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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102 views

Likelihood Function of Random Process

Given the following data: $$ x(t) = A + \omega(t) $$ where $ \omega(t) $ is an AWGN with zero mean, what would be likelihood function $p(x(t);A)$? I know it could be proven to be: $$ p(x;A) = C ...
2
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93 views

Random walks - two questions

Let us suppose that a person are throwing a coin. He'll get one dollar if he win, and he'll pays one dollar if he loses. I understand that the winning will trend to zero in the case of unlimited ...
2
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168 views

Family of measure that admits a continuous density

This question is a generalization of an example provided in Absolute continuous family of measures. Consider a metric space $(X,\rho)$ with a Borel $\sigma$-algebra $\mathcal B(X)$. Consider a ...
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109 views

Probability metric spaces for which $r \leq R$ implies $\frac{Pr(B(x,r))}{Pr(B(x,R))} \geq \frac{r}{R}$

I'm looking for examples of spaces $X$ such that: $X$ is a probability space. $X$ is a metric space. If $x \in X$ and $0 < r \leq R$ then $\frac{Pr(B(x,r))}{Pr(B(x,R))} \geq \frac{r}{R}$. I ...
2
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78 views

Optional stopping theorem for Hilbert valued martingales

Suppose $X_n$ is a Hilbert-valued martingale. Does the optional sampling theorem apply in this case? Does anyone know where I can find a proof? Thank you.
2
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222 views

How to obtain tail bounds for a linear combination of dependent and bounded random variables?

I am looking for tail bounds (preferably exponential) for a linear combination of dependent and bounded random variables. consider $$K_{ij}=\sum_{r=1}^N\sum_{c=1}^N W_{ir}C_{rc}W_{jc}$$ where $i ...
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2k views

Expected value of max/min of random variables

I am trying to solve the following problem. Let there be $n$ urns and let us have $k$ balls. Assume we put every ball into one of the urns with uniform probability. Denote by $X_i$ the random ...
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158 views

Are Cox point process and Poisson random measure the same concept?

Reading Wikipedia articles for Cox point process and Poisson random measure, I was wondering if they are actually the same concept? If they are not, I wonder how to understand the concept for Cox ...
2
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984 views

Maximum of Correlated Gaussian Random Variables

Let $x_{1},x_{2},\ldots, x_{n}$ be zero mean Gaussian random variables with covariance matrix $\Sigma=(\sigma_{ij}^{2})_{1\leq i,j\leq n}$. In other words, $$ \mathrm{cov}(x_i,x_j)=\sigma_{ij}^{2} ...
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103 views

Measurability of a point process or random measure at a measurable subset

Suppose $\xi$ is a point process on $(S, B(S))$, where $S$ be locally compact second countable Hausdorff space equipped with its Borel σ-algebra $B(S)$. I was wondering if $\xi(A), \forall ...
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530 views

Definition of vague convergence of distibution functions

I am reading Kai Lai Chung's A Course in Probability Theory. I understood the concept of vague convergence of a sequence of probability measures in the book. But it seems to me the book uses the ...
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22 views

Computing the expected value of $f$ where $\mu$ is a function of $f$

I am curious about the following scenario (This may not be the most sensible set up because I am puzzling this out independent of any homework etc). Suppose $X\sim\mathcal{N}(\mu,\sigma)$ and let ...
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0answers
42 views

Limit theorem for changed time

This post seems long, but its almost everything proofed except the last step. The unknown part is marked especially. Given a Levy-Process $U_{t}$ with with $E(U_t)=0$ (then $U_t$ is a martingale). ...
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0answers
38 views

Unfair coins connected in a game

I would like to ask the following question. There are 3 coins ($A,B$ and $C$) that are biased with probability of tails equal to $t_a, t_b$ and $t_c$ respectively.   The coins are tossed: ...
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28 views

Weak convergence of Banach space valued random variables

In the book 'Probability in Banach Spaces: Isoperimetry and Processes', available here http://michel.talagrand.net/, in the second chapter on the page 34, above the Theorem 2.1 there is a statement. ...
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27 views

Statement of the strong Markov property in Norris' book

In J.R.Norris' Markov chains book, the strong Markov property for discrete-time, Markov chains is stated and proved as follows: Let $(X_n)_{n \geqslant 0}$ be a Markov chain with transition ...
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0answers
31 views

Nonanticipativity constraint (filtration/measure theory)

I am trying to show that stochastic process must attend the nonanticipativity constraint using filtration in measure theory. Adaptability of a stochastic process tell us that: $$\sigma(X_t)\subset ...
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20 views

Find probability dense function of multiple random variable

Suppose we have random variable such that $S=g(X_{1},X_{2},...,X_{n})$ and $X_{i}, 0\leq i\leq n$ are all independent and uniformly distributed. I have done my best to find the cumulative distribution ...
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18 views

