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 ...

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384 views

complex integration over the whole plane

I am trying to solve this integral: $H(z)=\int_{\mathbb{C}}{p(z)\log_2{p(z)}dz}$, where $z$ is a complex number with complex normal distribution $p(z)$, and $\mathbb{C}$ denoted the complex plain. ...
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62 views

Spectral expansion of a discrete function of a set of continuous random variables

Let $\mathbf{X}(\omega)=(X_1(\omega), \dots, X_n(\omega))^T$, $\omega \in \Omega$, be a vector of independent continuous random variables defined on a probability space $(\Omega, \mathcal{F}, ...
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50 views

Stationarity of an Integral process

Let $f$ be a continous deterministic function defined on $[0,c]$ and $(B^{H}_{t})_{t\geq 0}$ be a fBM with $H\in(0,1)$. We define a Process $ (X_{t})_{t\geq 0}$ with ...
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601 views

What is the significance of error function?

Here's a Wiki article on the subject. Sadly it doesn't do a good job of explaining the significance of the function. Of course it may mean different things to different people (for mathematicians it ...
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169 views

Demonstrating independence and expectation without Fubini's Theorem

Suppose X and Y are independent random variables, and let f and g be measurable functions, which are either bounded or non-negative. The problem is to show that: $E(f(X)g(Y))=E(f(X))E((gY))$ but I ...
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91 views

Geometry of log-concave density functions and its distribution

Suppose $f:\mathbb{R}^n\rightarrow \mathbb{R}$ is log concave (density function). Consider now the antiderivative (distribution function) $F(t)=\int_{{ x\le t}}f(x)dx$, which is also log concave. We ...
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50 views

Markov chain supplementary litterature

I'm studying markov chains through Durrett and I'm finding it quite hard to read. Does anyone have a good idea to supplementary book, preferably one on his level of generality and which studies some ...
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191 views

Constructing Ito integral for adapted process

I am trying to construct Ito integral for adapted process. However, I am stuck at some point. Let $X^n(t)$ be a sequence of simple processes convergent in probability to the process $X(t)$. Then the ...
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59 views

Equation Involving Bilateral Laplace Transform

Assume that $f(x,y)$ is a compactly supported, joint probability density function on $\mathbb{R}^2$ and nice enough for the following function to make sense: $$P_t(y):=e^{ty}-\int_{-\infty}^\infty ...
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108 views

Hilbert Spaces and Projections

Suppose that $\{Y_{t}: t \in \mathbb{Z} \}$ is a stationary zero mean time series. Consider the Hilbert space $\mathcal{H}$ generated by the random variables $\{Y_t: t \in \mathbb{Z} \}$ with inner ...
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100 views

Compactness of the set of densities of equivalent martingale measures

Consider an incomplete market $(\Omega,\mathcal F,\mathbb P)$ driven by a semimartingale $S=(S_t)_{t\in[0,T]}$. Under the no free lunch under vanishing risk (NFLVR) assumption, the set $\mathcal ...
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133 views

Under certain condition, a local martingale is a martingale

It's well known that a local martingale of is a uniformly martingale if and only if it is of class D. I want to show the following: Let $L$ be a continuous local martingale, null at zero such that ...
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122 views

Uniform Integrability on Compact sets

Let $m$ be a probability measure on the compact set $W \subset \mathbb{R}^m$, so that $m(W)=1$. Consider $f: X \times W \rightarrow \mathbb{R}_{\geq 0}$ locally bounded, $X \subseteq \mathbb{R}^n$, ...
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69 views

Question (3) on Uniform Integrability (simpler)

Let $m$ be a probability measure on $W \subseteq \mathbb{R}^m$, so that $m(W)=1$. Consider $f: X \times W \rightarrow \mathbb{R}_{\geq 0}$, $X \subseteq \mathbb{R}^n$ such that $\forall w \in W$ $\ ...
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233 views

Question (2) on Uniform Integrability

Let $m$ be a probability measure on $W \subseteq \mathbb{R}^m$, so that $m(W) = 1$. Consider $f: X \times W \rightarrow \mathbb{R}_{\geq 0}$ locally bounded, $X \subseteq \mathbb{R}^n$, such that ...
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196 views

Determining a square integrable martingale

I'm preparing for an exam in my course Martingales & Stochastic Integrals. Currently I'm having a look at some old exams, and there's a question on one of them that I'm not able to figure out. The ...
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64 views

uniform integrability allows to apply limits

If I have the following expression: $$E[M_{t\wedge n}|\mathcal{F}_s]$$ Where the set of random variables $M_{t\wedge n}$ is bounded in $L^2$, i.e. $$\sup_{t} E[M^2_{t\wedge n}]<\infty$$ Hence ...
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58 views

A correction to confidence interval.

