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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19 views

How to do kernel density estimation for data stored in polar system?

I want to do a kernel density estimation for a wind speed dataset. The data are stored in speed and angle, like this ...
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15 views

Does $n$-dimensional convolution integral always exist

If $X_1,X_2,\ldots,X_n$ are independent random variables with densities $f_1,f_2,\ldots,f_n$, then we know that the density of their sum $X_1+\cdots+X_n$ is given by $$ g(z)=\idotsint f_1 ...
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17 views

Interchange intersection and union in proof of Blumenthal’s zero-one law

I am trying to prove Blumenthal's zero-one law using Kolmogorov's zero-one law. I use that $B_t$ Brownian $\iff$ $tB_{1/t}$ Brownian. Can I change the intersection and union as follows? Start of my ...
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25 views

differential stochastic equation and solutions

Let E(K):$Z_0=0$ and $dZ_t=K_t(B_t-Z_t)dt+\alpha K_td \beta_t$ with B and $\beta$ two brownian motions, $\alpha>0$ and K a continue function. 1)Show that E(K) has a solution Z with $E(\int_0^1 ...
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30 views

Proving that if $X_n$ is uniformly bounded, and if $X = \lim_{n}X_n$ with probability 1, then $E[X] = \lim_{n} E[X_n]$?

In Billingsley's Probability and Measure Book, he has a result otherwise known as the Bounded Convergence Theorem: Normally we would use the Lebesgue integral in lieu of the expectation and also ...
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45 views

Balls and Bins [Raab 1998 proof]

I cannot work out the proof in one of the steps. The following is copied from the original paper “Balls into Bins” — A Simple and Tight Analysis: The case when $n\log n \ll m \leq n ...
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22 views

Convergence of Moment Generating Function Given a Bound on the Random Variable, ie $\Bbb{E}(e^{\lambda X} | X \le K) \uparrow \Bbb{E}(e^{\lambda X})$

I'm trying to show that, given an integrable random variable in $\Bbb R$, for all $\lambda \ge 0$, $$\Bbb{E}(e^{\lambda X} | X \le K) \uparrow \Bbb{E}(e^{\lambda X}).$$ I can show that the left hand ...
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40 views

Is there a stable probability distribution on the rational numbers?

Does there exist a (non-trivial) probability distribution on the rational numbers $$\sum_{r\in\mathbb{Q}}p_r=1$$ with $0\leq p_r$, which is stable, meaning that the sum of two i.i.d. random variables ...
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83 views

Reference for inequalities involving limsup and liminf

What book contains proofs of those or contrapositives of those (either given or in the exercise sections)? I can't seem to find any in Williams' Probability w/ Martingales, Royden and Fitzpatrick's ...
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28 views

Finding the score vector and Fisher information matrix for multivariate case

In my review questions I have a question that indicates $X_{i}$ follows a $Bin(1,p_{i})$ distribution where $p_{i} = (1+e^{-\alpha - \beta q_{i}})^{-1}$ and the $q_{i}$'s are known constants. I am ...
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30 views

Tightness of probability measures

Let $\mu_\alpha$ be an uniform distribution on $[0, a_\alpha]$ ($a_\alpha >0$ for each $\alpha$). Prove that {$\mu_\alpha$} is tight if and only if $\sup_\alpha a_\alpha < \infty$. I've proved ...
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24 views

Proving almost sure finiteness of a stopping time

Let $M_t$ be a local martingale and $S_t = \sup_{0 \leq s \leq t}M_s$ its running supremum. How can I show that the stopping time $T=\inf\{t \geq 0 : S_t - M_t = a\}$ for an arbitrary $a>0$ is ...
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16 views

Superposition of two Renewal Processes

Assume we have a renewal process $P_1$ with inter-renewal distribution $p_1(x)$ and rate $\lambda_1 = \lim_{t \to \infty} \frac{N_1(t)}{t}$, where $N_1(t)$ is the total number of renewals of $P_1$ ...
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20 views

Independence of two random variables 2

Assume we put points on the two dimensional $x-y$ plane according to Poisson distribution, consider $r_1$ is the location of the closest point to origin and $r_2$ is the location of the second ...
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22 views

Convergence in distribution of $n(1-\xi_n^{(n)})$ and $n \dot \xi_k^{(n)}$ where $\xi_n \sim U(0,1)$

Let $\xi_1$, $\xi_2$, $\xi_3$,... a sequence of independent random variables with a uniform distribution on $(0,1)$. For each $n>1$ we range first $n$ the random variables (it means $\xi_1$, ...
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36 views

Probability of an sequence of independent random 2-dimensional vectors

Let $(X_n\mid n=1,2,\ldots)$ be a sequence independent of random $2$-dimensional vectors, where, for each $n$, $X_n$ is uniformly distributed on the square with vertices $(\pm n,\pm n)$. Calculate the ...
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26 views

Continuity of Median

There are three arbitrary random variables on $\mathbb{R}$: $X$, $Y$, $Z$. The median is defined as $\mathrm{Me}(X) = \mathrm{sup} \left\{t: F_X(t) \le \frac12\right\}$ (so it's hopefully always ...
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20 views

