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 [tag:probability-...

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0
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0answers
39 views

What is the almost sure limit of the normalized sum of these random variables? [on hold]

What is the almost sure limit of $\displaystyle\sum_{i=1}^n \frac {X_i} n$ if $$\displaystyle\mathbb{P}(X_n=n^2)=\frac 1 {n^2}$$ and $$\displaystyle\mathbb{P}(X_n=0)=1-\frac 1 {n^2}?$$ My guess is ...
1
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0answers
19 views

Return probability of a SRW in an even number of steps

I am looking for some references for the following problem. Consider a graph $G$ and a simple continuous time random walk $(X_t)_{t\geqslant 0}$ on this graph. Consider the family of events $(e_t)...
1
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0answers
9 views

Numerical scheme and boundary condition for 2D Fokker Planck equation

$\newcommand{\P}{\mathbb{P}}$ I have a 2D stationary Fokker-Planck equation $$\frac{\partial^2 \P(A,B)}{\partial A^2}+\frac{\partial^2 \P(A,B)}{\partial B^2}=\frac{\partial f_1(A,B) \P(A,B)}{\partial ...
1
vote
1answer
34 views

$X_1, X_2, \dots$ uncorrelated, $\frac{Var[X_i]}{i} \rightarrow 0$, then $\frac{S_n}{n} - \frac{\mathbb{E}[S_n]}{n} \rightarrow 0$ in $L^2$

Let $X_1, X_2, \dots$ be uncorrelated random variables with $\mathbb{E}[X_i]= \mu_i$ and $\displaystyle\frac{Var[X_i]}{i} \rightarrow 0$, when $i \rightarrow +\infty$. Show that $\displaystyle\frac{...
1
vote
1answer
49 views

Probability for a leading candidate to eventually win

Two candidates contest a close election. Each of the $n$ voters votes independently with probability $\frac12$ each way. Fix $\alpha \in (0,1)$. Show that, for large $n$, the probability that the ...
1
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0answers
20 views

One question regarding independence of $\pi$ systems.

Let G and H be sub-sigma-algebras of F and I and J are $\pi$ systems such that $\sigma(I)=G$ and $\sigma(J)=H$ Can anyone explain the following quote? Suppose I and J are independent, for fixed ...
3
votes
1answer
29 views

Minimal value of probability according to the difference of a Levy-process

Can we conclude for a Levy-Process, that for all $\epsilon>0$ it holds that $\min_{s\in [0,t]} \mathbb P\left(\left|X_t-X_s\right|\leq \epsilon\right)>0$? Stochastic continuity doesn't seem to ...
1
vote
1answer
6 views

Independent events from any other

In $(\Omega,\mathcal{F},P$) probability space, how can I show that $\forall A\in \mathcal{N}=\left\{ A\in\mathcal{F}: \mathbb{P}(A)=0,or\,\, \mathbb{P}(A)=1 \right\}\Rightarrow$ $\forall E\in\Omega$...
2
votes
1answer
34 views

Does there exist a random variable $\xi$ and a constant $c \neq 0$ such that $\xi + c \stackrel{d}{=} \xi$?

Does there exist a random variable $\xi$ and a constant $c \neq 0$ such that $\xi + c \stackrel{d}{=} \xi$? For context, I'm re-reading Kallenberg and in Chapter 3, on page 49, in his proof of Lemma ...
0
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1answer
22 views

Sum of Random Variables i.i.d. with $\mathbb{E}[|X_n|]=+\infty$

Let $(X_n)$ be a sequence of IID RVs (independent, identically distributed random variables) with $\mathbb{E}[|X_n|]=+\infty,\forall n$. Prove that $\sum_n \mathbb{P}[|X_n|>kn]=\infty$ with $k\...
2
votes
1answer
28 views

Transformation of random variables that preserves the distribution

Suppose we have a random variable $X$ with distribution $F_X$. Let $X_1$ and $X_2$ be two independent copies of $X$. My question: can we find a transformation $Z=g(X_1,X_2)$ such that the ...
1
vote
0answers
29 views

