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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4
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0answers
54 views

Characterization of point process, given the number of points

For a point process with independent and identically distributed (i.i.d) inter-renewals, with distribution $p(x)$, we observed $N$ points on $[0,T]$. What is the joint probability distribution ...
2
votes
1answer
23 views

Pdf of $Z=X/Y$ given a joint pdf

Find the p.d.f. of $z=x/y$ $f(x,y)= 2(x+y)$ for $0\le x\le y\le1$ I first did the simple way, transformation, then derivative, and multiply joint p.d.f by absolute value of the derivative. Then ...
2
votes
2answers
12 views

$X_n\to 0$ in probability implies $E[f(X_n)]\to f(0)$ for $f$ uniformly cts and bounded

Let $f$ be a uniformly continuous and bounded function. I've shown that if $X_n\to 0$ in probability, then $f(X_n)\to f(0)$ in probability as well. Now I want to say that $$\lim_{n\to\infty} ...
0
votes
0answers
12 views

Specific Radon-Nikodym Derivative Interpretation

Suppose $(\Omega, \mathcal{F}, P)$ and $(\Omega, \mathcal{F}, Q)$ are two probability spaces. The Radon-Nikodym theory says there exists a function $f$ on $\Omega$ that satisfies $$ P(A) = \int_A f ...
3
votes
0answers
33 views

a dynamical systems view of the central limit theorem?

I have seen many heuristic discussions of the classical central limit theorem speak of the normal distribution (or any of the "stable distributions") as an "attractor" in the space of probability ...
0
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0answers
6 views

Total variation relates to a norm: how about some metric inducing the weak topology?

Let $\mu,\mu'$ be Borel probability measures on a Polish space $S$. There's a straightforward definition of the signed measure $\mu-\mu'$ and on the space of signed measures, we have the total ...
-2
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0answers
23 views

How is Mathematics and Space related?

I've seen scientists come up with equations and proofs in mathematics of some incident happened or happening in space. How do they relate mathematics with that? For Example How do you start ...
1
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0answers
12 views

Uniqueness of Predictable Quadratic Covariation

In order to prove that $\langle M,N \rangle$ is the only process which is continuous and has bounded variation such that \begin{align} M_tN_t - \langle M,N \rangle_t \end{align} is a continuous ...
0
votes
1answer
14 views

Sufficient conditions for a probability measure to be characterized by finite number of moments

Suppose $Y(X)$ is a random variable defined on the probability measure $X$ with density, $f$, with respected to the Lebesgue measure and distribution function $F_y$. Consider the following ...
1
vote
0answers
46 views

On a Probability notation - $\mathbb{E}[X(.)|\mathcal{F}]_G$

What could mean this notation : $\mathbb{E}[X(.)|\mathcal{F}]_G$ ? where G : $\Omega \rightarrow \mathbb{R}$ is a random variable on a probability space $(\Omega,P, \mathcal{F})$. X could be a ...
2
votes
1answer
61 views

Show $P\left[A<Z \mid \mathcal{G} \right]=e^{-A}$ for $Z$ standard exponential and $A$ nonnegative $\mathcal G$-measurable

I have a question about exponential distribution and conditional probability. Let $(\Omega, \mathcal{F}, P)$ be a probability space and $\mathcal{G}$ be a sub $\sigma$ algebra of $\mathcal{F}$. Let ...
2
votes
2answers
32 views

Is covariance preserved under transformation?

Let $X_1,X_2$ be normally distributed random variables with $\rho = 0.5$, mean equal to $0$ and variance equal to $1$. Let $U_i = \Phi(X_i)$ where $\Phi$ is the marginal distribution of $X_1,X_2$. We ...
0
votes
1answer
6 views

Sufficient Estimator How to proceed?

Let $X$ be a random variable with exponential density function. Show that the mean of $X$, denoted $\overline{X}$, is the sufficient estimator of $\lambda$ but not an unbiased estimator of $\lambda$
1
vote
1answer
24 views

Sequences of random variables converging in probability to the same limit a.s

Let $(X_n)_{n \geq 1}$ and $(Y_n)_{n \geq 1}$ be two sequences of random variables s.t. $X_n$ converges to X and $Y_n$ to $Y$ both in probability. Furthemore, $X$ = $Y$ a.s. How can I prove that, for ...
2
votes
0answers
21 views

Hypothesis Testing on Renewal Processes

We have time $[0,T]$ to observe a renewal point process, where the inter-renewal timings are i.i.d, and then decide whether the observation is according to a renewal process in which the pdf of ...
1
vote
1answer
378 views

Increasing/Decreasing Sequence of Events Intuition

having trouble moving from the concept of increasing/decreasing sequence of real numbers to increasing/decreasing sequence of events. I have no problem with this concept regarding real numbers but ...
0
votes
1answer
27 views

Prove $X_{\infty} < \infty$

From Williams' Probability with Martingales: How exactly do we prove $X_{\infty} < \infty$ a.s.? $$E[|X_{\infty}|] = E[|\lim X_n|] = E[|\liminf X_n|] = E[\liminf |X_n|]$$ $$ \le \liminf ...
1
vote
1answer
23 views

Hidden Markov Models in part of speech tagging

I'm trying to prove the following whose context I will describe, but which is not strictly necessary to solving the problem. In natural language processing, specifically in part of speech tagging, ...
0
votes
1answer
19 views

Density of a distribution given by a Gaussian copula and a set of marginals

Suppose the distribution of an $n$-dimensional random vector $X$ is characterized by a Gaussian copula $C_R$ with correlation matrix $R$ and a set of marginal $\{(F_{X_i}, f_{X_i})\}_{i=1}^n$ (pairs ...
2
votes
0answers
23 views

What is the asymptotic value of the smoothed probability in a HMM model?

