User byron schmuland - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-24T23:34:01Z http://mathoverflow.net/feeds/user/6096 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/91195/reference-request-martingale-decompositions-positive-negative-and-u-i-singular/91340#91340 Answer by Byron Schmuland for Reference request: Martingale decompositions (positive/negative and u.i./singular) Byron Schmuland 2012-03-16T00:11:46Z 2012-03-16T00:11:46Z <p>These are the <em>Krickeberg</em> and <em>Riesz</em> decompositions, respectively. A good reference is section 4 of Chapter V in <strong>Probabilities and Potential B</strong> by Claude Dellacherie and Paul-Andre Meyer. </p> http://mathoverflow.net/questions/89243/gluing-markov-processes/89353#89353 Answer by Byron Schmuland for Gluing Markov processes Byron Schmuland 2012-02-24T01:34:47Z 2012-02-24T01:34:47Z <p>In his book <em>General Theory of Markov Processes</em>, Michael Sharpe has a chapter on Transformations. One of the transformations described there is "concatenation of processes" which may be what you want. </p> http://mathoverflow.net/questions/87948/markov-processes-with-given-marginals/87956#87956 Answer by Byron Schmuland for Markov Processes with Given Marginals Byron Schmuland 2012-02-09T03:24:58Z 2012-02-09T03:24:58Z <p>If you are willing to drop continuity in the parameter $t$, then you could let $(X_t)$ be independent with distribution $\mu_t$. </p> http://mathoverflow.net/questions/87265/symmetric-feller-processes-and-dirichlet-forms/87470#87470 Answer by Byron Schmuland for Symmetric Feller processes and Dirichlet Forms Byron Schmuland 2012-02-03T19:24:51Z 2012-02-03T19:24:51Z <p>I've decided to post an incomplete preliminary answer. </p> <p>I ran into your problem when I was writing [1]. On page 258 you will see my resolution. </p> <p>I should point out that in my case, the underlying space $X$ was compact, and that $m$ was a finite measure with full support. Thus, $C(X)$ embeds into $L^2(X;m)$ with a continuous, linear injection in the obvious way. This may not hold in the locally compact case, and I'm not sure how serious a problem that is.</p> <p>Translated into your notation, and letting $\tilde G$ be the Friedrichs extension we note that $\bar G$ and $\tilde G$ agree on $\cal D$ and so the resolvent operators $\bar R_\lambda$ and $\tilde R_\lambda$ agree on $(\lambda-G)({\cal D})$. We deduce that $\bar R_\lambda= \tilde R_\lambda$ on $C(X)$ and using the Yosida approximation conclude the same about the semigroup operators $\bar T_t$ and $\tilde T_t$.</p> <p>I hope this is of some help. If anything is unclear, let me know.</p> <p>[1] A result on the infinitely many neutral alleles diffusion model. <em>Journal of Applied Probability</em> 28, 253-267 (1991). </p> http://mathoverflow.net/questions/79373/some-questions-concerning-a-random-number-process/79415#79415 Answer by Byron Schmuland for Some questions concerning a random number process Byron Schmuland 2011-10-28T17:23:29Z 2011-10-30T23:44:32Z <p>Using Markov chain theory, it is not hard to show that the expected time until the process hits state "1" starting at $N_0$ is $1+1+1/2+\cdots+1/(N_0-1)$.</p> <hr> <p><strong>Correction</strong> </p> <p>In my previous answer, I mistakenly chose a new state uniformly from $[1,\dots, N_i]$ instead of $[1,\dots, N_{i-1}]$. The expected hitting time of state "1" for the OP's model starting at $N_0$ is $1+1/2+\cdots+1/(N_0-1)$. This is one less than my first answer. </p> http://mathoverflow.net/questions/78706/probability-one-event-for-markov-chain/78721#78721 Answer by Byron Schmuland for Probability-one event for Markov chain Byron Schmuland 2011-10-20T23:49:14Z 2011-10-24T15:12:16Z <p>If I've understood your problem correctly, an argument along these lines may help:</p> <hr> <p>Let ${\cal F}_n=\sigma(X_0,X_1,\dots,X_n)$ and define $S_n=\left(X_n\in S\right)$, so that $S_n\in {\cal F}_n$. We will use Levy's generalization of the Borel-Cantelli Lemma which states that $$\left( S_n\mbox{ i.o.