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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46 views

higher order uncertainty

Context: As a general rule of thumb parameter estimation (in data analysis) works better the fewer the biased assumptions one makes. For instance, when presented with a coin and trying to estimate the ...
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105 views

about birth and death process

My question is about a homework question that I found interesting. It gives another proof (without using martingales) for that the critical Galton Watson tree dies out eventually. But it has given a ...
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77 views

Multiplicative version of Mcdiarmid's inequality?

Suppose you have $n$ i.i.d. random variables taking values in $\{0,1\}$, and $X$ represents their sum. Then you can use a Chernoff bound to control the deviation of $X$ from its expectation. The ...
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344 views

The application of Doob's inequality and Doob's decomposition theorem

1) What is the application of Doob's inequality? Can we use Doob's inequality ($L^1$) to prove the convergence (maybe almost surely) of a martingale? Doob's inequality: Let $X$ be a submartingale ...
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274 views

Does Kolmogorov 0-1 law apply to every translation invariant event?

Let $\Lambda$ be a lattice in $\mathbb{R}^d$, $d \ge 2$, thought of as an infinite graph. In percolation theory we consider properties random subgraphs of $\Lambda$. In site percolation $\Lambda^s_p$ ...
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605 views

When can we interchange the derivative with an expectation?

Let $ (X_t) $ be a stochastic process, and define a new stochastic process by $ Y_t = \int_0^t f(X_s) ds $. Is it true in general that $ \frac{d} {dt} \mathbb{E}(Y_t) = \mathbb{E}(f(X_t)) $? If not, ...
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773 views

Uniform distribution on the unit circle (in the complex plane)

I was trying to prove that for a standard complex Gaussian variable $Z$ it holds that $|Z|^2$ is exponentially distributed with parameter 1, $\frac{Z}{|Z|}$ is uniformly distributed on the unit circle ...
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66 views

Suboptimal confidence for bounds on the extreme eigenvalues of random matrices: can they be improved?

Typical results from the literature of Random Matrix Theory (RMT) (e.g., Tropp'11, Vershynin'12) provide probabilistic guarantees for large deviation of the extreme eigenvalues of random matrices like ...
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456 views

Convergence in Probability to infinity

i was lately reading the book of Kallenberg "foundations of modern probability". I have a problem with understanding one of this thoughts(p. 70, Theorem 4.17): Let $\xi_1,\xi_2,\ldots$ be independent ...
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253 views

Cover time and intersection time for lazy random walks on graphs

Consider a simple lazy random walk on an $n$-vertex undirected, connected graph: this is the Markov chain which transitions from $i$ to $j$ with probability $p_{ij}=1/(2d(i))$ where $d(i)$ is the ...
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161 views

Easy probability theory proof

$A$ and $B$ are random occurrences in $\Omega$. Prove that if $P(A)=0{,}9$ and $P(B)=0{,}7$, then $P(A\cap B')\leq0{,}3$, where $B'$ is a complementary event of $B$. I thought of something like this: ...
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169 views

Bayesian inference on partitioned multivariate Gaussian

My question is on Bayesian inference of partitioned multivariate Gaussian. To make things easier, suppose there is a 2-dimensional Guassian, $$ X_1 \sim N(\mu_1, \sigma^2_1) \\ X_2 \sim N(\mu_2, ...
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644 views

Definition of convergence in distribution

My question is convergence in distribution seems to be defined differently in Wikipedia and in Kai Lai Chung's book. My view is that the one by Wikipedia is a standard definition of convergence in ...
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148 views

How to solve equation involving binomial coefficient?

I'm reading this paper which says If we have $$ \binom n d p^{\binom d 2} = 1 $$ where $ 0 < p \le 1$, then $$ d = 2 \log_bn - 2 \log_b \log_b n + 2 \log_b\left(\frac 1 2 e\right) + 1 + O(1) ...
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148 views

Convergence of marginal distribution in the case of convergence in distribution

Suppose a random variable $Z_n=(X_n,Y_n)$ takes a value on $A= [0,1] \times B=[0,1]$. We can consider a conditional distribution of $Y_n$ given the realized value of $X_n$. Now suppose $Z_n$ converges ...
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273 views

An unconscious statistician's higher central moments

Recall the 'law of the unconscious statistician' (wikipedia), namely that one can calculate the expectation of the transform $g(X)$ of a random variable $X$, given the transformation and the ...
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157 views

An identity about a continuous local martingale

Let $M_t$ be a continuous local martingale with $M_0=0$ and define $I_t^0=1$ and $I_t^n= \int_0^t I_s^{n-1}\; dM_s$. Prove that we have $$n I_t^n= I_t^{n-1}- \int_0^t I_s^{n-2} \;d((M)_t) $$ where ...
3
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344 views

somewhere between sub-gaussian and central limit theorem

(problem statement revamped.) Definition: A random variable $X$ is a sub-gaussian if $$\Pr(|X|>t) \leq \exp(-t^2/k^2) \quad \text{for all $t\geq 0$ and for some constant $k$.}$$ I want to compute ...
3
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428 views

Quadratic variation of a Brownian motion up to time $T$ converges to $T$ in $L^2$?

