# Tagged Questions

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### Determining the semigroup and relating it to the generator via the functional calculus

In the theory of Hille Yoshida, we have a (definitely real) banach space $L$ and on it a contractive, strongly continuous semigroup of operators. ($T_tT_s=T_{t+s}$ and $T_0=I$.) We then define the ...
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### Total boundness of Lipschitz densities

In the article Almost Sure Testability of Classes of Densities by Devroye and Lugosi in 1999. They claim in Example 10 (page 9) that Lipschitz densities on [0,1] with Lipschitz constant bounded by ...
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### Banach Fixed Point Theorem. Measurable version.

The Banach fixed point theorem has the following statement THEOREM ( Banach contraction principle). Let $(Y,d)$ be a complete metric space and $F:Y\to Y$ be contractive . Then $F$ has a uniqe ...
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### Can any measure be made into a bounded measure?

Is it possible to derive a bounded measure from any measure on a measure space? For example can the Lebesgue measure be made into a probability measure?
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### Return time Markov chain

I have been wondering about this for quite a while now that I found in a textbook in the proof that an irreducible positive recurrent markov chain $(X_n)$ has a stationary distribution Let $t_i$ ...
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### What is the proper definition of cylinder sets?

in class we defined the terminal $\sigma$-algebra for a sequence of random variables $(X_i)$ with $X_i:\Omega \rightarrow \mathbb{R}$ as $G_{\infty}:=\bigcap_i G_i$, with ...
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### Two notions of conditional expectation

