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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1answer
220 views

Pointwise convergence almost surely of an Approximation sequence

Unfortunately I've trouble to see the following: If you work with stochastic process $X$ you often want to approximate this in the following sense, define: $$ X^{(n)}(s,\omega) = ...
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1answer
242 views

How does this follow? Markov Chain and conditional expectation question.

I have the following from a book: Assume that $$ P_x(\tau_C \circ \theta_{(k-1)N} > N|F_{(k-1)N}) = P_{X_{(k-1)N}}(\tau_C > N). $$ Integrating over $\{ \tau_C > (k-1)N\}$ using the ...
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2answers
314 views

Distribution of Maximum of Sum of Sum of Gaussians

Let $X_i$ be a sequence of i.i.d. standard normal random variables. Let $Y_i=\sum_{k=1}^iX_k$ and $Z_i=\sum_{k=1}^iY_k$. I am interested in upper and lower bounds for $P(\sup_{1\leq i\leq m}|X_i|\leq ...
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0answers
587 views

Definition of convergence in distribution

My question is convergence in distribution seems to be defined differently in Wikipedia and in Kai Lai Chung's book. My view is that the one by Wikipedia is a standard definition of convergence in ...
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1answer
100 views

Prove that the ball is in the bag

In a bag there are $15$ balls.Five are green, five are yellow and five are white. The balls of each color are numbered from $1$ to $5$. Now,suppose that in the bag there are only some of the ...
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0answers
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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1answer
81 views

Is the conclusion(last line) correct? If not, what's wrong with this proof ? (Probability Density Estimation, Bayesian inference)

(Probability Density Estimation, Bayesian inference) $x_1$ and $x_2$ are independent random variables, so $p\left(x_1,x_2\right)=p\left(x_1\right)\cdot p\left(x_2\right)$ and ...
2
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1answer
117 views

How are Tr(AB) results restricted?

In quantum mechanics one postulates that for each state $i$ there is a matrix $A_i$ and for each measureable dynamical variable $j$ (velocity etc.) there is a matrix $B_j$. Both are Hermitian matrices ...
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1answer
270 views

Convergence of moments and absolute moments of random variables

Let $X$ and $(X_n)_{n\geq 1}$ be random variables such that $X_n\to X$ in distribution. Assume that $\sup_n E[|X_n|^r]<\infty$ for some $r>0$. Then how do I show that $E[|X|^r]<\infty$ and ...
2
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2answers
1k 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 ...
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1answer
4k views

Martingale that is not a Markov process

1. On the internet, it is suggested that $$ X_t=\left(\int_0^t X_s \;ds\right)\;dW_{t} $$ is a martingale, but not a Markov process. I understand that the process $$ I_t(C)=\int_0^t C_s \; dW_s$$ ...
2
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1answer
182 views

Some basic questions on Markov chains (Durrett)

If you have a state space $S$, usually I think of a Markov chain $X_n$ on it as $X_n$ takes values in $S$ and satisfies the obvious Markov property and so on. In Durrett's book, he says one should ...
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0answers
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 ...
6
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1answer
733 views

Convergence in law and uniformly integrability

I'm looking for an elementary way of showing the following. If $(X_n)$ and $X$ are random variables such that $X_n \to X$ in distribution and such that $\{X_n\mid n\geq 1\}$ are uniformly integrable, ...
3
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2answers
88 views

Unique continued fraction

If $x$ is a uniformly random number in $[0,1]$, what distribution should the $n$-th term in its continued fraction expansion follow? What is the expected vale of $a_n$ in $[a_0;a_1,a_2,\dots]$? Here ...
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0answers
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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1answer
567 views

spectral norm of random matrix

Suppose $A$ is a $n \times n$ random matrix with centered Gaussian (real) i.i.d entries with variance $\sigma^2/n$. What to we know about the spectral norm $s(A)$ of $A$, that is $\sqrt{\rho(A^t A)}$ ...
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2answers
136 views

Small question on uniformly distributed random variables

The question says the following: let $X$ be a random variable with uniform distribution on $[-1,1]$. Does $X^{-1}$ have a finite expectation? I was just working it using the definition but I'm ...
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2answers
184 views