} \right)=\left(\sum_n \mathbb{P}(S_{n+1} | {\cal F}_{n})=\infty\right).$$</p> <p>Let's calculate the conditional probability. Letting $E(x)={ X_{n}=x_{n},X_{n-1}=x_{n-1},\dots,X_0=x_0}$ be a generic partition set, we get \begin{eqnarray*} \mathbb{P}(S_{n+1}\,|\,{\cal F}_n)&amp;=&amp;\sum_x\mathbb{P}(X_{n+1}\in S\,|\,E(x))1_{E(x)}\cr &amp;=&amp;\sum_x\mathbb{P}(X_{n+1}\in S\,|\,X_n=x_n)1_{E(x)}\cr &amp;=&amp;\sum_x P(x_n, S)1_{E(x)}\cr &amp;=&amp;P(X_n, S), \end{eqnarray*} where $P$ is the transition kernel for the Markov chain.</p> <p>The definition of nice" set gives $P(X_n,S)\geq \varepsilon_K 1_K(X_{n}),$ and since $(X_n)$ visits $K$ infinitely often, we have $$\sum_n P(X_n,S)\geq \varepsilon_K \sum_n 1_K(X_{n})=\infty$$ almost surely.</p> http://mathoverflow.net/questions/78397/markov-property-determined-by-just-the-law-or-also-the-realization/78459#78459 Answer by Byron Schmuland for Markov Property: determined by just the law or also the realization? Byron Schmuland 2011-10-18T14:17:48Z 2011-10-18T14:17:48Z <p>I've posted a solution <a href="http://www.stat.ualberta.ca/people/schmu/preprints/markov_law.pdf" rel="nofollow">on my webpage</a>.</p> http://mathoverflow.net/questions/76347/distribution-of-the-biggest-gap/76361#76361 Answer by Byron Schmuland for Distribution of the biggest gap Byron Schmuland 2011-09-25T20:07:33Z 2011-09-25T20:15:47Z <p>This is the answer to a slightly modified version of the problem. I hope that it would also lead to a solution of the original version.</p> <p>As I point out in my answer to <a href="http://math.stackexchange.com/questions/66430/" rel="nofollow">Math StackExchange question 66430</a> ("What is the distribution of gaps?"), if, in addition to the gaps $G_1=a_1$and $G_j:=a_j-a_{j-1}$ for $2\leq j\leq n$, you introduce final gap $G_{n+1}=(m+1)-a_n$, the random vector $(G_1,G_2,\dots, G_{n+1})$ gives a random composition of the number $m+1$. That is, all outcomes $(g_1,g_2,\dots, g_{n+1})$ with $$g_1+g_2+\cdots+g_{n+1}=m+1,\quad g_j\geq 1$$ are equally likely. There are $m\choose n$ such compositions, as found using stars and bars. </p> <p>Then $Pr(a_{\max}\leq k)$ (where <em>my</em> maximum includes the final gap) is just the proportion of compositions using numbers from $1$ to $k$. By inclusion-exclusion and stars and bars, this probability is $$Pr(a_{\max}\leq k)={\sum_{x} (-1)^x {m-xk\choose n}{n+1\choose x}\over{m\choose n}}.$$</p> http://mathoverflow.net/questions/62747/support-of-probability-measures-on-separable-metric-spaces/62751#62751 Answer by Byron Schmuland for Support of Probability Measures on Separable Metric Spaces Byron Schmuland 2011-04-23T15:34:25Z 2011-04-23T15:45:33Z <p>A separable metric space is <a href="http://en.wikipedia.org/wiki/Lindelof_space" rel="nofollow">strongly Lindelof</a>, that is, every open cover of an open subset has a countable subcover. The complement of the support is the union of all open balls with zero measure. By reducing to a countable subcover, we see that this set has measure zero. So the support has full measure. </p> http://mathoverflow.net/questions/62720/probability-and-math-puzzle-books-references/62743#62743 Answer by Byron Schmuland for probability and math puzzle books/references Byron Schmuland 2011-04-23T13:57:59Z 2011-04-23T13:57:59Z <p><a href="http://books.google.ca/books?id=KCsSWFMq2u0C&amp;printsec=frontcover&amp;dq=blom+holst+sandell&amp;source=bl&amp;ots=R8Dj0_qoQg&amp;sig=Evx4thEukrKZ0BZXfyVIzS-9TYI&amp;hl=en&amp;ei=CtqyTbKVJ460sAP_i53kCw&amp;sa=X&amp;oi=book_result&amp;ct=result&amp;resnum=1&amp;ved=0CBkQ6AEwAA#v=onepage&amp;q&amp;f=false" rel="nofollow">Problems and Snapshots from the World of Probability</a> by Gunnar Blom, Lars Holst, and Dennis Sandell is