0
votes
1answer
22 views

Total variation inequality [on hold]

In an article I've stumbled upon this inequality: $$ V(f) \leq C(f)s(f) $$ where $f$ is a Lipschitz$(1)$ density with Lipschitz constant $C(f)<\infty$ and support $s(f)$. How can this be derived? ...
0
votes
1answer
55 views

Measure extension theorem(unique) [closed]

Please give an example of two probability measures $\mu \not = \nu$ on $\cal{F} $= all subsets of {1, 2, 3, 4} that agree on a collection of sets C with $\sigma(C)=\cal{F}$ . thanks in advance.
0
votes
0answers
25 views

differential equation with random coefficient

I am confused with a problem I encountered at hand, not on how to work on it but rather understanding the problem itself: Let $A(x;\omega)$ be a random field taking values in $[a,b]$ where $a,b < ...
1
vote
1answer
30 views

What can we say about correlation coefficients?

If we are looking at sales and inventory data with a correlation of 50%, what can I conclude? My intuition tells me that 50 % of the movement in one of the variables is attributed to the others ...
2
votes
1answer
115 views

How to Show the following converges to $e^{\frac{t^2}{2}}$

How to prove that $$\lim_{n\to\infty}\left[\left[e^{t\sqrt{\frac{1-p}{np\vphantom{()}}}}-1-t\cdot\sqrt{\frac{1-p}{np}}-\frac{1}{2}t^2\left(\frac{1-p}{np}\right)\right]\cdot p ...
0
votes
0answers
39 views

Card Shuffling and Convergence in Probability

There are $4n$ cards, and we denote the set of cards with number $4k,k \in \{1,2,\ldots,n\}$ as $S$. The we shuffle the whole cards randomly, which means that each permutation will happen with the ...
0
votes
1answer
35 views

How to show that the following limit converges to some order of 1/n term .

How to prove that $$[e^{t\sqrt{\frac{1-p}{np}}}-1-t\cdot\sqrt{\frac{1-p}{np}}-\frac{1}{2}t^2(\frac{1-p}{np})]\cdot p ...
2
votes
1answer
33 views

law of iterated logarithm

Wikipedia claims see this link that the law of the iterated logarithm marks exactly the point, where convergence in probability and convergence almost sure become different. It is apparent from the ...
1
vote
1answer
21 views

Is the characteristic function of a multivariate normal distribution a real analytic function?

The characteristic function of a multivariate normal distribution with mean $\mu \in \mathbb R^n$ and covariance $\Sigma \in \mathbb R^{n \times n}$ is given by \begin{align*} e^{it^T\mu - ...
0
votes
1answer
40 views

Prove that a Modified Cantor Distribution is Atomic.

Consider a measurable space $\{\mathcal{I},\mathcal{B}\}$, where $\mathcal{I} = [0,1]$ and $\mathcal{B}$ are the Borel sets on $\mathcal{I}$. And also, denote $\mathcal{C}$ as the cantor set on ...
0
votes
1answer
22 views

Return time Markov chain

I have been wondering about this for quite a while now that I found in a textbook in the proof that an irreducible positive recurrent markov chain $(X_n)$ has a stationary distribution Let $t_i$ ...
3
votes
3answers
79 views

convergence to exponential with order 1/n

We know that limit $\left(1+\dfrac{x}{n}\right)^n$ converges to $e^x$ but how can we prove that limit $\left(1+\dfrac{x}{n}+o(\frac{1}{n})\right)^n$ converges to $e^x$.
5
votes
1answer
186 views

Properties of Markov chains

We covered Markov chains in class and after going through the details, I still have a few questions. (I encourage you to give short answers to the question, as this may become very cumbersome ...
0
votes
1answer
21 views

Two notions of conditional expectation

For a randomn variable $Y$ and an event $B$ we can define: $$E(Y \mid B) = \frac{E(1_B\cdot Y)}{P(B)}$$ as the conditional expectation. Now, for a sigma algebra $\mathcal{B}$ and sets $B$ in it you ...
1
vote
0answers
21 views

Spectral Representation for a real valued process

So I just finished reading a section in a book which discusses how every stationary stochastic process $\xi(t)$ can be expressed as $\xi(t)=\int_{\mathbb{R}}e^{it\lambda}\,dZ(\lambda)$ where ...
1
vote
0answers
35 views

Central limit theorem does not converge to random variable

Recently, we investigated whether the expression in the central limit theorem converges to a random variable pointwise almost sure? The answer was negative due to $P ( \text{limsup} ...
0
votes
2answers
35 views

evaluating an integral with complex exponential (spectral density)