Distributions of Stochastic Processes

I learned that Brownian motion is any stochastic process $(W_{t})_{t\geq0}$ that satisfies four well-known properties. But I still don't understand how these four properties uniquely determine ...
4
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1answer
214 views

Weak convergence of a triangular array of Bernoulli-RV's

I am looking at the series $$X_{1,1},$$$$X_{2,1}, X_{2,2}$$ $$X_{3,1},X_{3,2},X_{3,3}$$ $$\dots$$ of independent r.v's with $p_n:=P(X_{n,k}=1)=n^{-\frac{1}{4}}$ and ...
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1answer
163 views

find a mapping of two probability mass function to another probability mass function

Let $p$ and $q$ be arbitrary probability mass functions of two discrete random variables. I need examples of functions $F(p,q)$ such that $r = F(p,q)$ and $r$ is a probability mass function for some ...
2
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1answer
46 views

Probability/ Counting question about Error Rates

This is probably a simple question to most of you but I wasn't seeing a clear solution. I was programming the other day and in one part there is a batch insert into a database of about 30 items at a ...
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1answer
308 views

How do define the entropy of a probability measure?

How do define the entropy of a probability measure? What is the motivation to define it? What is the importance of a probability measure to minimize or maximize the entropy? In a finite sample space ...
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2answers
2k views

Independence of sample mean and variance through covariance

I have seen the text book derivation where the independence is established through factoring the joint distribution. But has anyone tried to prove that the covariance is zero?. Let $Z_{i}$ come from ...
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1answer
322 views

Definition of the Brownian motion

The way I understood the definition of a Brownian motion $B_t$ in $\mathbb R$ is that it consists of two parts: We first define the finite-dimensional distributions $$ ...
7
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2answers
826 views

Is this local martingale a true martingale?

Using the Ito's formula I have shown that $X_t$ is a local martingale, because $dX_t=\dots dB_t$, where $$X_t = (B_t+t)\exp\left(-B_t-\frac{t}{2}\right),$$ $B_t$ - is a standard Brownian motion I ...
2
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1answer
72 views

The property preserved under the perturbations of the kernel

For a measurable space $(E,\mathcal E)$ and a Markov kernel $P:E\times \mathcal E\to[0,1]$ there is a unique homogeneous Markov chain $X$. The first return time is defined as $$ \tau_A = \inf\{k\geq ...
3
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1answer
138 views

Levy processes with no positive jumps

Let X be a Levy process with no positive jumps and $\tau_y:=\inf\{t> 0: X_t > y\}$ then we have $$X_{\tau_y}=y\text{ on }\{\tau_y <\infty\}.$$ Could you explain that why? and does it hold ...
0
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1answer
1k views

How the sum of 2 random variables is also a random variable…

Say I have two random variables $X$ and $Y$ defined on a probability space ($\Omega,\mathbb{F},\mathbb{P}$). To prove $X+Y$ is also a random variable, I need $\{\omega:(X+Y)(\omega)\le x\}\in ...
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0answers
146 views

How to solve equation involving binomial coefficient?

I'm reading this paper which says If we have $$ \binom n d p^{\binom d 2} = 1 $$ where $ 0 < p \le 1$, then $$ d = 2 \log_bn - 2 \log_b \log_b n + 2 \log_b\left(\frac 1 2 e\right) + 1 + O(1) ...
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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 $X_n$ ...
2
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2answers
115 views

Computing $E[X\| \mathscr{G}](x)$ from knowledge of which $G$ contain $x$

In Section 34 (page 445) of Billingsley's Probability and Measure (3rd ed.), he says, regarding conditional expectation, that since $E[X \| \mathscr{G}]$ is $\mathscr{G}$-measurable and integrable, ...
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2answers
168 views

Defining the distribution for a complicated random variable

I want to come up with at least the expectation, and at best, the cdf, for a variable $Z$ that I think of as the result of a process and am not quite sure how to translate into equations. Let $F(x) = ...
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2answers
112 views