Lindeberg condition's counterexample (central limit theorem)

My aim is to find an example where the CLT is true but not the following (equivalent to Lindeberg's) condition: Find a sequence of independent $(X_k)\sim\mathcal{N}\left(0,\sigma^2_k\right)$, so ...
1
vote
1answer
26 views

X random variable in $\mathbb{N}$ independence of events

If I have a random variable $X$ with values in $\mathbb{N}$, $$\mathbb{P}(X=n)=\frac{1}{n^s\zeta(s)}$$ where $s>1$ and $\zeta$ the Riemann zeta function, then how can I show that $$A_i=E_{p_i^2}=\...
-1
votes
0answers
23 views

conditional expected value and not mutual indipendent events [on hold]

$\newcommand{\P}{\mathbb{P}}$ Let be $E,G,H$ pairwise independent events but not mutual (e.g. $P(E\cap H)=\P(E)\mathbb{P}(H),\,\P(G\cap H)=\P(G)\P(H), ...but \,\P(E\cap G\cap H)\ne\P(E\cap G)\P(H)$ ...
0
votes
2answers
97 views

Probability Of four events

I have an issue with calculating probability of union of four events, formula listed below \begin{align*} P(A \cup B \cup C \cup D) & = P(A) + P(B) + P(C) + P(D) - P(A \cap B) - P(A \cap C)\\ &...
6
votes
2answers
170 views

Corollaries of the Yoneda Lemma in Analysis?

I am looking for some simple examples of how the Yoneda Lemma can be applied in analysis and probability theory and related fields. A simple candidate example that I can think of and somewhat ...
1
vote
1answer
40 views

Distributional equality

Let $(W_t)_{t\geq0}$ be a standard Brownian motion. I have to show that the following equality holds in distribution. Does someone has a good hint to show this? $\sup_{t \geq 0}( |W_t| -t) = \sup_{t \...
0
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0answers
49 views
+50

Calculation of $\ln\left( \frac{S_{1}(t)}{S_{2}(t)}\right)$ where $S$ are stocks

Assume we have a probability space $(\Omega,\mathcal{F},\mathbb{P})$ where $\mathcal{F} =(\mathcal{F}_t)_{0 \leq t \leq \tau}$ is a Filtration of an incomplete finance market with stocks $S_j(t)$ for $...
0
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0answers
14 views

Uniform Boundary for S.D.E with Lipschitz Coefficients

Let $X_t^x$ be a solution to the SDE: $dX_t=b(X_t)+\sigma(X_t)dW_t$ with $X_0=x$. Assume that $b$ and $\sigma$ are Lipschitz Continuous. I want to proof that there exists $0<L$ such that $E\left|...
0
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1answer
45 views

Using the Baire Category Theorem to prove $\mu$ is trivial

Suppose we have a probability measure space $(X,\mathcal A,\mu,T)$ where $T$ is measure-preserving. Then if for every $A,B\in\mathcal A$ we have $\mu\left(A\cap T^{-n}B\right)=\mu(A)\mu(B)$ for all $n\...
0
votes
1answer
60 views

Probability for a random permutation

Given a set of $N$ elements and a uniformly distributed random number generator, which always generates values between $0$ and $N-1$. Then the probability to get a random permutation (without re-draws)...
1
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0answers
17 views

Shifting the mean of a composite function of deterministic and random variables

For a project I am involved in relating to communication, I have the following model: $L = f(r).X$ where $X$ is a lognormal random variable with zero mean in the logarithmic scale and standard ...
0
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0answers
20 views

Expectation of Martingale [on hold]

I am reading an article of American Put option here and I want to understand the proof of Theorem 1 (Main Decomposition of the American Put). It is stated that: $$\exp(-rT) \max[0,K-S_T]=P_0 - rK\...
0
votes
0answers
14 views

probability measures vs. probability distributions vs. measure of probability density