I am having a hard time figuring out how to evaluate this integral from a book that I am reading. Here's the background info but I doubt it's highly relevant to the problem at hand: $X$ is a real ...
1
vote
0answers
28 views

monotonicity of a complex function referring normal distribution

In my research I need to make clear the following point: Suppose that a random variable $\theta\sim N(\mu, \sigma^2)$. There are two imperfect signals about $\theta$: $X=\theta+\sigma_x\xi$ and ...
0
votes
1answer
21 views

Sub sigma algebra and probability spaces — definition

I am reading this book and I am a bit lost with the definitions because they are not provided and I can't seem to find it online: Let $L_2(\Omega,A,P)$ be a probability space such that $f \in L_2$ ...
1
vote
1answer
55 views

Eigenvalue markov chain

I have a questions: We said that if we have a positive recurrent Markov chain, then there is a unique stationary distribution. 1.) Does this mean that if I have several positive recurrent classes, ...
1
vote
0answers
39 views

Transient/Recurrent Markov chain

I am currently studying the concept of recurrent and transient states and was wondering about the following: Is this concept dependent on the initial distribution? Let me take this example: You can ...
1
vote
1answer
60 views

Probability of event in normal distribution

Let $X$ be a random variable that is normally distributed and $X_1,\ldots,X_n$ be (independet) copies of $X$, then we can estimate this probability by using a simple Monte-Carlo estimator: $p := P (X ...
0
votes
1answer
29 views

Is this a Markov chain property

For $A,B$ measurable sets and $(X_n)_n$ a Markov chain. Do any of the following properties hold? $$P(X_2 \in B | X_1=x_1,X_0 \in A) = P(X_2 \in B|X_1=x_1)$$ or $$P(X_2 \in B|X_1 \in A,X_0=x_0) = ...
0
votes
0answers
27 views

Strong Markov property and its meaning

Given a sequence of random variables $(X_n)_n$ (fulfilling the Markov property) and a stopping time $\tau$ such that $P(\tau < \infty)=1$, we have that ...
2
votes
1answer
66 views

Convergence almost sure pointless?

A very common type of convergence in probability theory is 'almost sure convergence'. I don't understand why this type is used at all. In principle, we should always be able to substitute it by a ...
1
vote
3answers
34 views

Methods to distinguish continuous probability distributions

I read in the Wikipedia article for Variance The variance is one of several descriptors of a probability distribution. In particular, the variance is one of the moments of a distribution. In that ...
0
votes
2answers
37 views

Brownian Bridge conditional probability

The problem is to show that the density $P[W_{t_1} \in dx_1,...,W_{t_n}\in dx_n | W_T = b]$ is the density of a Brownian bridge from $a$ to $b$. $W$ is Brownian motion. The density of a Brownian ...
0
votes
1answer
39 views

Fubini Question in context of Independence

I am trying to show that if $X_t$ is some process and there is a function $p$ such that $$P[(X_{t_1},...,X_{t_n}) \in A_1 \times...\times A_n] = \int_{A_1 \times...\times A_n} ...
1
vote
1answer
32 views

American Put question

If the interest rate is zero. Then show that the optimal exercise for an american put option is always the terminal time. That is, it is equivalent to a european put option.
0
votes
1answer
53 views

Difference between Borel Sigma algebra and Cylindrical sigma algebra?

I see that there are two differen concepts for Sigma Algebras on cartesian products over the real numbers. The first one is the Borel Sigma Algebra created by the product topology. The other one is ...
0
votes
1answer
36 views

Multivariate normal distribution independet iff uncorrelated

I found a few threads about this but none of them answered my question. I am supposed to show that if you have random variables $X_1$,$X_2$ that are gaussian distributed and they fulfill that ...
1
vote
1answer
76 views

a generalization of normal distribution to the complex case: complex integral over the real line

How to prove $\int_{\mathbb{R}} e^{-\frac{(x+it)^2}{2}}dx=\sqrt{2\pi}$ for any $t\in \mathbb{R}$? I only obtained the case that $t=0$, $\int_{\mathbb{R}} e^{-\frac{x^2}{2}}dx=\sqrt{2\pi}$. Thanks.
0
votes
0answers
12 views

Trouble with Largrange Multipliers and expectation.