For a randomn variable $Y$ and an event $B$ we can define: $$E(Y \mid B) = \frac{E(1_B\cdot Y)}{P(B)}$$ as the conditional expectation. Now, for a sigma algebra $\mathcal{B}$ and sets $B$ in it you ...
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Recently, we investigated whether the expression in the central limit theorem converges to a random variable pointwise almost sure? The answer was negative due to $P ( \text{limsup} ... 0answers 29 views ### Interpretation of a tail event I am currently reading about tail events wikipedia. And I was wondering: Where does the interpretation come from that events in this sigma algebra are independent from the behaviour of any finite set ... 1answer 39 views ### How to show convergence in distribution Let$([0,1],B,\lambda)$(B Borel Sigma-algebra) and$\lambda$the Lebesgue measure. I want to show that this sequence converges in distribution. $$X_n(\omega)= \left(\begin{matrix} 1, & ... 1answer 55 views ### Eigenvalue markov chain I have a questions: We said that if we have a positive recurrent Markov chain, then there is a unique stationary distribution. 1.) Does this mean that if I have several positive recurrent classes, ... 0answers 40 views ### Transient/Recurrent Markov chain I am currently studying the concept of recurrent and transient states and was wondering about the following: Is this concept dependent on the initial distribution? Let me take this example: You can ... 1answer 29 views ### Is this a Markov chain property For A,B measurable sets and (X_n)_n a Markov chain. Do any of the following properties hold?$$P(X_2 \in B | X_1=x_1,X_0 \in A) = P(X_2 \in B|X_1=x_1)$$or$$P(X_2 \in B|X_1 \in A,X_0=x_0) = ... 0answers 28 views ### Strong Markov property and its meaning Given a sequence of random variables$(X_n)_n$(fulfilling the Markov property) and a stopping time$\tau$such that$P(\tau < \infty)=1$, we have that ... 1answer 39 views ### Fubini Question in context of Independence I am trying to show that if$X_t$is some process and there is a function$psuch that $$P[(X_{t_1},...,X_{t_n}) \in A_1 \times...\times A_n] = \int_{A_1 \times...\times A_n} ... 1answer 32 views ### American Put question If the interest rate is zero. Then show that the optimal exercise for an american put option is always the terminal time. That is, it is equivalent to a european put option. 0answers 59 views ### Law of iterated logarithms for BM The law of iterated logarithms for the standard Brownian motion asserts that (\ast) \limsup\limits_{h \downarrow 0} \frac{B(h)}{\sqrt{2h\log\log(\frac{1}{h})}} = 1 I'm trying to prove the ... 0answers 21 views ### Understanding certain parts of the proof of Helly's Selection Theorem I have read through the following proof of Helly's Selection Theorem. There are just two parts, which I have highlighted, that are left for the reader to fill in, and I would like to know how to prove ... 1answer 28 views ### Does weak convergence of \nu_{n} imply convergence of \int{f_{n}(x)d\nu_{n}(x)}? Suppose that we know that \int{ |f_{n}(x) - f(x)| d\mu(x)} \longrightarrow 0 \qquad (1) for every probability measure \mu \in \mathcal{A} in a certain class. Also, suppose that \{\nu_{n}\} ... 0answers 23 views ### Hellinger Integral properties Let \mu , \nu be two probability measures on (\Omega , \mathcal{F}). Suppose we have a probability measure \lambda such that both \mu , \nu are absolutely continuous with respect to ... 1answer 38 views ### Absolutely Continuous measures and Hellinger integral Let \mu , \nu be two probability measures on (\Omega , \mathcal{F}). Suppose we have a probability measure \lambda such that both \mu , \nu are absolutely continuous with respect to ... 0answers 105 views ### Question about Feller's book on the Central Limit Theorem My question concerns the proof of Theorem 1, section VIII.4, in Vol II of Feller's book 'An Introduction to Probability Theory and its Applications'. Theorem 1 proves the Central Limit Theorem in the ... 0answers 85 views ### Kolmogorov extension-type result I would like to prove the following, using the standard Kolmogorov extension theorem (e.g. http://en.wikipedia.org/wiki/Kolmogorov_extension_theorem): Let (\Omega, \mathscr{F}, P) denote our ... 1answer 65 views ### Topology of weak convergence Edited: Thanks to etienne. I start with a compact metric space (X,d). Then I consider the collection of finite measure \mathcal{M} on X and I equip \mathcal{M} with the topology of weak ... 1answer 86 views ### Convolution of a probability measure with a smooth function If f\in L^1(\mathbb{R}^n) and g\in L^p(\mathbb{R}^n) then by Young's convolution inequality we have the estimate:$$ \|f*g\|_{L^p}\leq \|f\|_{L^1}\|g\|_{L^p}.$$Question: Let \mu be a ... 1answer 22 views ### Find a asymptotic upper bound for \sum_{n=N}^{\infty}p_{ii}^{(n)} for a asymetric one-dimensional simple random walk For asymmetric one-dimensional simple random walk, that is$$P(X_n = X_{n-1} + 1) = p = 1 - P(X_n = X_{n-1} - 1)for some p \ne 1/2, provide an asymptotic upper bound for ... 1answer 25 views ### Set Difference Probability [duplicate] Here is the question: Prove that for every \epsilon>0 and every set A\in\mathcal{B}(\mathbb{R}^{n}) there is a compact set K\subset A such that P(A\setminus K)\leq\epsilon. --I have ... 