How can I prove this inequation $\Pr\{X+Y<t\} \le \Pr\{X<t\} \Pr\{Y<t\}$

Could you please help me to prove the inequality probability as follows: $\Pr\{X+Y<t\} \le \Pr\{X<t\} \Pr\{Y<t\}$ where $X$ and $Y$ are non-negative independent random variables with common ...
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1answer
335 views

Conditions for the disintegration theorem to hold

The disintegration theorem says that under certain conditions, a probability measure $\mu$ on a measurable space $$ the existence of Let $Y$ and $X$ be two Radon spaces (i.e. separable metric ...
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1answer
397 views

Conditional probability and the disintegration theorem

I was wondering how conditional probability and the disintegration theorem are related? How is the conditional probability given by the disintegration theorem? I don't quite understand what ...
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2answers
384 views

Continuity of the Characteristic Function of a RV

Defining the Characteristic Function $ \quad \phi(t) := \mathbb{E} \left[ e^{itx} \right] $ for a random variable with distribution function $F(x)$ in order to show it is uniformly continuous I say ...
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1answer
121 views

Transformation of a Bivariate Variable

Given the variables $X$ and $Y$ which are correlated and jointly Bivariate with corresponding probability distribution function denoted by $f_{X,Y}(x,y)$, and given the linear relationship $Z = ...
3
votes
1answer
728 views

Are functions of uncorrelated random variables still uncorrelated?

Suppose $X$ and $Y$ are real-valued random variables, and $f$ and $g$ are Borel measurable real-valued functions defined on $\mathbb{R}$. If $X$ and $Y$ are independent, then I know that $f(X)$ and ...
3
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1answer
466 views

Fubini's theorem and uncorrelated/independent random variables

Suppose $X$ and $Y$ are two real-valued random variables, and $f:\mathbb{R}^2\to \mathbb{R}$ is Borel measurable. I was wondering if $X$ and $Y$ being uncorrelated or independent implies that $$ ...
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2answers
3k views

Self-study on probability.

What book do you recommend for self-study of probability theory? I have a rather significant gap in that area (in lame terms sometimes I feel I don't get it) and need to try (strugle more likely) ...
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1answer
313 views

Tightness of distribution

Let's assume we have sequence $(X_n)$ of r.v. on a probability space $(\Omega,\mathcal{F},P)$ and we denote by $\mu_n$ the distribution of $X_n$. Now we assume that the sequence of distributions is ...
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2answers
483 views

Simple proof that stationary birth-death chains are reversible

A Markov chain with state space $\mathbb{Z}$ is a birth-death chain if the transition probabilities satisfy $p(x,y) = 0$ for $|x-y| > 1$. That is, the only possible transitions are to move one ...
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1answer
96 views

Existence of a general-purpose (almost) universal optimization strategy

From Wikipedia about interpretations of no free lunch theorem A conventional, but not entirely accurate, interpretation of the NFL results is that "a general-purpose universal optimization ...
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1answer
498 views

Application of Kolmogorov Zero-One Law

I'm having some trouble with an exercise. Suppose $(X_n)_n$ is a sequence of independent r.v with $\mathbb{P}(X_n > x) = e^{-x}$ for positive x and one otherwise. The exercise first asks for what ...
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2answers
98 views

Stuck at this probability problem

Let's say you have an 1/8 percent chance of getting shot, per day. The days are independent. Then let X be the number of days up to (and including) the next time you get shot. How would I go about ...
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0answers
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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1answer
279 views

Radical Applications of Algebraic Topology

Are there any radical applications of algebraic topology? For example, in probability theory we look at sample spaces. Suppose the sample space is a torus (for example). Would computing homology ...
2
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1answer
306 views

Understanding no free lunch theorem

From Wikipedia: $Y^X$ is the set of all objective functions $f$:$X$→$Y$, where $X$ is a finite solution space and $Y$ is a finite poset. The set of all permutations of $X$ is $J$. A random ...
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1answer
132 views

Whittle estimator and Hurst parameter

I'm in trouble finding the optimal estimator of the Hurst parameter in the fractional Brownian motion. Is there something better than the Whittle estimator? Thanks in advance