I am reading the following argument: Maximize $E[Log(X(T))]$ subject to $E[Z(T) X(T)]=x$, where $X(T),Z(T)$ are random variables, $x$ is a constant, and E is expected value (You can read this as ...
1
vote
1answer
43 views

LimSup of Random Variable

I have a seemingly trivial question. Why does $$\forall a\in\mathbb{R},\mathbb{P}(\limsup X_n>a)>0\Rightarrow \mathbb{P}(\limsup X_n=\infty)=1$$ Clearly, we don't have (at least trivially), ...
5
votes
1answer
137 views

Continuous probability distribution with no first moment but the characteristic function is differentiable

I am looking for an example of a continuous distribution function where the first moment does not exist but the characteristic function is differentiable everywhere. Cauchy distributions do not ...
0
votes
0answers
19 views

An online reference for proof of BIC approximation

I have been looking for an online note giving a proof of the BIC asymptotic approximation. It has been surprisingly difficult to find. In fact I only found the original paper by Schwarz 1978, which ...
0
votes
0answers
20 views

Understanding certain parts of the proof of Helly's Selection Theorem

I have read through the following proof of Helly's Selection Theorem. There are just two parts, which I have highlighted, that are left for the reader to fill in, and I would like to know how to prove ...
0
votes
1answer
46 views

Continuity of a function defined by means of the Lebesgue measure

Let $f:\mathbb{R} \rightarrow \mathbb{R}$ be a Lebesgue measurable function and $\phi(x)=\lambda ( \lbrace{ t: f(t) >x \rbrace} )$. Prove that $\phi$ is right-continuous but not necessarily ...
1
vote
0answers
43 views

Continuous transition kernel for Markov Chains

I am trying to show that the stationary distribution for a Markov Chain on a continuous state space can be obtained by building a transition density kernel, which obeys the detailed balance rule where ...
0
votes
1answer
41 views

solve the functional equation

Let $\phi : R-> C $ (complex numbers) $\phi(0)=1$ $ \phi(-t) = \overline{\phi(t)} $ ( continuous and bounded) solve the functional equation: $Re \phi(t)= \phi(t) \overline{\phi(t)}$ This is all ...
0
votes
0answers
17 views

Scaling model output to be between 0 and 1

I have fitted Cox model and the output is generated as: $e^{\beta x}$, where $\beta$ is the coefficient. Now, I would like to have the model output ranging between $0$ and $1$. I'm currently using ...
0
votes
1answer
40 views

Average of IID Cauchy RVs

Suppose that $X_i$'s are iid Cauchy RV's with pdf $f_u (x) = \frac{1}{\pi} \frac{u}{u^2+x^2}$. I am aware that the RV $Y:=\frac{1}{N}\sum_{k=1}^N X_k$ has the same density as the $X_i$'s. I am trying ...
0
votes
1answer
33 views

measurable function and composition of function

Show that if $f$ is a measurable function and $g$ is a continuous function on $\Bbb R$ then $g\circ f$ is measurable. please tell me how to prove it !
1
vote
0answers
19 views

Computing area/ space of intersection between a pair of Beta distribution/ Dirichlet distributions

I need to compute the area/ space of intersection between a pair of Beta distribution/ Dirichlet distributions. As I am a non mathematics guy, it will be great if someone helps me out with the ...
1
vote
1answer
45 views

Time Series Analysis.Calculate the variance mean and autocorrelation of the time series below.

For the following time series, calculate the mean, varia nce and autocorrelation function: (a) Y_t=5+Z_t+ 0.6Z_t-1
0
votes
0answers
10 views

Problem on Smoothness of Rate functions in Large Deviations Theory

I was reading Fields Institute Monographs, Large Deviations by Frank den Hollander, whereupon I encountered a few exercise problems. While solving one of them, I was required to prove a (possibly) ...
2
votes
0answers
97 views

A probability problem with multivariate Gaussian distribution

I am a computer science guy, not a mathematician so kindly excuse me if there is any ridiculous error in my problem description. I have two clusters $C_1$ and $C_2$ in a feature space spanned by $k$ ...
1
vote
1answer
23 views

Animal jumping equally likely to the left and right.

We have an animal that starts at the point zero and jumps equally likely to the left(-1) and right(+1). After 2k jumps, where $k \in \mathbb{N}$ ,it arrives back again at the point $0$. The question ...
1
vote
1answer
37 views

What is the name of this sequence/progression?

Does the following sequence form some special sequence/progression (such as arithmetic progression, geometric progression, hypergeometric progression, and more): $$ p_k: = \frac{\lambda^k}{k!} ...
0
votes
2answers
47 views

What is the meaning of Common Support here

I am reading a notes in statistical inference, and I am constantly being confused about the term 'common support', i hardly find any definition of this,here is an example, 'Suppose S is a space of ...