Difference equations and the characteristic polynomial

The context for this is solving the gambler's ruin problem using linear algebra. I haven't found a good explanation for why the linear combination of the eigenvalues for the matrix representing a ...
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0answers
45 views

Bayes' Theorem and Law of total propability for CDF

The calculation of conditional probability is the same for conditional PDF and CDF(according to a number of questionable sources: first, second) (I will use rough notation, with just $x$ and $y$): ...
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0answers
23 views

Generalized inverse of a function

It is well-known that if a function is strictly increasing, then it has an inverse function. I also see the concept of "generalized inverse" in the litarature, which has the definition ...
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0answers
24 views

Simulation of a diffusion on $[0,1]$

I have a diffusion process $X=(X_t)_{t \ge 0}$ with the generator $$Af(x)=\frac{1}{2}(a(1-x)-bx)f'(x)+\frac{1}{4}x(1-x)f''(x),$$ where $a,b >0$ are constants. I want to simulate $X$ to a ...
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29 views

Converges of measures.

Good afternoon, we have the following: Let $(Y,d)$ is a general metric space, $\mathcal{M}(Y)$ is the set of finite Borel measures on $Y$ and $C_B(Y)$ denotes the Banach space of bounded continuous ...
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32 views

Brownian Moment Generating Function and Hitting Times

Here is my question. I've done the first part, but I'm stuck on the second. If I can work out (/be advised) how to do the second, then I hope to be able to do the third similarly. Please note: While ...
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0answers
19 views

How should I calculate the MLE based on a random sample from $PAR(\theta,2)$

Consider a random sample of size $n$ from a Pareto distribution, $X_i \sim PAR(\theta, \kappa =2)$. I have to compute the MLE, $\hat \theta$, to three decimale places. So I started doing the ...
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0answers
27 views

Slutsky, Continuous mapping for uniform convergence

I have a question- suppose I have a function f(x,$\hat \theta$), $\hat \theta$ is a consistent estimate for $\theta$ and therefore it holds $\hat \theta \rightarrow \theta$ in probability. Suppose f ...
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0answers
29 views

Proof of discrete probability monotone convergence

I am trying to show that for a sequence of random variables defined on a sample space $\Omega$ $$0\leq X_1(\omega)\leq X_2(\omega \leq ......\leq X_{n}(\omega)...$$ for all $\omega\in\Omega$, with ...
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0answers
17 views

Intuition behind Mutual independence of sub-$\sigma$-algebras definition.

I was reading about Independence of sub-$\sigma$-algebras when I found the next definition: Let $\mathcal{B}_{1},\ldots,\mathcal{B}_{n}$ $n$ sub-$\sigma$-algebras of $\mathcal{A},$ let $H$ be a ...
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0answers
31 views

Bayesian Estimation Derivation

I am trying to understand Bayesian estimation and I come across this line in my lecture notes: θ(Bayesian) = E_θ|x[θ] = E[π(θ|x)] So it's meant to reader that ...
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0answers
27 views

Information in Filtrations

Is the “information” kept track of by filtrations the same as information-theoretic “information”? If not, is there some way the two concepts can be reconciled?
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47 views

Lim Sup and Measurability of one Random Variable with respect to Another

Here, there is a common proposition in probability theory : Let $X,Y: (\Omega, \mathcal{S}) \rightarrow (\mathbb{R}, \mathcal{R})$ where $\mathcal{R}$ are the Borel Sets for the Reals. Show that ...
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19 views

Ito's lemma in infinite dimensional spaces

I'm trying to use the ito's lemma in infinite dimensional spaces applicatte to $F(X)=\Vert AX\Vert^{2}$, where $A$ is a linear map. But i I have trouble calculating the integral $\int_{0}^{t}\langle ...
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0answers
50 views

If $X_n\to0$ in probability then $X_n/r_n\to0$ in probability, for some $r_n\to0$

Suppose that a sequence of positive random variables $X_n$ is such that for all $\epsilon >0$, $P(X_n>\epsilon)\rightarrow 0$ as $n\rightarrow\infty$, that is $X_n$ converges in probability ...
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0answers
22 views

Verifying data came from a Wiener Process

From the Wiki article a Wiener Process has the properties that $$E[W_t] = 0$$ $$Var[W_t] = t$$ According to A Standard Wiener Process the Wiener Process is given by: $$W(t) - W(s) \tilde{} ...
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31 views

Problem that may use Borel Cantelli Lemma

So, there is a sequence of identically distributed independent random variables taking values on the integers, and they have a positive expectation. The problem is to prove that with probability 1 the ...
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0answers
33 views