I have set of random values with the same distribution $y_1, \ldots, y_N$ , $N = mN_1$. $ m \ge 4$, $N_1$ is big enougth( $\approx 1000$ ). I want to to estimeat $E(x)$. How I do it: I make $m$ ...
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43 views

How to calculate P(W|F) empirically?

So say I have three files $(A, B,C)$ that are filled with words. Say for a specific word $W_{i i}$ want to determine the probability of the word given a file. that is Calculate: $$ ...
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1k views

What does normalization in math mean.

I have encountered something like this in a paper and was wondering what it really means normalize the local values in a manner that it leads to elegant probalistic interpretation Its not that ...
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93 views

How to calculate the new conditional probability in a baseyan network when an evidence on one attribute is provided?

I'm trying to understand bayesan networks also I created a simple bayesan network according to same sample date. This is the network (created with Hugin Lite) There is one class (Failure) and two ...
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445 views

Density of compound Poisson process

Is the probability density function (pdf) of the Compound Poisson $X(t)=\sum_{i=1}^{N(t)}Y$ known? Where $N(t)$ is a Poisson process and $Y$ is normally distributed with mean $\mu$ and variance ...
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101 views

Graph measurability (difficult but interesting)

Let $\mu(\cdot)$ be a probability measure on $W \subseteq \mathbb{R}^p$, so that $\int_W \mu(dw) = 1$. Consider a locally bounded function $f: \mathbb{R}^n \times \mathbb{R}^m \times W \rightarrow ...
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93 views

Polarisation in proving Kunita-Watanabe identinty

Kunita-Watanabe identity: Let $M,N$ be local martingales, $H$ be a locally bounded previsible process, then $$[H\cdot M,N]=H\cdot[M,N],$$ where $[M,N]$ is covariation. I am going though the proof, ...
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112 views

Induction on Uniform Boundedness

This question gives interesting insights on whenever uniform boundedness can be "iterated". Let $\mu(\cdot)$ be a probability measure on the closed set $Z \subseteq \mathbb{R}^p$, so that $\int_Z ...
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35 views

Uniqueness of function representation as a mean value

Let $f(x)$ be a smooth function from $\mathbb{R}^n_{+}$ to $\mathbb{R}_{+}$ (it may be analytic). Are there some sufficient conditions on such function $f$ such that $$ f(x_1 y_1, \ldots, x_n y_n) ...
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69 views

Nonlinear Optimization in Probability

Consider a locally-bounded function $f: X \times Z \rightarrow \mathbb{R}_{\geq 0}$, where $X \subset \mathbb{R}^n$ is compact and $Z \subseteq \mathbb{R}^m$ is closed. Let $m(\cdot)$ be a measure on ...
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289 views

Proving that a non-negative submartingale converges if the compensator has a finite limit

I'm stuck on the following problem: Let $X_n$ be a non-negative submartingale with Doob-Meyer decomposition $X_n = X_0 + M_n + A_n$, where $M_n$ is the martingale part and $A_n$ is a strictly ...
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34 views

Theorem for number of required samples

What is the name of the Theorem that ensures that it is enough to sample $O(\epsilon^{-2} \log \delta^{-1})$ if one wants with probability $1-\delta$ an estimate that is correct within $\pm ...
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133 views

Definition of probabilistic metric space

From Wikipedia Let $D+$ be the set of all probability distribution functions $F$ such that $F(0) = 0$: $F$ is a nondecreasing, left continuous mapping from the real numbers $\mathbb{R}$ into ...
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229 views

How is the Signal-to-noise ratio distance defined?

