le x<2\\ \frac{c(6x-x^{2}-1)}{2},\ if\ 2\le x\le3\\ 1,\ if\ x>3 \end{cases}$$ Find the value of ...
0
votes
1answer
37 views

sigma algebra of a stopping time

Let $N$ be a stopping time. i.e $\{N=n\} \in \mathbb{f}_n \forall n$. $\mathbb{f}_n$ is the filtration. $\mathbb{f}_N=\{A\in \mathbb{f}, A\cap \{N=n\}\in \mathbb{f}_n \forall n\}$ is the sigma ...
1
vote
1answer
22 views

Finding bivariate probability mass function (by counting?)

Suppose that we role $d$ dice. Let $X, Y$ be random variables, where $X = \#$ rolled by the die with the highest value. $Y = \#$ rolled by the die with the second highest value. By convention, we ...
2
votes
2answers
29 views

Prove: If $A \subset B$ then $P(A) \le P(B)$ and $P(B-A) = P(B)-P(A)$

I'm trying to prove the following theorem using the axioms quoted below. Theorem: If $A \subset B$ then $P(A) \le P(B)$ and $P(B-A) = P(B)-P(A)$ Axiom 1: For every event $A$ in the class ...
2
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0answers
21 views

Convergence of sum dependent random variables in $L^2$ with mean zero

Suppose that $X_1, \ldots$ were a sequence of random variables with $E(X_i)=0$ and $E(X_i^2)= \sigma_i^2$ and that $S_n := X_1 + \cdots + X_n$ converged in $L^2$. Does this imply $\sum \sigma_i^2 < ...
2
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0answers
24 views

A functional of a Lévy process

Does anyone know if there are any papers/results on functionals of the type : $$\int_0^tp(X_s)ds$$ where $X$ is a Lévy process and $p$ is a polynomial. For example, is the distribution of such an ...
0
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0answers
22 views

Probability density function above a given value. $\{ f(x) > c\}$

Say $X$ is a stochastic variable with a distribution $\nu$ and $f$ is the corresponding Lebesgue-measurable density. If I want to calculate a set $$A = \{ x \in \mathbb{R} \ | \ f(x) > c \}$$ for ...
0
votes
1answer
35 views

A doubt on a proof of a theorem of Durret's Probability Theory

Below is the text of the theorem: $\mathcal{F}_{i,j}$ are sigma algebras indexed by $i$ and $j$. I'm having some difficulties in understanding this proof. Do the $\mathcal{A}_i$ contain $\Omega$ ...
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0answers
23 views

What is a Tail Field and how to interpret it?

I cannot understand or form a good intuition in my head of what a tail field is. An introduction to rigorous probability theory by Rosenthal gives the following definition: Given a sequence of events ...
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0answers
61 views

Hypergeometric distribution with a priori probabilities of the balls

If we have a urn with $N$ balls of two colours ($D$ red and $N-D$ black balls respectively), then probability of having $k$ red out of $n$ balls drawn at once without replacement follows the ...
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0answers
10 views

How to calculate mean of busy-cylce

Let $\{X(t)\}$ be birth-death process on a finite state {0,1,2} with non negative birth rates $(\lambda_0,\lambda_1)$and death rates $(\mu_1,\mu_2)$. Suppose $\mathbb{P}(X(0)=0)=1$ and $s_0=\inf ...
2
votes
1answer
36 views

Disentangling $\int_Af(\mathbf{x})\ d\mathbf{x}$, using Fubini Theorem.

Let $\mathcal{B}^n$ be the borel sigma algebra generated by the rectangles in $\mathbb{R}^n$. I can write $f(\mathbf{x})=g_1(x_1)\cdots g_n(x_n)$. Let $\mu=\mu_1\times \cdots \times \mu_n$ be the ...
0
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0answers
16 views

Integrals of functions of statistics

Let $X: \Omega \to \mathbb{R}^n$ be a measurable random vector with law $\Lambda_X$ and probability density function (pdf) $f_X$. Let $T:\mathbb{R}^n \to \mathbb{R}^2$ a statistic (a ...
1
vote
2answers
31 views

Prove that if $X_n \xrightarrow{P} c$, then $E(X_n) \to c$ for $X_n$ uniformly bounded

I have been trying to prove that for a random variable that is uniformly bounded, i.e. $|X_n - c| <M$ for all $n$, convergence in probability to $c$ implies that $$E\left(X_n \right) \to c$$ ...
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votes
0answers
11 views

How to find expectation of birth-death process [closed]

Let $\{X(t)\}$ be birth-death process on two-state space {0,1}. Let birth rate $\lambda=2$ and death rate $\mu=12$. How to calculate $\lim_{t\to\infty} \mathbb{E}[X(t)]$?