a book that I like very much. Their problems are pretty challenging and assume a working knowledge of basic probability. </p> http://mathoverflow.net/questions/61664/number-of-required-trials-to-sample-all-possible-states-of-a-d-sided-loaded-die/61694#61694 Answer by Byron Schmuland for Number of required trials to sample all possible states of a 'd'-sided loaded die Byron Schmuland 2011-04-14T12:55:13Z 2011-04-14T12:55:13Z <p>You may also find the answers of <a href="http://math.stackexchange.com/questions/25568/a-question-about-dice/25576#25576" rel="nofollow">this Math SE question</a> helpful.</p> <p>The average number of unique observations $E(M)$ is given in Henry's answer <a href="http://math.stackexchange.com/questions/32800/probability-distribution-of-coverage-of-a-set-after-x-independently-randomly-s" rel="nofollow">here</a>, at least in the case of a fair die.</p> http://mathoverflow.net/questions/59713/probability-estimates-for-beans-boxes/59731#59731 Answer by Byron Schmuland for Probability estimates for "beans & boxes" Byron Schmuland 2011-03-27T15:33:19Z 2011-03-27T15:42:46Z <p>Your problem falls under the general category of "coupon collector's problems". There is a large literature on such problems, mainly concerned with formulas for the mean and variance of the random variable $T$.</p> <p>The exact distribution of the time $T$ to have at least one bean ($N=1$) in each box is known in terms of Stirling numbers of the second kind: see <a href="http://math.stackexchange.com/questions/26528/m-balls-n-boxes-probability-problem" rel="nofollow">Henry's answer here</a>, or below<br> $$P(\mbox{ every box has at least one bean after }T \mbox{ seconds}) = P^{-T}\ P\ !\ \left\lbrace {T\atop P} \right\rbrace.$$ </p> <p>In general, I think you should take Alekk's advice and use the Poisson approximation.</p> http://mathoverflow.net/questions/59244/what-is-the-cover-time-of-a-random-walk-on-a-cube/59261#59261 Answer by Byron Schmuland for What is the cover time of a random walk on a cube? Byron Schmuland 2011-03-23T02:02:21Z 2011-03-23T15:22:27Z <p>An additional reference:</p> <p>Chapter 12 in <em>Problems and Snapshots from the World of Probability</em> by Blom, Holst, and Sandell is devoted to an elementary exposition of such cover problems. </p> <p>A related problem: </p> <p>The solution to Problem 6556 in the <em>American Mathematical Monthly</em> (Vol. 96, No. 9, Nov. 1989, pages 847-849) looks at the average number of steps for a random walk to visit all the <strong>edges</strong> on the cube in dimensions $d=2$, $3$, and $4$. </p> <p>For $d=2$ the answer is easily computed to be 10.</p> <p>For $d=3$ a system with 387 equations in 387 unknowns is solved to give an answer of about 48.5.</p> <p>For $d=4$ the problem is declared hopeless. </p> http://mathoverflow.net/questions/35695/is-the-infimum-of-the-ky-fan-metric-achieved Is the infimum of the Ky Fan metric achieved? Byron Schmuland 2010-08-15T22:53:00Z 2011-02-24T18:53:47Z <p>Consider the probability space $(\Omega, {\cal B}, \lambda)$ where $\Omega=(0,1)$, ${\cal B}$ is the Borel sets, and $\lambda$ is Lebesgue measure.</p> <p>For random variables $W,Z$ on this space, we define the Ky Fan metric by</p> <p>$$\alpha(W,Z) = \inf \lbrace \epsilon > 0: \lambda(|W-Z| \geq \epsilon) \leq \epsilon\rbrace.$$</p> <p>Convergence in this metric coincides with convergence in probability.</p> <p>Fix the random variable $X(\omega)=\omega$, so the law of $X$ is Lebesgue measure, that is, ${\cal L}(X)=\lambda$.</p> <blockquote> <p><b> Question:</b> For any probability measure $\mu$ on $\mathbb R$, does there exist a random variable $Y$ on $(\Omega, {\cal B}, \lambda)$ with law $\mu$ so that $\alpha(X,Y) = \inf \lbrace \alpha(X,Z) : {\cal L}(Z) = \mu\rbrace$ ?