For two probability measures, or two random variables, I wonder how their Signal-to-noise ratio distance is defined? I encounter this concept in Wikipedia. For a probability measure on $\mathbb{R}$, ...
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247 views

normal random variable distribution

i have such problem in the book of Applied statistic and probability for Enigneering and need some help to solve it.problem is following: Let random variable X denote a measurement from a ...
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165 views

Continuous-time Markov process - need help with flux and discrete-time equivalent

I have a continuous-time markov process and I need to calculate the following transition frequencies matrix (aka intensity matrix) transition probabilities all parameters which define permanence ...
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170 views

Span of random variables equals the space of square integrable random variables

Supposing $Y_1,Y_2,\cdots, Y_n$ be random variables such that $Y_i \in \mathcal{L}^2(\Omega,\Sigma,P)$ for all $i$. What are the conditions under which $$\mathrm{span}(Y_1,Y_2,\cdots,Y_n) = ...
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483 views

Martingale transform question

I was reading my notes and I was having trouble understanding theorem 4.3 below. I understand essentially what it is saying, but to me its simplying stating something rather intuitive? That given ...
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70 views

Purpose of studying measures by some/all measurable mappings that induce them

I wonder what benefits or purposes can be obtained, through studying measures completely in terms of some or all measurable mappings that can induce the measures on their codomains? Since different ...
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136 views

Probability to hit a real number

Given the function $$y=mx$$ defined in $\mathbb{R^2}$ with $m\in\mathbb{R}$ is it possible to give a proof that the probability for a dart to hit the line defined by the previus function is zero? The ...
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119 views

Incrementally compute the conditional entropy

Is it possible to compute a conditional entropy (see the two following formulas) in an incremental manner ? That is, the sets C and K are not fix: each time we have a new element c, set K may increase ...
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233 views

Random walk with increasing step size

A random walk in the plane with step size $f(n)$ is cool if there is some constant $C$ such that it returns within a distance $C$ of the origin infinitely many times with probability $1$. At the ...
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603 views

So is this, finally, the difference between convergence in probability and almost sure convergence?

I've been trying to come up with a intuitive, practical distinction between convergence in probability and convergence almost surely. Can someone please tell me if the following is correct? Let $X_n$ ...
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55 views

Green kernel of Hunt processes

Let $X$ be a regular Hunt process on $R^+$ with starting at $x$ and $T_y :=\inf\{t>0: X_t=y\}, y\in R^+$. $G_q(\cdot,\cdot), q >0$ denotes Green kernel of the process $X$. We have the following ...
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54 views

Preference of Axioms

Why do some people prefer the following axiom (e.g. deFinetti) If $A,B \in \mathcal{B}$ where $\mathcal{B}$ is a $\sigma$-algebra of sets and $A,B$ are disjoint then $P(A \cup B) = P(A)+P(B)$ ...
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610 views

How is the law of a stochastic process defined?

Let $(Ω, F, P)$ be a probability space, $T$ some index set, and $(S, Σ)$ a measurable space. $X : T × Ω → S$ is a stochastic process. Let $S^T$ be the collection of all functions from $T$ into ...
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143 views

Proving that a sequence of random variables satisfies Lindeberg's condition

I have a sequence of iid random variables $X_n$ with zero mean and constant variance $\sigma^2_n$. Let $S_n=\sum_{j=1}^{n}{(j-1)X_j}$. In order to prove asymptotic normality I need to prove first ...
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72 views

Existence of measure factorization

Let $(\Omega,\mathcal F,\mathbb P)$ a probability space, $(X,\mathcal B)$ a measurable space and $m$ a probability measure on $\Omega\times X$ such that its projection on $\Omega $ is equal to ...
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130 views

Cesaro mixing time: Show $t_m(2^{-k}) \le k t_m(1/4), k \ge 1$

Let $(X_t)_{t \ge 0}$ be a finite Markov chain with state space $\Omega$, transition matrix $P$ and stationary distribution $\pi$. Let $\| \cdot \|$ denote the total variation distance and define  ...
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110 views

When can a measurable mapping be factorized?

Problem 13.3 of Probability and Measure by Billingsley states: $(\Omega, \mathcal{F})$ and $(\Omega', \mathcal{F}')$ are two measurable spaces. Suppose that $f: \Omega \rightarrow ...
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179 views

What is a $\chi^2$-test?

I solved part (a) of this problem but I don't understand what a $\chi^2$ test is in part (b) (Wikipedia did not help me): Let $X_1,\ldots,X_n$ be a random sample from a distribution with pdf ...
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91 views

Studying the maxima of columns of a random matrix as a point process

Consider a matrix, $S$, of i.i.d. real RVs : $X_{ij}$ for $1 \leq i \leq s$, $1 \leq j \leq n$. Let $F$ denote the distribution of $X_{ij}$. For $1 \leq j \leq n$, consider $Y_{j}^{(1)} = \max_{i} ...
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48 views

Bias in Resampling

I'm currently doing some work with Particle Filters, a sampling-based method for computing expectations of functions with respect to dynamic (ie: time-variant) random variables. For example, consider ...