If I have a HMM model with a hidden markov chain $(S_t)_t$ with 3 states and if I assume that the distribution of the observation knowing in which state it is, is a normal. Do I know what is the value ...
2
votes
0answers
15 views

Characteristic functions proof… $\mu\{x:|x|>r\}\le\tfrac r2\int_{-2/r}^{2/r}(1-\varphi(t))\,dt$

I am trying to prove Lemma 4.1 (1) from Olav Kallenberg's Foundations of Modern Probability: $\mu\{x:|x|>r\}\le\tfrac r2\int_{-2/r}^{2/r}(1-\varphi(t))\,dt$ From context, I believe that ...
1
vote
0answers
311 views

Hidden Markov Model, transition probabilities which are modeled with an exponential distribution

I'm looking at implementing an algorithm described in a paper, but I'm having trouble understanding how the transition probabilities for a Hidden Markov Model are defined. In the first sections, I ...
0
votes
0answers
9 views

Identification of Infinite Dimensional State in Hidden Markov Model

Consider a hidden markov model (HMM) where the state, $X_t(\alpha)$, is a stochastic distribution over $\alpha \in \mathbb{R}_+$ and one observes a signal $Y_t$, which is simply a moment of this ...
2
votes
2answers
49 views

Hidden Markov Model and Viterbi algorithm: Understanding the Casino Problem?

I am deeply struggling with understanding how to apply the Viterbi algorithm. From my course notes, I have the following simple(I'm told) example: If the sequence ...
0
votes
1answer
10 views

emmission probabilities in a hidden markov model with 2 states and an alphabet of 4 characters

I'm reading through a text that is describing how to use use hidden markov models to identify areas of biological sequences that correspond to specific biological features. It starts with a simple ...
1
vote
0answers
54 views

Hidden Markov Model Transition Probability

I am doing my assignment and I am asked to derive transition probability of a HMM. There are Three states. H, E and T. They initially gave me the information as follow. E is followed by an H 40% of ...
1
vote
1answer
23 views

$P($partial sums of $\sum X_k$ are bounded$)>0 \to \sum_{k=1}^{\infty} X_k < \infty\ \text{a.s.}$

From Williams' Probability with Martingales How is the remark deduced from the proof of $b$? I really don't see it.
0
votes
1answer
27 views

If $M_n \to M_{\infty}$ in $\mathscr L^{2}$, then inequality holds with equality

From Williams' Probability with Martingales I tried rewriting the RHS to: $$\sum_{k=n+1}^{\infty} E[(M_k - M_{k-1})^2] = \sum_{k=n+1}^{n+r} E[(M_k - M_{k-1})^2] + \sum_{k=n+r+1}^{\infty} ...
1
vote
0answers
21 views

Brownian Motion with Levy's Characterization 2

Let W be a $\mathbb{R}$-valued Brownian motion. To prove that $(B_t)_{t\geq 0}$, where: $B_t:=W_t-\int_0^t\frac{W_u}{u}du$, is a Brownian Motion with respect to $\mathcal{F}^B$, I showed $[B]_t=t$ and ...
0
votes
1answer
17 views

Prove $A_{\infty} < \infty$?

From Williams' Probability with Martingales How do we know that $A_{\infty} < \infty$? If $T = \infty$, then $$E[A_{T \wedge n}] \le (K+c)^2$$ $$\to E[A_{n}] \le (K+c)^2$$ $$\to ...
1
vote
0answers
30 views

Itô-formula proof, remainder term.

I have a question about the proof a a certain version of the Itô-formula. First the author defines an Itô-process and states the formula: My question is in regarding the proof. The proof uses ...
2
votes
1answer
36 views

How to solve for the expectation of the Ito Integral: $\int_0^4 B_t^2 dB_t$?

