# Tagged Questions

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 points on a sphere — expected angular distance

Suppose we randomly select $n>1$ points on a sphere (all independent and uniformly distributed). What is the expected angular distance from a point to its closest neighbor? What is the expected ...
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### Prokhorov's Theorem-Prove if tight subsubsequence, then tight sequence.

Let $P_n$ be a sequence of Borel probability measures on $\mathbb{R}$ has a subsequence $\{P_n\}_k$ has a further subsequence that is tight. Show that $P_n$ is tight. Clearly, this is Prokorov's ...
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### Could an average number misrepresent a likely scenario if there is no limit on one end? [on hold]

Boggling myself over this question since a friend asked me it. If you are trying to calculate your probability of sucess on a system from 0 to infinity on say a 1% rate of sucess with no failure cap. ...
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### Condition implying that $E(X|\mathcal{C}_1)=E(X|\mathcal{C}_2)$ when $\mathcal{C}_1\subseteq\mathcal{C}_2$

I have the following corollary in my notes but I don't see how it follows from law of total probability: Corollary. Assume that $\mathcal{C}_1\subseteq\mathcal{C}_2\subseteq\mathcal{F}$ are sub ...
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### Independence and expected value

I have a theorem that says If two random variables $X,Y$ are independent, then for any non-negative measurable functions $f:E\to\mathbb{R}$ and $g:E\to\mathbb{R}$ the following holds ...
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### Multiple sequences of random variables that converge in probabilty

I'm struggling with this exercise: For each $k\in \mathbb{N}$, let $(X^{(k)}_n)_{n\in\mathbb{N}}$ be a sequence of real random variables converging to $0$ in probabilty as $n\to\infty$. Define for ...
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### What does P(X=Y) mean?

Let X and Y be binary random variables, with $P(X = 0) = 1/4$, $P(Y = 0) = 1/4$ and $P(X = Y) = 1/2$ I want to calculate $P(X=x,Y=y)$ (i.e. probability of x and y) and P(X=x|Y=y) for all all x and y. ...
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### Maximal deviation from mean of a bounded random variable

Is there a non-probabilistic Hoeffding like inequality which tells me the deviation between a bounded random variable and its expectation? Let $X$ be a random vector such that $||X|| \leq c$. I am ...
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### Deriving the asymptotic distribution of a two-stage estimator

Suppose $X_i$ are iid, $Y_n = g(X_1, \cdots, X_n)$ is a statistic and $\sqrt{n}(Y_n -\theta) \stackrel{d}{\to} N(0,V)$, where $\theta$ is a constant. Define $Z_{i,n} = f(X_i,Y_n)$, where $f$ is a ...
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### Casting an expectation as an integral

I probably picked the most ambiguous title possible for the question I am about to ask. Sorry for that. I have two random variables, $X$ and $Y$. I am about to define conditional densities and I am ...
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### Monotone property of transition density of rotational $\alpha$-stable process

For a Brownian motion $B_t$ in $\mathbb R^d$, the transition density of $B_t$ is the normal distribution $$P_x[B_t\in dy]=(2\pi t)^{-d/2}e^{-\frac{|x-y|^2}{2t}}dy$$ and obviously the density is ...
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### Finding a function of a random variable that maximizes some expression

The following problem is part of my studies, so I would prefer hints or suggestions for self-study. Let $v_1$ be a random variable taking values in $[a,b]$ for $a,b\in \mathbb R$ and assume that the ...
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### Does absolute continuity imply no stochastic domination?

I have an interesting question which goes as follows: Let $F_0$ and $F_1$ be two (nominal) distributions defined on a measurable space $(\Omega.\mathscr{A})$, where $\Omega$ is continuous. ...
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### Marginal convergence in distribution plus independence imply joint convergence?

Suppose $X_n \stackrel{d}{\to} X$, $Y_n \stackrel{d}{\to} Y$, and $X$ and $Y$ are independent. Does it follow that $(X_n, Y_n) \stackrel{d}{\to} (X,Y)$? I don't think this is true, but am having ...
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### Definition of a random variable $\mathrm{Var}(X)$

So $\mathrm{Var}(X) = \mathrm{E}((X-\mu)^2)$, but how can you subtract a function $(X)$ by a value ($\mu)$? And does it make sense to square a function?
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### Can we apply an Itō formula to find an expression for $f(t,X_t)$, if $f$ is taking values in a Hilbert space?

