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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1answer
12 views

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 ...
0
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0answers
11 views

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 ...
3
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1answer
15 views

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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0answers
9 views

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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0answers
30 views

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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0answers
5 views

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 ...
2
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2answers
33 views

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 ...
0
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1answer
72 views

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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74 views
+50

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 ...
0
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1answer
30 views

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 ...
1
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0answers
17 views

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 ...
1
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0answers
29 views

Specific Radon-Nikodym Derivative Interpretation

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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0answers
13 views

Gaussian process via RKHS construction: joint measurability comes for free?

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 ...
1
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1answer
13 views

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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0answers
27 views

$N = Poisson(\lambda)$, $\{U_i\}$ iid $\implies (N_1, N_2) = Po(\lambda p_1)$x $Po(\lambda p_2)$

Let $\{N\}\cup\{U_i\}$ be independent random variables. $N = $ Poisson$(\lambda)$ $\{U_i\}$ iid, taking values in $\{1,2\}$, $\mathbb{P}[U_i = 1] = p_1$ and $\mathbb{P}[U_i = 2] = p_2$, $p_1 + p_2 ...
2
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0answers
31 views

Let $X$ and $Y$ be iid real-valued random variables. Show $P[|X-Y| \le 2] \le 3P[|X-Y| \le 1]$. [duplicate]

Found this question in The Probabilistic Method and tried for hours to prove it, but I'm not getting anywhere. Can anyone walk me through it? I see that if we can show $P[1 \le X - Y \le 2] \le P[|X ...
1
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0answers
21 views

Convergence of $\sum_{i \leq n} X_i/n$

I have a question like this: Let $(X_n)$ be an i.i.d sequence of random variables with values in {-1,1}, and define $Y_n:= \sum_{i \leq n} X_i/n$. Show that $(Y_n)$ converges almost surely and in ...
0
votes
1answer
33 views

Finite-state Markov chain

Suppose $X_n$ is a Markov chain with transition probability matrix $p$ where the set of possible states is $S = \{1,2, \ldots, k\}$. If we are given, $X_1$, $X_2, \ldots, X_{2000}$, can we say about ...
0
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0answers
28 views

Central limit theorem and the sequence with general term $e^{-n} ( 1+n+ \cdots + n^n/n!)$ [Proof check]

As an exercise I need to find the limit of the said sequence $$e^{-n} ( 1+n+ \cdots + n^n/n!)$$ using the toolkit of probability theory. Since no solution (only hints) is provided, I would appreciate ...
8
votes
1answer
134 views

Representation of a linear functional Lipschitz in total variation

Let $\Omega$ be a Borel space and let $\mathcal P(\Omega)$ be the space of all Borel probability measures on $\Omega$ endowed with the topology of weak convergence. Define the total variation metric ...
3
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0answers
17 views

Visualizing Convergence in Probability

Let $\{X_n\}_{n=1}^\infty$ be a sequence of random variables defined over a common underlying space $(\Omega, \mathcal F,P)$. We say that $X_n$ converges in probability to a real number $\mu$ iff: ...
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0answers
34 views
+50

Convergence of moments of a sequence of random variables

I encountered this problem in my study of time series. It seemed trivial at first but I don't see the finishing move to complete the proof. The problem is as follows. Let $(X_n)_n$ be a sequence of ...
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0answers
5 views

Norm estimates on Markov operator.

Let $G$ be a group, and let $S$ be it's finite symmetric generating set. Assume that Markov operator is defined as $$M(f)=\sum\limits_{s \in S} g.f $$ Obviously, $f$ can be a function in $l_p$. ...
3
votes
1answer
24 views

Is a subsequence of an exchangeable sequence exchangeable?

Consider a finite sequence of random variables $X_1,...,X_n$ (1) SUFF COND: Suppose $X_1,...,X_n$ are exchangeable, meaning that the joint probability distribution of $X_1,...,X_n$ is equivalent to ...
2
votes
1answer
19 views

convergence in distribution of exponential of a brownian motion

If $(B_t)_{t≥0}$ is a standard Brownian motion, show that, as $t \to \infty$, $$ \left(\int_0^t e^{B_s} \, ds\right)^{1/\sqrt{t}} \text{ converges in distribution to} \ e^{M_1}, $$ where $M_1 = ...
2
votes
0answers
24 views

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 ...
2
votes
1answer
23 views

$T$ 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 ...
1
vote
1answer
55 views

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 ...
1
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0answers
36 views

Does Strong Markov property need time homogenous property?