Gaussian measures from a finite dimensional space to an infinite dimensional space

It's well known that the characteristic function of a gaussian measure $\mu$ on $\mathbb{R}^d$ is given by: $$ \hat{\mu}(\xi)=\int_{\mathbb{R}^d}exp(i\xi . x)\mu (dx),\qquad \xi \in \mathbb{R}^d $$ ...
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31 views

Higher Order Estimation Errors

I well know estimation measure is the so called minimum mean square error (MMSE) defined as: \begin{align} E[|W-\hat{W}(V)|^2] \end{align} where $W$ is a random variable (that we want to estimate) and ...
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35 views

Law of large number for a product of uniform iid random variables (stick breaking)

Let $(X_n)_n$, $n = 1, 2, \dots$, be an iid sequence of random variables uniformly distributed on $(0, 1]$. Set $S_0 = 1$ a.s. and, for $n = 1, 2, \dots$, set $S_n = \prod_{k=1}^n X_k$. Compare $S_n$ ...
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41 views

Triangular inequality for n-th step distributions

Assume that $p_n$ is the $n$-th step distribution of a random walk with state space $\mathbb{Z}^d$, i.e. $p_n(x,y)=\mathbb{P}(S_{n+1}=y\mid S_0=x)$, where $S_n=S_0+\sum_{i=1}^nX_i$ with $X_i$'s i.i.d. ...
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16 views

Alternate Markvov Chain Model.

In class we are working on DTMCs for past couple of week, and in our last lecture we did an example. Question was: In a city of nowhere it only rains if there are clouds for successive m-days. Any ...
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8 views

How is an adjunction map of a probability space well-defined?

The book says something like: Let $(\Omega, \mathcal{F}, P)$ be a probability space. Let $\Delta \notin \mathcal{F}$, suppose that for all $F \in \mathcal{F}$ such that $F \supset \Delta$, we have ...
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19 views

Generating stochastic processes from distributions of random variables

A stochastic process is a sequence of random variables $\{Y_t : t=0,\pm 1,\pm 2,\pm 3,\pm 4,\dots \}$. How is this determined by the set of distributions of all finite collections of $Y$'s? I do not ...
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22 views

Approximating a probability distribution by the moments

In a version of Levy's theorem we know that if we have a sequence of characteristic functions $\phi_n$, such that $\phi_n\rightarrow \phi$ pointwise, then $\phi$ is the characteristic function of a ...
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30 views

Calculate sample size given Conditional Probability and level of significance

On the basis of a pilot study, it was found that the predation probability for dark-coloured moths on a dark background is 0.10, in contrast to 0.90 on a light background. What should be the sample ...
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32 views

Example of a $\sigma$-additive function over a $\sigma$-algebra

While working in measure theory I've decided to study each theorem using a concrete example of concepts I deal with. In this case, I want proof that: If ϕ is σ-aditive function on a σ-field, ...
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11 views

Auto-correlation of random process (integration)

Consider the following problem: Suppose we know that $f_X(x)=1$, now I want to calculate the autocorrelation: $E[\underbrace{X(t)X(t+\tau)}_{g(t)}]=\int_0^1 g(t)f_X(x)dx=\int_0^1 ...
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34 views

Uniqueness of Limit in Convergence in Distribution

If $X_n \rightarrow X$ in distribution and $X_n \rightarrow Y$ in distribution , show that X and Y have the same distribution. My Approach: Suppose the distribution of X and Y are not same, i.e., ...
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39 views

Sum of two families of uniform integrable random variables

Suppose $\{X_n\}$ and $\{Y_n\}$ are two families of u.i(uniform integrable) random variables defined on the same probability space. Is $\{X_n+Y_n\}$ u.i? Proof Given $$\mathbb{E}[|X_n|\,I_{|X_n|\geq ...
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34 views

Central Limit Theorem: Lindeberg condition application

Let $0 < a_1 < a_2 < · · ·$ be fixed real numbers and let $\{X_k\}_{k \geq 1}$ be a sequence of i.i.d. random variables with zero mean and unit variance. Let $T_n = \sum_{i=1}^{n} a_iX_i$. ...
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23 views

Lebesgue measure vs Lebesgue-Stieltjes measure

Reading advanced probability theory book I've come across Lebesgue-Stieltjes measure. Could someone explain what is the difference between it and "standard" Lebesgue measure on $\mathbb{R}$? Thank ...
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15 views

Independent increment property

$X=(X_k, k \geq 0)$ is defined recursively by $X_0=1$ and $X_{k+1}=(1+X_k)U_k$ where $U_0,U_1,\ldots$ are independent random variables each uniformly distriubted on $[0,1]$. Determine if the process ...
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17 views

How to prove that $cov(f(X_1, X_2, \ldots ,X_n), g(X_1, X_2, \ldots ,X_n)) \geq 0$ for $X_1, \ldots, X_n$ independent and $f,g$ increasing?