In Stochastic Calculus for Finance II: Continuous-time Models by Steve Shreve, Theorem 3.4.3. Let $W$ be a Brownian motion. Then $[W, W](T) = T$ for all $T > 0$ almost surely. where $[W, ...
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80 views

Multiplicative functionals for Markov process: discrete time

I read a theorem, stating Let $X_t$ be a Markov process w.r.t. to its natural filtration $(\mathcal F_t)$ on the space of cadlag functions on $\mathbb R_{\geq 0}$ and $(Z_t)_{t\geq 0}$ be a ...
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45 views

incorrect rejection of a true null hypothesis?

We have a contest 1 weeks ago. One question is a bit strange for us as follows: $X\sim B(4,p). $ for test $H_0:p=0.2$ versus $H_1:p>0.2$. if $X=4$, $H_0$ assumption is rejected. calculate ...
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19 views

An inequality for symmetric random walk

I need to show that if $(X_j)$ are symmetric i.i.d. random variables with partial sums $S_n:= \sum_{j=1}^n X_j$, then for all $x \geq 0$ $$P(|S_n| > x) \geq \frac{1}{2} P(\max_{1 \leq j \leq n} ...
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29 views

For any given value x, are there uncountably many (countably infinite) binary sequences (ones and zeroes) whose limiting relative frequency is x

I have the following question, and given few proofs (provided by friends, professors, and my myself) which seem to work, I suspect the answer is yes: But I am still not completely sure. The question ...
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43 views

2d random walk on the nonnegative quadrant using martingale techniques

I know the basics of (discrete time) martingales, and I'd appreciate any help and suggestions on how to prove the following using martingale techniques. Let $Z_n$, $n\ge 0$ be a random walk on the ...
2
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59 views

Name of a “factorial” distribution

Is there a common name for the distribution $P(m)=\frac{(m-1)}{m!}$, for integers $m\in \{1,\infty\}$? Its mean is $e$.
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26 views

A necessary condition for boundedness in probability

I understand that it is straightforward to show (via Markov's inequality and standard arguments) that \begin{equation} E(X_n)=O(a_n) \end{equation} implies \begin{equation} X_n=O_P(a_n) \end{equation} ...
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27 views

Marginal Distribution of the sum of Bernoulli rv

consider the conditional on probabilities $p_1, \ldots, p_n$, with independent Bernoulli random variables $Y_1, ..., Y_n$ given that $P(Y_i = 1\mid p_1, \ldots, p_n) = p_i, \ P(Y_i = 0\mid p_1, ...
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23 views

Conditional Borel-Cantelli lemma

Let $A_1, A_2, \ldots$ be events with $A_n\in\mathcal{F}_n$. Show that $$\biggl\{\sum_{n=1}^\infty \mathbf{P}[A_n|\mathcal{F}_{n-1}]=\infty\biggr\} = \limsup_{n\rightarrow\infty} A_n \text{ a. ...
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33 views

Variance of absolute value of brownian motion

Im wondering if anyone has this calculated, I cant seem to find it anywhere online. I am trying to find the variance of absolute value of BM. Here is my attempt: First, $f_{\lvert W_t \rvert} ...
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24 views

Proof that $\mathbb E[X]=\sum_{\omega \in \Omega}X(w)P(\{w\})$

The question is inspired by a theorem about the expected value of a discrete random variable: $$\mathbb E[X]=\sum_{x}xp(x)$$ if the series converges absolutely. The theorem says that $\mathbb ...
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20 views

Approximating an element of a product sigma algebra by rectangles independent to a sub sigma algebra

Let $(S,\mathcal{S},\mu)$ be a finite measure space and $(\Omega,\mathcal{F},\mathbb{P})$ be a probability space. Denote the product space $(S\times \Omega, \mathcal{S} \times \mathcal{F},\mu \times ...
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33 views

$L_p$ inequality for Conditional Expectations with Mismatched Distributions

Suppose, we define $E_F[X]= \int X dF$ and $E_P[X] =\int X dP$. We know that there exits the follow $L_p, \ p \ge 1$ inequality \begin{align*} E_P |E_P(X| \mathcal{B})|^p \le E_P[|X|^p] \end{align*} ...
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26 views

