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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Interchanging Malliavin derivative with Lebesgue integral

I am reading Oksendal's book "Malliavin calculus for Levy processes with application to finance". In the proof of Lemma 4.9 (page 47), the author interchanges the Malliavin derivative $D_t$ with the ...
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47 views

Which version of Taylor Theorem is this?

Suppose $X$ is a random variable and $\psi(t)=E[\exp(itX)]$ is its characteristic function. Let $K(t)$ be the principal value of the logarithm of $\psi(t)$. Suppose further that ...
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28 views

Probability of each state less than or equal to $2^{-n}$ for each $n$?

This relates to the hint to exercise 3.6.5 in Rosenthal's A First Look at Rigorous Probability Theory. ($\Omega, \mathscr{F}, P$) is a probability triple, where: $\Omega$ is countable (in this book, ...
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65 views

Characterization of renewal processes, and examples with exactly one of stationary or independent increments

A continuous-time process $\{N(t) : t\in[0,\infty)\}$ is a counting process if it satisfies $N(t)\geqslant0$ a.s. (nonnegative) $\mathbb P(N(t)\in\mathbb N\cup\{0\}) = 1$ (integer-valued) If ...
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65 views

Suppose $X$ and $Y$ have joint density $f(c,y)=6xy^2$ for $x,y\in(0,1)$. What is $P(X+Y<1)$?

My attempt: $$f(x,y)=6xy^2$$ $$P(X+Y<1)=\int_{0}^1\int_{0}^{1-y} 6xy^2dxdy$$ $$=\frac{1}{10}$$ However, the answer was supposed to be $\frac{3}{5}$, and the bounds on the integral were supposed ...
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The boundedness of a certain sequence of expectations

In Bálint Tóth's paper, "No More Than Three Favourite Sites for Simple Random Walk", while proving one of the many technical lemmas in his theorem's proof, he makes the following claim: suppose for ...
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47 views

Find $P\left(X ≤ \frac12, Y≥\frac34\right)$. Joint Probability Density Function

Textbook: Mathematical Statistics with Applications Wackerly Let $X$ and $Y$ have the joint probability density function given by $$f(x,y) = \cases{6(1-y), \quad 0≤x≤y≤1 \\ 0, \quad ...
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70 views

Decomposition of mutual information for conditionally independent variables

I have a question regarding the mutual information of conditionally independent random variables (observations). Given $p(x,y|z) = p(x|z)p(y|z)$ where $z$ corresponds to a latent variable, I was ...
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44 views

If $X_n \rightarrow^{P} 0$ then for any $p >0$ $\frac{|X_n|^p}{1+|X_n|^p} \rightarrow_{P} 0$

Let $\{X_n\}$ be a sequence of random variables If $X_n \rightarrow^{P} 0$ then for any $p >0$ $$\frac{|X_n|^p}{1+|X_n|^p} \rightarrow_{P} 0$$ and $$E(\frac{|X_n|^p}{1+|X_n|^p}) \rightarrow 0$$ . ...
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26 views

Convergence of generalized inverses

I copy-paste the post from Math Overflow - maybe someone can give me a tip. During the reading about Fisher–Tippett–Gnedenko theorem (it can be found easily on wiki - I don't have enough reputation ...
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57 views

Limiting sequence of exponential random variables

Let $\eta_k$ be i.i.d. random variables having an exponential distribution, $$F_\lambda(x) = P(\eta_k \leq x) = 1-e^{-\lambda x}$$ for $x \geq 0$. Consider a sequence $\xi_k = ...
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41 views

Concentration inequalities for $P(\sum_{i=1}^n \epsilon_i X_i > t)$

Let $\epsilon_i \sim \text{Bernoulli}(p)$ and $X_i \sim \text{Normal}(0, \sigma^2 / n)$ for $i=1,\ldots,n$. I am interested in getting a sub-Gaussian type upper bound for $$ P\left(\sum_i \epsilon_i ...
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31 views

$S_n \in [-a,a]$ for some $a$ infinitely often

Suppose we have iid r.v.s $X_n \in \mathbb{R}$ with mean $0$ variance $\sigma^2$, I wonder is it true that we have $\exists a>0,$ $$P(S_n \in [-a,a] \text{ infinitely often} )=1,$$ where $S_n ...
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64 views

$X,Y$ independently uniformly distributed in $[-1,1]$. Find $P[XY<1/2]$

If $X,Y$ are independently uniformly distributed in $[-1,1]$. Find $P[XY<1/2]$ The following is my answer: We know the total region is a square with area $4$, centered at origin. Find the ...
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54 views

Understanding Markov's inequality

Here is a statement of Markov's inequality. Suppose that $X$ is a random variable and that $g: \mathbb{R}\to [0,\infty]$ is Borel measurable and non-decreasing. Then, for any real $c$, $$ ...
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24 views

Conditional expectation and stopping time $\mathbb{E}(X1_{T\leq m}|\mathcal{F}_{T\wedge m})=\mathbb{E}(X1_{T\leq m}|\mathcal{F}_T)$

Let $X$ be a random variable and $T$ a stopping time in a filtrated probability space. If $m > 0$ is it true that: $$\mathbb{E}\left(X1_{T\leq m}|\mathcal{F}_{T\wedge ...
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37 views

Why the case of independence of random variables is more important than any other specific type of dependence?

Maybe a stupid question but why is the case of independence of, say, two random variables $X$ and $Y$ is in some ways considered to be more ``central'' or more important than any other type of fixed ...
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38 views

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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46 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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92 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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49 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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49 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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216 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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86 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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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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44 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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86 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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74 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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46 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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59 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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73 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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46 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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70 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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49 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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83 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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223 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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69 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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49 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} ...