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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2
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2answers
39 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
votes
0answers
7 views

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 ...
-4
votes
0answers
19 views

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. ...
0
votes
0answers
25 views

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 ...
0
votes
3answers
32 views

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 ...
0
votes
1answer
19 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
votes
2answers
37 views

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. ...
1
vote
1answer
13 views

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 ...
3
votes
1answer
34 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, ...
0
votes
0answers
34 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 ...
0
votes
1answer
21 views

Recurrence of 0 in a random walk

Assume $\mathcal{S} := \{0, 1, \cdots \}$, $p(0,1)=1$ and $p(n,0)=p(n,n+1)=\frac{1}{2}$ for $n=1,2, \cdots$. Is $0$ recurrent or transient? So, basically this is an irreducible, closed but infinite ...
0
votes
0answers
6 views

Reformulate a SPDE parameterized by space and time as an SDE parameterized by time (as it is possible for PDEs)

Let $d\in\mathbb N$ $\mathcal V_t\subseteq\mathbb R^d$ be a bounded domain for $t\ge 0$ $\Phi_t:\mathcal V_0\to\mathcal V_t$ be bijective for $t\ge 0$ with $$\Phi(\;\cdot\;,x_0)\in C^1\left(\mathbb ...
0
votes
0answers
14 views
+50

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 ...
0
votes
0answers
14 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
votes
1answer
17 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 ...
0
votes
0answers
10 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 ...
0
votes
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. ...
0
votes
0answers
10 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 ...
0
votes
1answer
75 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?
5
votes
0answers
75 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
votes
1answer
32 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
vote
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
vote
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 ...
1
vote
0answers
17 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
vote
1answer
14 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 ...
2
votes
0answers
33 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
vote
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
46 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
votes
0answers
29 views

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

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
138 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
votes
0answers
21 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: ...
2
votes
0answers
35 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 ...
0
votes
0answers
6 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
21 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
33 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
25 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 ...
5
votes
1answer
73 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 ...
1
vote
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 ...
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 ...
-3
votes
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
vote
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
votes
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
votes
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
votes
1answer
28 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
votes
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
vote
1answer
29 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 ...