</p> </blockquote> <p><em>Notes:</em></p> <ol> <li><p>By Lemma 3.2 of <a href="http://arxiv.org/PS_cache/math/pdf/0601/0601524v1.pdf" rel="nofollow">Cortissoz</a>, the infimum above is $d_P(\lambda,\mu)$: the L&#233;vy-Prohorov distance between the two laws.</p></li> <li><p>The infimum is achieved if we allowed to choose both random variables. That is, there exist $X_1$ and $Y_1$ on $(\Omega, {\cal B}, \lambda)$ with ${\cal L}(X_1) = \lambda$, ${\cal L}(Y_1) = \mu$, and $\alpha(X_1,Y_1) = d_P(\lambda,\mu)$. But in my problem, I want to fix the random variable $X$.</p></li> <li><p><em> Why the result may be true: </em> the space $L^0(\Omega, {\cal B}, \lambda)$ is huge. There are lots of random variables with law $\mu$. I can't think of any obstruction to finding such a random variable.</p></li> <li><p><em> Why the result may be false: </em> the space $L^0(\Omega, {\cal B}, \lambda)$ is huge. A compactness argument seems hopeless to me. I can't think of any construction for finding such a random variable. </p></li> </ol> http://mathoverflow.net/questions/23478/examples-of-common-false-beliefs-in-mathematics/23933#23933 Answer by Byron Schmuland for Examples of common false beliefs in mathematics. Byron Schmuland 2010-05-08T14:39:53Z 2011-02-24T18:40:57Z <p>From the Markov property of the random walk $(X_n)$ we have</p> <p>$$P(X_4>0 \ |\ X_3>0, X_2>0) = P(X_4>0\ |\ X_3>0).$$</p> <p>To paraphrase Kai Lai Chung in his book "Green, Brown, and Probability",</p> <p>"The Markov property means that the past has no after-effect on the future <em>when the present is known</em>; but beware, big mistakes have been made through misunderstanding the exact meaning of the words "when the present is known"." </p> http://mathoverflow.net/questions/51139/differentiation-of-a-series-of-increasing-functions/51164#51164 Answer by Byron Schmuland for Differentiation of a series of increasing functions Byron Schmuland 2011-01-05T00:02:18Z 2011-01-05T00:02:18Z <p>This is also Theorem 17.18 (page 267) of <em>Real and Abstract Analysis</em> by Hewitt and Ross. The result is credited there to Fubini.</p> http://mathoverflow.net/questions/50154/reachability-for-markov-process/50241#50241 Answer by Byron Schmuland for Reachability for Markov process Byron Schmuland 2010-12-23T13:41:34Z 2010-12-23T13:41:34Z <p>It is right, and not so obvious.</p> <p>The question of whether or not a Markov process hits particular sets is usually studied using the concept of capacity.</p> <p>For a continuous time parameter Markov process taking values in a general topological state space, this leads to non-trivial problems of measurability. For instance, for a Borel $A$ there is no guarantee that the set $R(T,A)\in{\cal F}$ where $(\Omega,{\cal F},\Pr)$ is the probability space. However, under suitable conditions, capacity theory can be used to show that $R(T,A)$ is universally measurable, and hence that $\Pr[R(T,A)]$ makes sense.</p> <p>Let's assume that the state space and process are "nice"; say, the state space is a locally compact, separable metric space, and the process has right continuous sample paths. For fixed $T&lt;\infty$, the formula $\phi(A)=\Pr[R(T,A)]$ defines a Choquet capacity on the Borel sets $A$. Therefore, $$\phi(A)=\sup(\phi(K): K\subseteq A,\ K\mbox{ compact}).$$</p> <p>For a compact $K$, define the stopping time $\tau(\omega):=\inf(t\geq 0: X_t(\omega)\in K)$. Since the sample paths of $(X_t)$ are right continuous and $K$ is closed, we have $R(T,K) = (X_{\tau\wedge T} \in K).$</p> <p>Therefore, $$\Pr[R(T,K)]\leq \mathbb{E}[I_A(X_{\tau\wedge T})]\leq \Pr[R(T,A)].$$</p> <p>Taking the supremum over compact subsets of $A$ gives $$\Pr[R(T,A)]=\sup_{\tau}\ \mathbb{E}[I_A(X_{\tau\wedge T})],$$ which gives your desired result. Letting $T\to\infty$ gives the infinite version.</p> <p>The result hinges on the fact that, as far as the process goes, the Borel set $A$ can be well approximated from the inside by compact sets.