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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Random variable, diagonalization type argument?

Let $d_j$ be i.i.d. random variables, with$$P(d_j = 0) = P(d_j = 1) = {1\over2}.$$Define$$X = 0.d_1d_2 \dots = \sum_{j = 1}^\infty {{d_j}\over{2^{j}}}.$$I know how to show that $X$ is uniformly ...
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45 views

Convergence in Probability of a sequence whose variance is going to 0

Let $X_n \ge 0$ be a sequence of random variables (which are not necessarily independent) satisfying the following conditions: $E(X_n) = \mu_n$ where $l \le \mu_n \le u$ for some constants $l, u ...
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82 views

Probability of combinations of independent events

Disclaimer: I'm almost 70 and trying to learn math concepts I didn't learn earlier in life. So don't expect this to be a deep or particularly difficult question. I've always understood the statement ...
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47 views

Repetition in digits on a normal number

Consider a normal real number $0< n < 1$. Let $n_i$ denote the $i^\text{th}$ digit of $n$. What is the probability that there exists some $k \in \mathbb{N}$ such that $n_1=n_{1+k}, n_2=n_{2+k}, ...
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94 views

Show the probability of liminfA is 1

Consider a random set $S$ constructed as follows. For any $n>2$, $\mathbb p(n\in S) = 1/\log n$, all independently. Show that $S$ almost surely satisfies the twin prime property: there ...
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48 views

Is the term $E\left[Z^3E[Z\mid Y] \right]$ positive or negative?

Let $Z$ be standard normal and $Y$ some other random variable. Is the $E\left[Z^3E[Z\mid Y] \right]$ positive or negative? I tried a few things like using orthogonality principle \begin{align} ...
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36 views

Conditional expectation and Joint distribution

Given $(X,Y)$ and $(X',Y')$ pairs of rvs on different probability spaces $\Omega_{1},\Omega_{2}$ respectively but with equal joint distribution show that ...
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37 views

Wireless networks on two sequential office floors: Random partitions of a finite interval via a point process on a line

Construct a Poisson point process of density one on a line of length $L$. Allow each point in the process to "see" part of the line to their left, and part of the line the their right (such that the ...
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203 views

The mathematics of a drinking game called Spoon

This question is about the drinking game Spoon. It was asked on reddit.com/r/math : http://www.reddit.com/r/math/comments/3i9790/drinking_game_turned_mathematical/ The question is whether the person ...
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83 views

Show that $\frac1n\log X_n$ converges almost surely

Let $X_0$ follow $\mathrm{Uniform}(0,1)$. Define $X_{n+1}$ iteratively as $X_{n+1}$ follows $\mathrm{Uniform}(0,X_n)$, $n\geq0$. Show that $\dfrac{\log X_n}{n}$ converges almost surely and find the ...
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53 views

Intuition behind almost sure limit of $\frac{|S_n|}{n^{1/p}}$

Suppose $X_i$'s are non-degenerate i.i.d. Then (1) If $E|X_1|^p=\infty$ we've $\limsup_{n\rightarrow \infty} \frac{|S_n|}{n^{1/p}}=\infty$. And this is true $\forall p>0$ (2) However for $p=2$ ...
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57 views

Doob's inequalities for not necessarily right-continuous martingales

In Revuz and Yor, they denote $\mathbb{H}^2$ the space of $L^2$-bounded martingales, and $H^2$ the space of continuous $L^2$-bounded martingales. They state "... by Doob's inequality ... ...
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72 views

A Markov Chain probability, conditioned on a random time.

My question: Upon reading theory about diffusion processes, i came across an argument which i believe simplifies to this: Say we have a Borel measurable set $A$ (if it matters you can set $A=\lbrace ...
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53 views

Can Monotone Class Theorem be easier to check than $\pi$-$\lambda$ Theorem?

I've been working on problem 14.4 in Billingsley's "Probability and Measure", which says: "Let $C$ be the set of continuity points of $F$. Show that for every Borel set $A$, $P(F(X) \in A, X \in ...
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43 views

Ito's formula and Infinitesmal generator

Consider an Ito process $$ dX_t = \sigma_t dB_t $$ where $\sigma_t$ is a two-state continuous-time Markov chain with state space $\{ \sigma_1, \sigma_2 \}$ that switches state with Poisson ...
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84 views

Conditional Distribution absolutely continuous w.r.t Lebesgue measure?

Let $X,Y$ be integrable random variables. Then the condition expectation of $Y$ given $X = x$ is defined as $$ \mathrm E[Y \mid X=x] := \int_\Omega Y(\omega) \, P^X(\mathrm dw \mid x), $$ where ...
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70 views

Generalizing the pull-out property in conditional expectations

Let $(\Omega, \mathscr{F}, P)$ be a probability space and $\mathscr{G}$ a sub-$\sigma$-algebra of $\mathscr{F}$. If $X$ and $Y$ are (integrable) random variables with $X$ being ...
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61 views

A Simple Inquality with Expected Value

Given $X(k) \ge -1, \;\;k=1,2,3...,N$ be discrete random variable with distribution $f_X$ and assume $a \in [0,1]$ so that $aX(k) \ge -1$ for all $k$ and $$ 1 - E \left[ \prod_{i=1}^N(1+aX(i)) ...
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45 views

Theorem of Portmanteau: It suffices to show it for a base?