Let $U$ and $H$ be separable Hilbert spaces $Q\in\mathfrak L(U)$ be nonnegative and symmetric with finite trace $U_0:=Q^{1/2}U$ $(\Omega,\mathcal A,\operatorname P)$ be a probability space ...
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### Sum of random variables goes to infinity

I'm trying to show the following: Let $(X_n)_{n\in\mathbb{N}}$ be a sequence of i.i.d random variables with $\mathbb{E}[|X_1|]<\infty$ and $\mathbb{E}[X_1]=\mu$. Consider ...
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### issue on conditional-expectation with crossed filtration

Why we have this equality ? $$\mathbb{E}[\ \mathbb{\hat{E}}(X(.)|\mathcal{F}_t)_G K(G) |\mathcal{F}_t] = \int_{\mathbb{R}}\mathbb{\hat{E}}(X(.)|\hat{\mathcal{F}}_t)_u K(u) dP_t^G(u)$$ For all ...
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Suppose $(\Omega, \mathcal{F}, P)$ and $(\Omega, \mathcal{F}, Q)$ are two probability spaces. The Radon-Nikodym theory says that if $P$ is absolutely continuous with respect to $Q$, then there exists ...
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Billingsley's probability and measure and others show the joint measurability of the Brownian motion using the continuity of paths. Makes me wonder if we can say it before saying the continuity, if we ...
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### Showing that $\mathbb{E}[ \frac{S'_n}{n \log_2 n}]$ converges to 1 for a problem related to geometric distribution

We define independent random variables $X_i$ which follow the law $P(X_i = 2^k)=\frac{1}{2^k}$. We set $S_n = X_1+ \cdots +X_n$. Since we cannot apply the law of large numbers to $S_n$, we define ...
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### Strong Markov property proof

Let $X$ be a Markov chain with state space $\mathcal{S}$ and denote $\mathbb{N} := \{0,1, \cdots\}$. I need to show that for any stopping time $\tau < \infty$ and any bounded measurable function ...
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### $T$ can be $\infty$ with positive probability

From Williams' Probability with Martingales How exactly do we know $T$ can be $\infty$ with positive probability or $$P(T = \infty) > 0 \text{ ?}$$ I'm guessing that that means there ...
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### Itô-formula proof, remainder term.

I have a question about the proof a a certain version of the Itô-formula. First the author defines an Itô-process and states the formula: My question is in regarding the proof. The proof uses ...
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### About the random $\pm 1$ matrices

I was reading the paper "On the probability that a random $\pm 1$ matrix is singular". In the paper the author defined the following notations: $M_n$: a random $n\times n$ matrix with i.i.d entries ...
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### When convergence a.s. implies convergence in mean?

Can someone help me with proving the following: Assume that $X_n$ converges almost surely to $X$, where $X_n$ is a sequence of non-negative random variables. Furthermore, assume that the sequence ...
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### How well can the maximum of a Gaussian process be approximated by a finite-dimensional Gaussian variable?

Consider a compact set $K$ in $\mathbb{R}^p$, and let $W$ be a mean zero continuous Gaussian process on $K$, meaning that $W$ takes its values in the space of continuous functions from $K$ to ...
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### Prove that : $X_n \xrightarrow{\mathrm{a.s.}}0\iff \sum_n P(X_n>0) <\infty$

Let $(X_n)$ be a sequence of independent integer-valued (nonnegative integers) random variables Prove that $X_n\xrightarrow{\mathrm{a.s.}} 0\iff \sum_n P(X_n>0) <\infty$ For the ...
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### Binomial distribution with independent failure and success probabilitis [on hold]

We have the probability distribution $f(k,p_1,p_2) = \binom{n}{k} p_1^k (p_2)^{n-k}$, known as Binomial distribution for $p_2=(1-p_1)$. It is often used to model errors in binary symmetric channel ...
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### Uniform integrability of a sequence of random variables defined by a recursive relation

I have an i.i.d sequence $(u_j)_{j\in \mathbb{Z}_+}$ with zero mean and finite variance, say $\sigma^2$. Furthermore, I have another random variable $X_0$ (defined on the same probability space) which ...
### local martingale $\exp(\lambda X_t-\frac{\lambda^2}{2}t)$ is stochastic exponential
I have an $\mathbb{R}$ valued process $X$ which is an $\mathcal{F}^X$ Brownian motion if and only if for all $\lambda \in \mathbb{R}$ $M_t:=\exp(\lambda X_t -\frac{\lambda^2}{2}t)$ is a ...