I am learning probability theory right now and am confused about some basic concepts. I have a few questions and am wondering if you can also check if the following is correct: Suppose we have a ...
0
votes
1answer
19 views

Weak convergence implies convergence on continuous functions

Let $X$ be a metric space, and let $\mu_n$ be a sequence of measures on $X$ converging weakly to a measure $\mu$, meaning for all bounded continuous functions $f$, we have $\int_{X}fd\mu_n \rightarrow ...
1
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2answers
53 views

Proofs related to chi-squared distribution for k degrees of freedom

I was reading a proof related to chi-squared distribution for k degrees of freedom from wiki. https://en.wikipedia.org/wiki/Proofs_related_to_chi-squared_distribution I think I might understand the ...
1
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0answers
15 views

Variance computed using Taylor series does not agree with numerical experiment [migrated]

I would like to estimate an angle $\theta\in\left(-\frac{\pi}{2},\frac{\pi}{2}\right)$ given the noisy observations of its sine and cosine (this is related to my earlier question). My estimator is ...
0
votes
2answers
17 views

Independence of Random Variables From Expectation Counter Example

I know that if $X$ and $Y$ are independent random variables, then $\mathbb{E}(XY) = \mathbb{E}(X)\mathbb{E}(Y)$. I also know that the converse is not true, although I cannot seem to find an easy ...
-1
votes
0answers
18 views

Communicating classes of a power of the irreducible transition matrix? [on hold]

Suppose $P$ is an irreducible transition matrix, with period $d$. Consider the transition matrix $P^k$. In terms of $d$ and $k$, how many communicating classes does $P^k$ have, and what is the period ...
1
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0answers
18 views

Convergence in law and sequence of reals

How to show that if I have $Y_n$ a sequence of random variables and $a_n$ a sequence of reals such that (i) $Y_n\to Y$ in law (ii) $a_n\to a$ i have $a_nY_n\to aY$ in law? I know that if I have (i)...
15
votes
2answers
9k views

Expectation of Minimum of $n$ i.i.d. uniform random variables.

$X_1, X_2, \ldots, X_n$ are $n$ i.i.d. uniform random variables. Let $Y = \min(X_1, X_2,\ldots, X_n)$. Then, what's the expectation of $Y$(i.e., $E(Y)$)? I have conducted some simulations by Matlab, ...
14
votes
1answer
864 views

Compact set of probability measures

I think I can solve the following exercise if $X$ is assumed to be separable, otherwise I can't. Let $X$ be a (Hausdorff) locally compact space, $\pi\colon X \to Y$ a continuous map into a ...
-1
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0answers
34 views

Possible master thesis [on hold]

I am searching a topic for my master thesis. My interests are especially probability theory (something with brownian motion would be nice) and fourier analysis (also in an abstract Hilbert space ...
0
votes
1answer
72 views

weak L1 convergence

Given a sequence $Y_{un}$, where $Y_{1n},Y_{2n},\ldots$ have the same domain. Assume for every $u\in \mathbb{N}$ we have $e^{itY_{un}}\rightarrow \mathbb{E}[e^{it M}]$ weakly in $L_1$ as $n\rightarrow ...
-1
votes
0answers
29 views

Entropy and Markov chain [on hold]

Assume that $X_n$ is a discrete Markov chain and $H$ is entropy function. I want to prove $$H\left(X_0\mid X_n\right) \geq H\left(X_0\mid X_{n-1}\right)$$ but I have no idea how to prove it. please ...
0
votes
2answers
69 views

Is it right to give equal chances?