I have a question to the Theorem of Portmenteau, see here. Two equivalent statements to $P_n\to P$ weakly, are (1) $\limsup_n P_n(C)\leq P(C)$ for all closed sets $C$. (2) $\liminf_n P_n(O)\geq ...
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31 views

Expected value of sorted subsequence

Consider the following discrete random variable: Given an array of size n containing random unique integers, what is the maximal length of a sorted subsequence from that array. What is the expected ...
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55 views

Charaterize the $\mathcal{F}_\tau$ a sigma algebra for the stopping time $\tau$

Consider a stochastic process $X: [0, \infty) \times \Omega \to \mathbb{R}^d$ We define $\mathcal{F}_t = \sigma(X(s), 0 \leq s \leq t)$ The sigma algebra generated by the sets $\{\omega: X(s,\omega) ...
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51 views

Correct steps in rewriting expectation to a probability

My knowledge of measure theory and probability spaces is limited, so please keep it relatively simple. Let $\{X(t), ~ t \ge 0\}$ be a Markov process on the countable state space $\mathbb{N}_0$ with ...
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48 views

How to show that two probability generating functions are equal?

From Grimmett's Probability and Random Processes: Let $G_a(s) := \sum_0^\infty a_is^i$ where $a = \{a_i : i \geq 0\}$ is a real sequence. Uniqueness. If $G_a(s) = G_b(s)$ for $|s| < R'$ ...
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68 views

Hitting time is a stopping time

Can somebody help me proving that the following hitting time is a stopping time? Let $\{X_t\}_{t\ge 0}$ be a real-valued, right-continuous process, adapted to a filtration $\mathfrak{F}$ which ...
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44 views

Interpretation of composite of random variable

Let $~f:[0,1] \to[0,\infty]$ be a measurable function bounded by $c \in \mathbb R$. Let $X_1,X_2,..,X_n \sim i.i.d ~\text{uniform}(0,1)$. How do I interpret the following statement: $$ Var(f \circ ...
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67 views

Why is the black-scholes model arbitrage free when $\sigma >0$?

I want to show that: if $σ$ is positive then there is no arbitrage in the model, even if $r > µ$. Whilst I have satisfied this for $ r > \mu$, I cannot see why the conditioning on $\sigma>0 $ ...
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48 views

Probability Generating Function of Compound Poisson Process

Let $(N_t)_{t\ge 0}$ be a poisson process with intensity $\alpha > 0$. Let $(X_n)_{n \in \mathbb N}$ be iid real valued random variables that are independent of $N_t$. Let $Y_t = ...
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44 views

Estimation of the Hermite Polynomials using Plancherel-Rotach asymptotics

Suppose $H_n(x)$ is a Hermite Polynomial such that $$\int_{\mathbb{R}} H_n(x) H_m(x) e^{-x^2} dx = \delta_{m,n}.$$ I want to show for $ \phi_n(x) = H_n(x)e^{-\frac{X^2}{2}}$ $$ \left( ...
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82 views

Confidence Interval of Information Entropy?

Information entropy, $IE$, is defined as: $$IE = \sum_{i} p_i log\frac{1}{p_i}$$ Where $p_i$ is the probability of event $i$ (and we are summing over all possible events). Let's say I have data ...
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221 views

Joint pdf of N > 1 i.i.d. random variables isotropic if and only if they are centered gaussian?

Are centered Gaussian densities given by $$f_X(x) = \frac{1}{\sigma \sqrt{2\pi}} e^{-x^2/(2 \sigma^2)}$$ the unique densities such that the joint pdf of $N > 1$ independent and identically ...
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68 views

Support of distribution functions in copula theory

Suppose I have a distribution function $$C(u,v)$$ with domain $I^2$. Let us define the support of this function as the complement of the union of all open subsets of $I^2$ with C-measure zero. Based ...
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48 views

Quadratic Variation of Increasing Process?