I read in a talk that a consequence of the FKG inequality is that: $$ cov(f(X_1, X_2, \ldots ,X_n), g(X_1, X_2, \ldots ,X_n)) \geq 0 $$ for $X_1, X_2, \ldots ,X_n$ independent and $f,g$ increasing ...
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26 views

Showing that $E[X|G_1,G_2]=E[X|G_1]$, where $\sigma(X,G_1)$ and $G_2$ are independent

Suppose that $X$ is an integrable r.v. on probability measure space $(\Omega,F,P)$. Show that $E[X|G_1,G_2]=E[X|G_1]$, where $G_1,G_2$ are sub $\sigma$-algebras and $\sigma(X,G_1)$ and $G_2$ are ...
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28 views

Weak convergence function

I am studying for an exam and this was one of the hard problems in my textbook. Let $f_n(x) = 1 −\cos(2\pi nx)$ for n $\in$ N and x $\in$ $[0,1]$. Verify that $f_n$ is the density of a probability ...
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28 views

Convergence in distribution for continuous functions of random variables

If two sequences of random variables $\{X_n\},\{Y_n\}$ are such that $X_n \xrightarrow{d}X,\ Y_n \xrightarrow{d}Y $, i.e, they converge in distribution to $X$ and $Y$ respectively, where $X_n$ and ...
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18 views

Finding the marginal distributions of a binormal random variable

Let $\overline X$ be a binormal random variable with distribution $N_{\overline X}(\overline m, \Sigma)$ where, $\overline X = \left( \begin{array}{c} x \\ y \end{array} \right)$, $\overline m ...
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21 views

Normal $(\frac{n-1}{n})$-percentile asymptotic to $(2\log n)^{1/2}$?

I am working from Durrett's Probability: Theory and Examples, and I have encountered the following question: Suppose that $X$ is normally distributed, and $b_n$ is defined by ...
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57 views

Probability that Consecutive Partial Sums of Normals are Positive

Let $\{X_i\}$ be i.i.d. standard normals and let define $S_n = \sum_{k = 1}^n X_k$ to be their partial sums. Find $\mathbb{P}(S_1,S_2,S_3 >0)$. What I've Tried: We can set up the integral in a ...
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14 views

Questioning convergences in respect to $0$: almost surely, in probability, in distribution of random variable with $\varphi$ density given.

Questioning convergences in respect to $0$: almost surely, in probability, in distribution of random variables with $\varphi_{n}(x)$, density given. $$\varphi_n(x)=\frac{n}{\pi(1+n^2x^2)}, x\in ...
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20 views

Limit of expectation of L2 norm unbounded implies function ill defined

Given a random process $A(t,x) := \lim_{N\rightarrow\infty} A^N(t,x)$ where $ x \in (0,1)$ and $t \geq 0$ and given that \begin{equation} \mathbb{E}\left[ \int_0^1 A^N(t,x)^2 dx \right] = tN ...
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14 views

Why condition like j!=i is imposed on PDF of hypoexponential distribution?

I am currently working on network project in which i have to choose a predetermined number of nodes from entire network as Central Nodes. And this selection is based on the inter contact time between ...
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34 views

Composition of two probability kernels is measurable

Let $p$ be a probability (or Markov) kernel with source $(X,\mathcal{A})$ and target $(Y,\mathcal{B})$, and $q$ a probability kernel with source $(Y,\mathcal{B})$ and target $(Z,\mathcal{C})$. We can ...
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20 views

Proof of an expected value for a continuous random variable?

One more question I've been having trouble with, hopefully someone can point me in the right direction. The question is: For any positive continuous random variable $X$ with pdf $ f_X(x)$, show that ...
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34 views

Question about Slutsky's Theorem/Convergence of ratio of sequences of random variables

I have question about Slutsky's theorem, specifically regarding the property that $X_n\xrightarrow{p}X$, $Y_n\xrightarrow{p}c$, then $\frac{X_n}{Y_n}\xrightarrow{p}\frac{X}{c}$ (provided $c\neq0$ and ...
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17 views

Distribution of $L(\text{inf}\{t: |B_t| \ge x\})$.

Let $B_t$ be a standard Brownian motion. Let $L_t = L(t)$ denote the Brownian local time. Can anyone supply a reference as to the distribution of $L(\text{inf}\{t: |B_t| \ge x\})$? I know that ...
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27 views

If a sequence of Poisson RV's converges to $0$ show their parameters converge to $0$

Let $X_n$ be a sequence of random variables such that $X_i \sim Poi(\lambda_i)$. If $X_n$ converges to $0$ almost surely show that $\lim_n \lambda_n = 0$. If $X_n$ converges to $0$ almost surely, ...
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53 views

Integrating path of a diffusion process

Let $X_t$ be a process that satisfies the Ito diffusion process: $$dX_t = a(X_t) dt + b(X_t) dW_t$$ I am interested in approximate numerically $$I(\omega) = \int_0^t X_s ds$$ and eventually also ...
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18 views

Continuous I.I.D.'s on the interval?

Just something I am curious about. I know of infinite families of discontinuous I.I.D's on the unit interval (certain step functions), but I am curious if there are infinite families of continuous ...