1answer 68 views ### The probability distribution function of uniform random variables is as given Given U_1, U_2, \dots, U_n where each U_i \sim U[0,1], then use uniqueness theorem to show probability distribution function of X = U_1 + U_2 + \ldots +U_n (sum of independent uniform random ... 1answer 78 views ### Relation between convergence in distribution and weak convergence If (X_n: n\in \mathbb{N}), X are a sequence of random variables in \mathbb{R}, I wish to show that X_n \to X weakly if and only if X_n \to X in distribution. By 'converging weakly' I mean that ... 0answers 54 views ### A naive question about “random” probability distributions So I've come up with this idea, which may be mathematically unprecise, but it goes as follows. Example: Suppose you have a random variable X taking values in the interval [-1,1]. Then, say, ... 1answer 114 views ### Expected value, I do not get this “wikipedia triviality” on wikipedia are two definitions of the expected value for some random variable E(X)=\int t f_X(t) dt and E(X)=\int X dP. I do not see how they are equivalent. In cases, that X would be ... 2answers 118 views ### Too stupid to understand random variable questions? I have two excercises: 1.) Let X_1,X_2,X_3 be independent uniformly distributed random variables on [0,1]. What is the density function of X_1+X_2+X_3? 2.) Let X_1,...,X_4 be independet ... 1answer 66 views ### Inequalities of the quantile function [closed] I'm trying to rigorously prove the following inequalities involving the quantile function Y(a) = \inf \{x \in\mathbb{R} : a \leq F(x)\} where F is the distribution function: 1) F(x) < a \iff ... 0answers 24 views ### Problem with an inequality from probability theory (Random matrix theory) I read the following notes on random matrix theory http://www.umpa.ens-lyon.fr/~aguionne/cours.pdf . While reading Wigner's proof for the semi-cicular law I encoutered the following inequality on page ... 1answer 29 views ### Convergence of the limit of a sum of |P_{nk}-P_{k}|, where P_{nk} and P_{k} are sequences of nonnegative numbers summing to 1 Let (P_{nk})_{k \geq 1}, n=1,2,\cdots and (P_{k})_{k \geq 1} be a sequence of nonnegative numbers satisfying \sum_{k=1}^{\infty}P_{nk}=1 and \sum_{k=1}^{\infty}P_{k}=1, and let \lim_{n ... 1answer 48 views ### supremum norm and convergence. Suppose f:[0,\infty) \rightarrow \mathbb{R} is continuous. Suppose that for some \epsilon > 0, \max_{t \in [0,n]} |f(t)| < \epsilon for all n \in \mathbb{N}. Is it then true that ... 1answer 47 views ### IFF conditions for convergence in probability and almost surely I am working on a bunch of problems in preparation for an exam in Probability Theory. I have come across two similar questions that I need some assistance with. Suppose we have a sequence of ... 1answer 66 views ### Law of large numbers for Brownian Motion Let \{B_t: 0 \leq t < \infty\} be standard Brownian motion and let T_n be an increasing sequence of finite stopping times converging to infinity a.s. Does the following property hold? ... 1answer 38 views ### Definition of a random variable in the context of a hypergeometric distribution We defined a random variable in a probability space (\Omega, E, P) as a map X: \Omega \rightarrow \mathbb{R}. Unfortunately, I somehow have the impression that this term "random variable is used ... 1answer 163 views ### Convergence of events in a probability space with respect to L^2 Define for events X, Y that d(X,Y) = P((X-Y) \cup (Y-X)) = P(X \bigtriangleup Y) , show that d(X_n,X) \rightarrow 0 if and only if \chi_{X_n} converges in L^2 to \chi_X (these are ... 1answer 57 views ### Does convergence of \lim_{n \rightarrow \infty} \int h d\mu_n for all continuous, bounded h imply weak convergence of (\mu_n)? Let (\mu_n) be a sequence of positive finite Borel measures on \mathbb{R}. Suppose \lim_{n \rightarrow \infty} \int h d\mu_n converges for every bounded, continuous function h on \mathbb{R}. ... 1answer 186 views ### Does weak convergence with uniformly bounded densities imply absolute continuity of the limit? Suppose (f_n) is a sequence of probability density functions on \mathbb{R} such that \begin{align*} f_n &\leq M \text{ for all }n \\ f_n(x) &= 0 \text{ for all } |x| > 1 \end{align*} ... 1answer 105 views ### Proving \int^{\infty}_0 \cos(tx)\left (\frac{\sin(t)}{t} \right )^n \, dt = 0 I've been asked to prove that \int^\infty_0 \cos(tx)\left (\frac{\sin(t)}{t} \right )^n \, dt = 0, \space \forall x > n \geq 2.$$My approach so far has been to use a theorem proved in class ... 1answer 75 views ### Techniques for determining whether the weak limit of an absolutely continuous sequence of probability measures is absolutely continuous? Let (\mu_n) be a sequence of probability measures on \mathbb{R} that converges weakly to a probability measure \mu on \mathbb{R}. So$$ \lim \int hd\mu_n = \int h d\mu$whenever$h$is a ... 1answer 160 views ### Show using dominated convergence that summation and differentiation can be interchanged. Problem: Define$\{p_k: k \ge 0 \}$as a probability mass function on a discrete random variable X taking values on$\{0,1,...\}$then define the generating function$P(x) = E(x^X) = ...
The goal of this problem is to prove that $P(S_n = k)\sqrt{2\pi n} \rightarrow e^{-\frac{x^2}{2}}$ by using Stirling's formula. Here is what is given: 1) $S_n = \sum_1^n X_i$, where $\{X_i\}$ are ...