I know convergence in probability does not imply convergence in measure. I would like to see some simple example. Do you have any ideas please?
121382
Can somebody provide me with a sequence of (real-valued) functions, say on $[0,1]$ with the Lebesgue measure, such that the sequence converges in probability, or maybe in $\Vert \cdot \Vert _{L^2}$, but does not converge almost surely?
121971
I was wondering if someone could please give me an example of a sequence of decreasing sets where the first set has infinite Lebesgue measure; i.e., $\{B_{n}\}_{n=1}^{\infty}$ such that $m(B_{1}) = \infty$ but $m(\cap_{n=1}^{\infty} B_{n}) \neq \lim_{n \to \infty}m(B_{n})$?  thank you.
120568
Prove the following statement:   $ X_n \Rightarrow 0 $ (convergence in distribution) if and only if $ (\exists\; \epsilon&gt;0: |t|&lt;\epsilon) \;\; \phi_n(t) \rightarrow 1  $, where $\phi_n(t)$ is the characteristic function of the random variable $X_n$.  We could write $X_n \rightarrow 0 $ in probability since convergence in probability to a constant and convergence in distribution to a constant are the same concept.  I guess Lévy's continuity theorem implies the 'only if' part.  Thank you very much for your help in advance!
46068
I am looking for a book (English only) that I can treat as a reference text (more colloquially as a bible) about probability and is as complete - with respect to an undergraduate/graduate education in Mathematics - as possible. What I mean by that is that the book should contain and rigorously address the following topics:   Measure Theory (As a mathematical foundation for probability)   It is of course fine if this theory is addressed with an emphasis on probability and not only for the sake of mathematical measure theory, although the latter would be great too.  Introduction to Probability, i.e. the most common theory a student is exposed to when taking a first course in theoretic Probability. For example: distributions, expected value, modes of convergence, Borel Cantelli Lemmas, LLN, CLT, Gaussian Random Vectors More advanced topics such as: Conditional Expectation (defined through sigma-Algebras), Martingales, Markov Processes, Brownian Motion   I want it to be one book so I can carry a physical copy of it with me and work through the material in my spare time.   Examples:   The book by Jean-Francois Le Gall which can be found here:  but (unfortunately for me) is written in French.  Rick Durrett's book on Probability which can be found here  - the  for this book seems a bit mixed, uncertain about how to weigh that.   I am well aware that it's not easy to meet all of the above criteria simultaneously, but I would be grateful for any recommendation.
40818
I've encountered a question I couldn't answer, and I would appreciate any help: Is it true that $$f_n \xrightarrow{m}0\,\,\implies \,\,\frac{1}{n} \sum_{k=1}^{n}f_k \xrightarrow{m}0$$ where $(X,{\mathcal{B}},m)$ is a probability space, and $\xrightarrow{m}$ means: $\forall \epsilon&gt;0,\,  m\left(|f_n|\geq \epsilon\right) \rightarrow 0.$  This is similar to Cesaro convergence, which is weaker than regular convergence, but it is not the case here. I've tried to prove the proposition, or to give a counter example, but didn't manage to succeed in either. Any insights will be appreciated!
42992
Is there an example where convergence in mean does not imply convergence in quadratic mean. $E|X_n-X|\rightarrow 0$ but $E|X_n-X|^2 \nrightarrow 0$.  It seems like this should be fairly straightforward, but I cannot really come up with an answer.  I'd imagine the answer would look something like   $$X_n=\begin{cases}a &amp; \text{with probability } \frac{1}{n} \\\  0&amp; \text{with probability } 1-\frac{1}{n} \end{cases}$$
21227
Any suggestion/hint, not the whole solution, how to determine convergence/divergence of $$ \sum_{n=1}^{\infty}\dfrac{e^n \cdot n!}{n^n} $$ I'm currently stuck.
77932
I am trying to solve following exercise from Folland,     If $\mu$ is a semifinite measure and $\mu(E) = \infty$, for any $C &gt; 0$, $\exists$ $F \subset E$ with $C &lt; \mu(F) &lt; \infty$.   It seems to follow from definition of semifinite measures, which you can find , but I couldn't prove it.
103506
I want to consider a space of probability measures on some set $X$, such that the space of measures is complete, not in the sense of  (though probably that too), but as in the standard analysis meaning. I haven't studied far enough in probability measures, so I can't tell -- what conditions do I need on my space such that it is complete? And how should I metrize it?   Thanks!  Edit: A reference/citation to a text would also be super helpful!
120937
Prove, or give a counter example: Let $\mu$ be a finite positive borel measure on $\mathbb{R}$. Then $\int (x-y)^{-2} d \mu (y) = \infty $ almost everywhere on $\mu$ (for the selection of x's).  This is a question I had in an exam, and the answer is supposed to be presented in less than 30 words, so there must be something quite simple I'm missing.
3412
I don't know how to begin this. My thoughts are that since there are three convergent subsequences, we could use some knowledge about their limits to construct our convergent sequence but I'm not sure how to prove $x_n$ also converges. Is there a property of metric spaces I should use, or something about Bolzano-Weierstrass theorem (for metric spaces)?  EDIT: As noted in a comment I am not allowed to assume they all converge to the same limit.
92296
Most of the books I read, they only state some examples of tail events. One of them is  $[\sum_n X_n \ converges]$. My main problem is to show that this is indeed a tail event.
39420
Can somebody tell me why the counting measure (so, if $S=P(X)$, then $\mu(A)$=infinity if $A$ isn't finite and $\mu(A)=$#$A$ if $A$ is finite) is a measure? (The second property of a measure isn't clear for me).
47091
My question is pretty simple.  Since $n! \gt n!!$, it's clear by the comparison test that $\sum_{n=0}^\infty \frac {1}{n!!}$ converges.      But what value does the sum converge to? How does one go about determining its value (if possible)?
71569
So one of my professors proposed a problem to me and it has stumped me for some time now. Here's how it goes:  Suppose you have a sequence $a_n$ of real numbers such that $$\lim_{n\to\infty} a_{n} = 0$$ and suppose the sequence of partial sums $s_n$ is bounded. Prove that $s_n$ converges or give a counterexample.  I'm hoping to figure this out without anyone handing me the complete solution, so if someone could point me in the right direction with a hint, it would be much appreciated.
4693
what are some good books on probabilities  and measure theory? I already know basic probabalities, but I'm interested in sigma-algrebas, filtrations, stopping times etc, with possibly examples of "real life" situations where they would be used  thanks
32443
Exercise 5.3.1 in Durrett's "Probability Theory and Examples" states  Let $X_n$, $n\ge 0$, be a submartingale with $\sup X_n &lt; \infty$. Let $\xi_n=X_n-X_{n-1}$, and suppose $E(\sup \xi_n^+)&lt;\infty$. Show that $X_n$ converges almost surely.  Here are my thoughts: It looks like we need to use theorem 5.2.8 (Martingale convergence theorem), which states that $X_n$ converges almost surely if $\sup EX_n^+&lt;\infty$. I am trying to understand why we can just do the following (or at least I'm guessing we can't):  Suppose $X_n &gt; 0$ (edit: on some set with positive measure), then $0&lt; \sup X_n &lt; \infty$, and therefore $EX_n^+ \le \sup X_n &lt; \infty$. So we're done.  Otherwise, if $X_n \le 0$ a.e., then $\sup X_n^+ = 0$ a.e. Once again, we're done.  Am I missing something? Since I didn't use all the information, I probably am.  Thanks for the help.