I would like to find the expectation of the Ito Integral: $\int_0^4 B_t^2 dB_t$. My strategy is to use Ito's general formula with: $$ f(t, B_t) = f(0,0) + \int_0^t \frac{df}{dx}(s, B_s) dB_s + ...
0
votes
0answers
14 views

Prove $|M_{T \wedge n}| \le c + K$

From Williams' Probability with Martingales Is $\sigma_k^2$ random (and not constant)? How can that be? As far as I know unconditional variance and unconditional expectations are supposed ...
0
votes
0answers
14 views

Clarifications on proof of Doob's Forward Convergence Theorem, warning related to it and proof of a corollary

From Williams' Probability with Martingales: $X_n(\omega)$ does not converge to a limit in $[-\infty,\infty]$ --> Is this supposed to be stronger than $\lim X_n$ does not exist because it's ...
2
votes
1answer
49 views
+50

On the linear combination of $\pm 1$ random variables

Let $X_1,\dots, X_n$ be i.i.d symmetric $\pm 1$ random variables, i.e. $X_j$ takes values in $\{-1,1\}$ with $$\mathbb{P}(X_j=1)=\mathbb{P}(X_j=-1)=\frac{1}{2}.$$ Let $a_1,\dots,a_n\in\mathbb{Z}$, ...
3
votes
0answers
42 views
+100

About the random $\pm 1$ matrices

I was reading the paper "On the probability that a random $\pm 1$ matrix is singular". In the paper the author defined the following notations: $M_n$: a random $n\times n$ matrix with i.i.d entries ...
1
vote
0answers
31 views

Integrals with erf^N

Can anyone help with integral of type. In general, what to do if erf is in power higher than 1? $$g(S|S<L)=\frac{1}{\sqrt{2 \pi \sigma^2}} \int_{-\infty}^{+\infty} \left [ \frac{1}{\sqrt{2 \pi ...
0
votes
1answer
31 views

Does finite expectation imply finite essential supremum?

I have a real valued function $f$ with the property that $$\mathbb{E}\big[f(X)\big] = \int f(x)\ d\mathbb{P}(x) \leq c$$ for some $c > 0$. Does this imply $$ \operatorname{ess sup } \|{f(X)}\|^2 ...
0
votes
1answer
18 views

Given Conditional Expectation check if it satisfies the density function. [on hold]

I have this math problem that I can't seem to solve. Let $X$ be a continuous random variable that only takes non-negative value that satisfies $\mathbb{E}(X\mid X \ge t) = t + ...
2
votes
2answers
2k views

The subadditivity of a measure.

I'm reading Probability: Theory and Examples by Rick Durrett. Theorem 1.1.1 states that Let $\mu$ be a measure on $(\Omega, \mathcal F)$ (i) monotonicity. If $A \subset B$ then $\mu(A) \le ...
0
votes
1answer
20 views

Is sharing the same support a necessary condition for exchangeability?

I am confused on the meaning of exchangeable random variables. The question is: consider the random variables $X_1,X_2,X_3$ defined one the same probability space $(\omega, \mathcal{F}, P)$; is ...
1
vote
0answers
28 views

Expectation formula

Let $F(z)=P\{ Z \leq z\}$. Assume $F(c)=0$. It is well known that: $$E(Z)=\int_c^{\infty}(1-F(z))dz$$ and more generally: $$E(g(Z))=\int_c^{\infty}g(z)dF(z)$$ Is it also true for: $\tilde{F}(z)=P\{ Z ...
-3
votes
1answer
257 views

Prove $S \doteq \sum_{n=1}^{\infty} p_n < \infty \to \prod_{n=1}^{\infty} (1-p_n) > 0$ assuming $0 \leq p_n < 1$. [duplicate]

Prove $S \doteq \sum_{n=1}^{\infty} p_n < \infty \to \prod_{n=1}^{\infty} (1-p_n) > 0$ assuming $0 \leq p_n < 1$. Hint: Show that $S < 1 \to \prod_{n=1}^{\infty} (1-p_n) \geq 1 - S$. ...
1
vote
0answers
30 views

Riemman-Stieltjes Integral Exercise

The truth is that I have no experience with the integral of Riemann-Stieltjes and developing a Bayesian inference problem in the book "Mathematical Statistics" by Shao, appears one of these steps, I ...
0
votes
2answers
33 views

Probability of a number in the real line

I have read that the probability to pick a rational number in the real line is null. My problem is: If $S$ is a dense set in the real line, what is the probability to pick an element of $S$? There ...
1
vote
1answer
22 views

Convergence of sequence of random variables 2

If I know $\lim\limits_{n \to \infty} \mathbb{P}(X_n<c-\gamma)=0$ for all $\gamma>0$, how can I prove supremum of all reals $\alpha$ for which $\lim\limits_{n \to \infty} \mathbb{P}(X_n\leq ...
3
votes
2answers
88 views

Origin of the notation for statistical divergence

The unusual notation $D(P||Q)$ seems to be universally used for statistical divergences (e.g. KL divergence). What is the origin of this notation, and do the double bars (pipe symbols) have any ...
3
votes
1answer
168 views

I need a textbook! Information theory and probability

I have posted some questions: Probability result - 3 discrete random variables Markov chain - a notation I don't understand Random variables identities - how to make a formal proof. These ...
0
votes
1answer
23 views

Is this martingale constant 0?

I have a martingale X where $X_0 = 0$ a.s. And for each $\omega$, the path $f(t)=X_t(\omega)$ is of bounded variation in the classical sense. That ...
11
votes
1answer
297 views

Why “One cannot construct more than countably many independent random variables”?

I'm reading the book "Large Networks and Graph Limits" by László Lovász. On the page 18 he said the following: One cannot construct more than countably many independent random variables (in a ...