I am looking through my notes and I came across the following statement: Let $X_s$ be a positive local martingale and let $M_t = max_{0 \le s \le t} X_s$. Then since $M_t$ is an increasing process, ...
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96 views

Image of probability measure

Let $(\Omega,\Sigma,P)$ be a probability space. What is known about $P(\Sigma)$, the set of probabilities of events in $\Sigma$ by $P$? Clearly, $P(\Sigma)$ contains $0$ and $1$ since ...
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36 views

Showing that the space of bounded measurable functions can't be used to characterize convergence in distribution

An exercise I am doing from Chung's book asks me to find some asbsolutely continuous probability measures $\mu_n, \mu$ and some measurable and bounded (but not continuous!) function $f: \mathbb{R} ...
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58 views

probability measure on space of sequences

Let $\Omega=\{0,1\}^\infty$. For some $n$, let $B\subset \{0,1\}^n$. I have seen these two statements which make me confused little bit. (1) If $A\subset \Omega$, $A=B\times \{0,1\}^\infty$, and ...
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22 views

If $X_1,X_2$ are independent beta then showing $\sqrt{X_1X_2}$ is beta

Here is a problem that came in a semester exam in our university few years back which I am struggling to solve. If $X_1,X_2$ are independent $\beta$ random variables with densities ...
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221 views

Right-continuity of the augmented filtration

In most books about stochastic processes, the authors write about "the usual augmentation of a filtration". I am having trouble with proving that their construction is correct, i.e. that the augmented ...
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52 views

Aid to understand conditional expectation

I have a hard time to understand conditional expectation, that is $\mathbf{E}[X| \mathcal{G}]$ where $X$ is a random variable on the probability space $(\Omega, \mathcal{A}, \mathbf{P})$ and ...
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50 views

Measurable functions on complete space.

$(\Omega, \mathcal{E}, \mathbb{P})$ a probability space, $\mathcal{F}$ is $\mathcal{E}$ completed by the $\mathbb{P}-$null sets. Which conditions should meet a topological space $X$ (taken with its ...
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90 views

Sufficient condition for martingale property

Let $(\Omega,\mathcal{F},(\mathcal{F}_t)_{t \geq 0},\mathbb{P})$ be a filtered probability space and $M=(M_t)_{t\geq 0}$ an $\mathcal{F}_t$-adapted stochastic process. If $$ \forall t<s, \ ...
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68 views

Weak convergence of order statistics

I've encountered the following problem: Let $U_1,...,U_n$ be iid uniformly distributed on $[0,1]$ and let $U_{n(k_n)}$ denote the $k_n$-th order statistic where $k_n$ is chosen, s.t. ...
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91 views

Book on Convergence Concepts in Probability without Measure Theory

I am looking for a comprehensive book on Probability which discusses Convergence of Random Variables in detail, excluding portions of Measure Theory. Allan Gut's "Probability: A Graduate Course" seems ...
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150 views

Proof of Wald's Identity, is this valid?

So Wald says that assuming that $T$ a stopping time and $X_i$ i.i.d. variables are $L^1$, that $E[S_{T}] = E[T]E[X_1]$ given that $S_n = \sum_{i=1}^n X_i$. Consider the following proof that is ...
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79 views

Condition for $L^p$ convergence of backwards martingale

Is there any condition that is known to be sufficient for $L^p, 1<p<\infty$ convergence of a backwards martingale (and why is it sufficient)? I couldn't find anything else than the normal $L^1$ ...
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44 views

Calculating $\mathbf{P}[X < Y]$ for $X, Y$ exponentially distributed?

This is exercise 2.2.1 from Achim Klenke: »Probability Theory — A Comprehensive Course« Let $X$ and $Y$ be independent random variables with $X \sim \exp_\theta$ and $Y \sim \exp_\rho$ for certain ...
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127 views

How to prove product of measurable set is measurable using dynkin's $\pi-\lambda$ theorem?

My question: If $E_1$ and $E_2$ are measurable subsets of $R^1$, I want to show that $E_1 \times E_2$ is a measurable subset of $R^2$ and $|E_1 \times E_2|=|E_1||E_2|$. My attempt: First I tried to ...
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125 views

Estimating the support of a probability density function

The inverse moment problem deals with the reconstruction of a probability density function (PDF) of a random variable (RV) by means of its statistical moments. In the special case of the Hausdorff ...
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114 views

quadratic SDE solution

I have this SDE $dX_t=[a+bX_t+sX_t(1-X_t)]dt+\frac{1}{2}X_t(1-X_t)dW_t, \, X_0=0,$ where $a,b \in(0,1)$ and let's say that $s$ is a real constant (it's actually a function of $X$, but I think I can ...
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233 views

Donsker's Invariance Principle and Gambler's Ruin

Let $(S_{n})_{n\geq0}$ be a Random Walk (i.e. $S_{n}:=X_{1}+\cdots+X_{n}$, where $\mathbb{P}(X_{i}=1)=\mathbb{P}(X_{i}=-1)=1/2$). Define interpolated random walks $(S^{n}(t))_{t\in\left[0,1\right]}$ ...
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35 views

A basic question on spaces of probability measures

This problem is regarding the space of probability measures. For $N \geq 1$, let $\{e_i^N(.), i\geq 1\}$ denote a complete orthonormal basis for $L_2[0,N]$. Let $\{f_j\}$ be countable dense in the ...