$n$ times integrated Brownian motion martingale process

According to this post, we found that a $n$ times integrated Brownian motion could be expressed as, \begin{align} V_n(t) = \int_0^t V_{n-1}(s)\ ds = \frac{1}{n!} \int_0^t (t-s)^n\ dB_s, \end{align} ...
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0answers
35 views

Suppose $x_n$ is a sequence of positive monotonically increasing random variables converging to $X$. Show $\lim_{n \rightarrow\infty}E(x_n)=E(X)$

I am hoping to get some verification of the below proof. I am worried that I am missing something conceptually. That $\lim_{n\rightarrow \infty}E(x_n)\leq E(X)$ is clear since $E(x_n)\leq x$ for any ...
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0answers
35 views

Experiment by Bernoulli process

I have a question. Assume I carry out an experiment by Bernoulli process. I repeat the tests until the number of successful outcomes exceed the number of unsuccessful outcomes by m. What will be the ...
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0answers
22 views

Variance of Conditional Expectation From First Principles

Let $X : \Omega \to S \subset \mathbb{R}$ and $Y : \Omega \to T \subset \mathbb{R}$ be random variables on $(\Omega, \mathcal{F}, P)$. Form the conditional expectation $$ E(Y \mid X) := E(Y \mid ...
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0answers
26 views

subaddivity of VaR

It is known that the VaR (Value at risk) doesn't fulfill subadditivity, i.e. $VaR(X)+VaR(Y) \le VaR(X+Y)$. But for elliptical distributions subadditivity is true. Questions: (1) Which ...
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0answers
17 views

Counterexamples of Cumulative Distribution Function ( multidimensional )

For simplicity, we consider 2-dimensional. We consider the function $F:\mathbb{R}^2\to\mathbb{R}$. Let $F$ satisfies: $0\leq F(x_1,x_2)\leq1$ for $(x_1,x_2)\in\mathbb{R}^2$ ...
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0answers
22 views

Transition functions induced by Markov processes

Let $(\Omega,\mathcal{F},\mathbb{P})$ be a probability space and denote by $(X_t,\mathcal{F}_t)_{t\geq 0}$ a time-continuous Markov process with values in $(E,\mathcal{E})$. For $s<t\in ...
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0answers
59 views

Can this approximation result for stochastic processes be modified.(p=1 instead of p=2)?

In McKeans stochastic integrals from 1969 he proves this: You have a filtered probability space $(\Omega,\mathcal{F},P)$, where the filtration is based on a Brownian motion. Assume that $X_t$ is ...
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0answers
17 views

What is the chance that two random binary random variables are independent?

Consider the space of probability distributions on 4 letters. Now all the probability distributions on four letters do not represent a distribution of two independent binary random variables. But if ...
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0answers
25 views

Feedback of proof that $\lim \inf A_n \subseteq \lim \sup A_n$

I just wrote down the proof of the following easy proposition, and I was wondering about both the content (I would like to know if it is error-free), and the form of it. Proposition: $\lim \inf ...
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0answers
38 views

Sequence of random variables, mean zero, convergence to -infinity

What would be an example of a sequence $(X_k)$ of independent random variables with zero mean such that $$\frac{1}{n} \sum_{i=1}^{n} X_{i} \xrightarrow[\mbox{almost surely}]{n \to \infty}-\infty\ ...
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0answers
31 views

Proof that $\mathcal L^2 \supsetneq \mathfrak B^2$.

Let $\mathfrak L^d$ be the $\sigma$ -algebra of all Lebesgue-measurable subsets and $\mathfrak B^d$ the one of the Borel sets in $\mathbb R^d$. I want to prove that $\mathfrak L^2 \supsetneq \mathfrak ...
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0answers
41 views

Mixing convergence

Given a process $X_n \xrightarrow{d} X$ on some probability space $(\Omega,\mathcal{A},P)$. If for every $B \in \mathcal{A}$ it holds, that $$ \lim_{n\rightarrow \infty} P(X_n\in A,B)=P(X\in A)P(B) $$ ...
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16 views

Show the sample mean converges to minus infinity when ${X_n}$ are i.i.d. and $E(X_n^+)<\infty$ and $E(X_n{^-})=\infty$

Suppose ${X_n}$ are iid and $E(X_n^+)<\infty$ and $E(X_n{^-})=\infty$. Show that if $S_n := \sum_{j=1}^nX_n$ then $\frac{S_n}{n} \rightarrow -\infty$ almost surely. A hint in my book says to use ...
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0answers
23 views

Mean number of tosses to obtain a given sequence of heads and tails

What is the expectation of the number of tosses needed to obtain a given sequence of heads and tails if the coin is fair and the coin is tossed until we get this sequence? For exemple what is ...