Conditional expectation of a bounded random variable

Suppose we have a bounded random variable $X:(\Omega,\mathcal{F})\rightarrow(\mathbb{R},\mathcal{B})$ and some other random variable $Y:(\Omega,\mathcal{F})\rightarrow(\mathbb{R},\mathcal{B})$ . ...
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17 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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28 views

Convergence of derivative in L1

Let $F(x)$ be a distribution function over $\mathbb{R}$ with positive derivative at origin $f(0)$. Let $Q$ be a measure on $\mathcal{B}(\mathbb{R})$. Can we have following results under some ...
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25 views

How to prove the monotonicity of semi-norm?

In the book "Malliavin Calculus and related topics", the author states that $||F||_{k,p}=(E(|F|^p)+\sum_{n=1}^kE(||D^n F||^p_{H^k})^{\frac{1}{p}}$ has monotonicity property, i.e. $||F||_{k,p}\leq ...
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16 views

Disintegration of probability measures, reference request

Do you have a complete reference for disintegration of probability measures? Is there only the disintegration theorem, or is there a whole theory? Thanks.
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27 views

Supermartingales and optimal strategies for a game

Your winnings per unit stake on game $n$ are given by independent random variables $\epsilon_n$ such that $P(\epsilon_n=1)=p$, $P(\epsilon_n=-1)=q$ with $1/2<p=1-q<1$. Let $C_n$ be your stake on ...
2
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15 views

Exchangeability and conditional i.i.d. property

Consider a probability space $(\Omega, \mathcal A, P)$ and a sequence of random variables $(X_i)_i = X_1, X_2, \ldots$. $(X_i)$ is called exchangeable, if the joint distribution of every finite ...
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31 views

Is it possible to show that $E\left\|\sum_{i=1}^nX_i\right\|^2_s\leq Mdn$?

Let $X_i\in\mathbb{R}^{d\times d}$ be i.i.d. random matrices with zero-mean elements and variance bounded by some $C$. Then $$E\left\|\sum_{i=1}^nX_i\right\|^2_s\leq ...
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18 views

Formal argument on independence of consecutive hitting times of a Markov chain.

I'm refering to the question: Differences of consecutive hitting times. I'm interested in the independence of consequtive hitting times of certain values of a Markov chain. And I do "understand" the ...
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43 views

Probabilities related to Brownian excursion

I am reading a paper that uses a fact about Brownian excursion which I don't understand. Let $(E_t)$ be a standard Brownian excursion, i.e. $E_t = X_t + i R_t$, where $X$ is a standard real Brownian ...
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39 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 ...
2
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54 views

Pathwise integral of $W^{-a}$

Denote by $\tau(x) := \inf \{t \ge 0, W_t=x\},$ where $W_t$ is a Wiener process started at $W_0 = w_0 > 0$ and I would like to show that for any $a>1$ it almost surely holds that ...
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27 views

Find the distribution of the random variable given probability generating function

Let $X$ be a non-negative, integer valued random variable such that $\phi_X(t)=-\log(1-qt)$. Determine $P(X=k)$ where $k=0,1,2,...$. Now nothing is said about $q$, so maybe we can assume $q$ ...
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28 views

Covariance of minimum of independent random variables and a constant

I have two random variables $X \sim \min (k, X_1)$ and $Y \sim \min (k, X_2)$ where $X_1$ and $X_2$ are exponential random variables with same rate $\lambda$ independent of each other and $k$ is a ...
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34 views

Is there any standard way of analysing this integral?

I have a compound Poisson process $(X_t)$, with jump distribution $F$, which assigns mass only to $(0,\infty)$. In my working I have an expression of the following form: $$ \mathbb{E} \int_0^{\tau} ...
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42 views

a.s. for all $t$ or for all $t$ a.s.?

Assume that we have some equality, $$ X (t) = Y(t). \quad \quad \quad \quad \quad \quad \quad (1) $$ I imagine that if I say "(1) holds a.s. for all $t>0$" it means that $$ P\{X (t) = Y(t) ...
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33 views

Bounding the density of random variable

This is a followup to the question in Bounding the Density of the Maximum of N Random Variables I have a random variable, X, whose cdf is bounded as below: $ \Pr \{X \le x \} \le ...
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31 views

Stochastic control with stopping times

Given a wealth process that evolves as $$d w_t = r w_t dt + \theta_t ( \sigma dW_t + (\mu-r) dt) - c_t dt.$$ and smooth functions $u,F: [0, +\infty) \rightarrow \mathbb{R}$, how can we optimise the ...