In a certain town, the probability that it will rain in the afternoon is known to be $0.6$ Moreover, meteorological data indicates that if the temperature at noon is less than or equal to $ 25°C$, the ...
1
vote
1answer
23 views

For a one-dimensional Brownian motion $B_t$ $Ee^{\zeta B_t}=e^{t\zeta ^2/2}\; \text{for all} \; \zeta \in \mathbb{C}.$

A one-dimensional Brownian motion $B_t$ has exponential moments of all orders, i.e. $$Ee^{\zeta B_t}=e^{t\zeta ^2/2}\; \text{for all} \; \zeta \in \mathbb{C}. (2.6)$$ This is given as a corollary to ...
0
votes
0answers
23 views

Probability of $n$ numbers picked from set to have greater mean than set

If we have (N) varying non-negative numbers , with a mean equal to X, and a median less than X, if we pick (n) unique numbers from the set, what is the formula for the probability that the mean of ...
1
vote
0answers
59 views

Sum of random variables that are independent but not identical [on hold]

For a real number $t$, let $X_t$ be the random variable that is uniformly distributed in the interval $[t/2, 3t/2]$. If $\{t_n\}$ is a sequence of positive real numbers, is there anything we can say ...
0
votes
1answer
170 views

Prove random variable $Y_0$ and $\sigma$-algebra $\mathscr{R}$ are independent. [closed]

Let $Y_0, Y_1, ...$ be independent and identically distributed random variables with $P(Y_n = 1) = P(Y_n = -1) = 1/2$ for n = 0, 1, 2 ... Define random variables $X_n = Y_0Y_1Y_2...Y_n = \prod_{i=0}^...
0
votes
0answers
19 views

Prove $\lim M_{S(k) \wedge n}$ exists a.s. if $S(k) = \infty$. Is $N_n \ge 0$?

Probability with Martingales: Why does $\lim M_{S(k) \wedge n}$ exist a.s.? Is it connected to $$\sup E[M_{S(k) \wedge n}^2] < \infty$$ ? What I tried: My approach is to use: If $\lim ...
-2
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0answers
18 views
1
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0answers
41 views

Prove $A^{S(k)}$ is previsible

Probability with Martingales: I have a different attempt in mind, but I'm guessing it's wrong because if it were right, the book would've used it. It seems that we must show that $$A_{S_k \...
1
vote
0answers
34 views

If a process is previsible, is the stopped process previsible? [closed]

Assume we have a filtered probability space $(\Omega, \mathscr F, \{\mathscr F_n\}_{n \in \mathbb N}, \mathbb P)$ where $A = \{A_n\}_{n \in \mathbb N}$ is an $\{\mathscr F_n\}_{n \in \mathbb N}$-...
1
vote
2answers
634 views

Tossing a coin with at least $k$ consecutive heads

Toss a coin with $\Pr(\text{Heads})=p$ repeatedly. Let $A_k$ be the event that $k$ or more consecutive heads occurs amongst the tosses numbered $2^k,2^k+1,...,2^{k+1}-1$. Show that, $\Pr(A_k\ i.o.)=...
0
votes
1answer
27 views

I am trying to find answer to this bivariate normal problem. Can anyone help. [on hold]

The distribution of the heights of husband-wife pairs in a particular population is modelled by a bivariate normal distribution. The mean height of the women is 165cm and the mean height of the men is ...
0
votes
1answer
25 views

For an invertible measure preserving system, $\lim_NA_f^T(N)=\lim_N A_f^{T^{-1}}(N)$

For an invertible measure preserving system, show that $\lim_NA_f^T(N)=\lim_N A_f^{T^{-1}}(N)$. Here we consider the measure preserving system $(X,\mathcal A,\mu,T)$ where $T$ is invertible and $\mu$...
0
votes
0answers
17 views

If $(W_t)_{t\ge 0}$ is a $L^2(D)$-valued Wiener process, then $W_t(x)$ is normally distributed

Let $d\in\mathbb N$ $\lambda$ denote the Lebesgue measure on $\mathbb R^d$ $D\subseteq\mathbb R^d$ be a domain $U:=L^2(D)$ and $\langle\;\cdot\;,\;\cdot\;\rangle:=\langle\;\cdot\;,\;\cdot\;\rangle_U$...