If $X$ is a Markov chain, we know that for any bounded measurable function $f$, $E[f(X_{n+1}) \vert \mathcal{F}_n] =E[f(X_{n+1}) \vert \sigma(X_n)] = g(X_n)$ where $\sigma(X)$ is the sigma-algebra ...
5
votes
1answer
72 views
+100

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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0answers
25 views

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 ...
3
votes
1answer
24 views
+50

Linear transform of a strictly stationary time series

First, let me clarify what I mean by a strictly stationary time series. Let $(X_t)_{t\in \mathbb{Z}}$ be a sequence of random variables on some probability space. If it holds that $$(X_t, ...
7
votes
1answer
238 views

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 ...
2
votes
1answer
37 views

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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0answers
17 views

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 ...
1
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0answers
21 views

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 ...
0
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0answers
16 views

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 ...
2
votes
1answer
275 views

Progressively Measurable for Rigth Continuous Adapted Processes

Any adapted and right continuous process $X_t$ is progressively measurable. For the above statement, I found proof in several books. They all have similar argument as follows. For a given $t > ...
2
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0answers
35 views

Theoretical interpretation of simulating from a distribution

Suppose there is a random variable $X$ with marginal density $p_X$. However only the conditional densities $\{p_{X\mid\Theta}(\cdot\mid\theta):\theta \in \mathbf{T}\}$ are known directly, where ...
0
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1answer
27 views

Long-term probability question

I am in intro probability class, and I know the basics, such as conditional probability, and how to solve simple problems. However, how does one solve this problem (below) without knowing whether or ...
2
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2answers
46 views

References on probability theory, stochastic processes, Monte Carlo and convex optimisation, with similar writing style to Terence Tao

I learned a lot from prof Tao's notes and books because unlike many authors, he seems to prefer writing more words, explanations and intuitions rather than just mathematical formulae. His approach is ...
1
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1answer
28 views

Uniform Integrability of Random Variables

$\{X_n\}$ is uniformly integrable if $\lim_{M \rightarrow \infty} (\sup_n \mathbb{E}(|X_n| \chi_{|X_n| > M}) = 0$ I would like to know if $\{X_i\}$ uniformly integrable $\implies \sup_n ...
3
votes
1answer
103 views

Can we characterize the probability generating function as a linear operator?

For a nonnegative integer-valued random variable $X$ with $\mathbb P(X=j)=p_j$, we define the probability generating function of (the distribution of) $X$ by $$P_X(s):=\mathbb E\left[s^X\right] = ...
2
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0answers
29 views

Tauberian theorem when limit is zero

Let $h \geq 0$ be a non-negative increasing function with Laplace transform $H$. Let $\rho \geq 0$ be a constant. A simple Tauberian theorem says that the following two statements are equivalent: I. ...
0
votes
2answers
32 views

Correct notation for Continuous random variables

Assume two random variables $X \in \mathcal{X}$ and $Y \in \mathcal{Y}$. Then, if $X$ and $Y$ are discrete random variables $P(X \in E) = \sum_{y \in \mathcal{Y}} P(X \in E, Y=y)$. I want to know ...
0
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0answers
21 views

How can I calculate definite integral of chi-squared pdf with one degree of freedom

enter image description here I need a calculating process of the above definite integral please help me.. (sorry for my poor English)
4
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0answers
57 views

Characterization of point process, given the number of points

For a point process with independent and identically distributed (i.i.d) inter-renewals, with distribution $p(x)$, we observed $N$ points on $[0,T]$. What is the joint probability distribution ...
2
votes
1answer
28 views

Pdf of $Z=X/Y$ given a joint pdf

Find the p.d.f. of $z=x/y$ $f(x,y)= 2(x+y)$ for $0\le x\le y\le1$ I first did the simple way, transformation, then derivative, and multiply joint p.d.f by absolute value of the derivative. Then ...
2
votes
2answers
16 views

$X_n\to 0$ in probability implies $E[f(X_n)]\to f(0)$ for $f$ uniformly cts and bounded

Let $f$ be a uniformly continuous and bounded function. I've shown that if $X_n\to 0$ in probability, then $f(X_n)\to f(0)$ in probability as well. Now I want to say that $$\lim_{n\to\infty} ...
5
votes
0answers
47 views

a dynamical systems view of the central limit theorem?

I have seen many heuristic discussions of the classical central limit theorem speak of the normal distribution (or any of the "stable distributions") as an "attractor" in the space of probability ...