</p> <p>You can find more details in Chapter I, Section 10 of Blumenthal and Getoor's <em>Markov Processes and Potential Theory</em>, or in Section 3.3 of Kai Lai Chung's <em>Lectures from Markov Processes to Brownian Motion</em>.</p> http://mathoverflow.net/questions/46011/is-the-space-of-continuous-functions-from-a-compact-metric-space-into-a-polish-sp Is the space of continuous functions from a compact metric space into a Polish space Polish? Byron Schmuland 2010-11-14T03:50:59Z 2010-12-06T04:01:09Z <p>Let $K$ be a compact metric space, and $(E,d_E)$ a complete separable metric space. Define $C:=C(K,E)$ to be the continuous functions from $K$ to $E$ equipped with the metric $d(f,g)=\sup_{k\in K}\ d_E (f(k),g(k))$. Is the space $C$ separable?</p> <p>The result is true when $E$ is the real line; this is Corollary 11.2.5 in Dudley's book <em>Real Analysis and Probability</em>. </p> <p>The result is also true when $K=[0,1]$ (if I'm not being too careless) by considering $C$ as a subspace of the Skorohod space $D_E[0,1]$, which is complete and separable by Theorem 5.6 in Ethier and Kurtz's book <em>Markov Processes: Characterization and Convergence</em>. </p> <p>For general $K$, it is not so obvious how to find an explicit countable dense set in $C$, but I suspect one could modify Ethier and Kurtz's approach and get a proof. </p> <p>But surely this result is known, and stated in some book? I've searched through my library without success. </p> <hr> <p><strong>Update:</strong> This result is also <strong>Theorem 2.4.3</strong> of S. M. Srivastava's book <em>A Course on Borel Sets</em>. His proof is the same as Kechris's. I have also found an alternative, but false, published proof using the "fact" that $C(K,E)$ is $\sigma$-compact. Beware! </p> http://mathoverflow.net/questions/46957/right-continuity-of-natural-filtrations/46962#46962 Answer by Byron Schmuland for Right-continuity of natural filtrations Byron Schmuland 2010-11-22T15:46:34Z 2010-11-22T15:46:34Z <p>Right continuity fails even for canonical continuous processes. </p> <p>The natural filtration on $C([0,\infty))$ is not right continuous. For example, the event $\{\omega: {d^+\over dt}\ \omega_t\mbox{ exists at }t=0\}$ belongs to ${\cal F}_{0+}$ but not ${\cal F}_0$. In words, you can tell whether the function $\omega_t$ has a right derivative at $t=0$ with an infinitesimal peek <em>beyond</em> time 0, but you cannot tell just from the value of the function $\omega_t$ <em>at</em> time 0. </p> <p>Right continuous filtrations are nicer to work with, and since it fails for the natural filtration,<br> we often use the right continuous version instead. Fortunately, many of the nice properties of right continuous processes carry over even with this enlarged filtration. For example, Brownian motion is still Markov with respect to ${\cal F}_{t+}$ which leads to interesting results like Blumenthal's zero-one law. </p> http://mathoverflow.net/questions/46094/if-h-is-a-separable-hilbert-space-is-l2h-separable/46097#46097 Answer by Byron Schmuland for If $H$ is a separable Hilbert space, is $L^2(H)$ separable? Byron Schmuland 2010-11-15T01:25:19Z 2010-11-15T01:37:52Z <p>By Example 7.14.13 in Volume 2 of Bogachev's <em>Measure Theory</em>, every Radon measure on $H$ is separable, so that $L^2(H,\gamma)$ is also separable. It is not necessary that $H$ is a Hilbert space, just that every compact subset of $H$ be metrizable. </p> http://mathoverflow.net/questions/45581/is-there-an-extension-of-the-arzela-ascoli-theorem-to-spaces-of-discontinuous-fun/45632#45632 Answer by Byron Schmuland for Is there an extension of the Arzela-Ascoli theorem to spaces of discontinuous functions? Byron Schmuland 2010-11-11T01:16:07Z 2010-11-11T01:41:43Z <p>Billingsley's book <em>Convergence of Probability Measures</em> has a section on the geometry of the Skorohod space $D=D[0,1]$, and in Theorem 14.3 gives an analogue of the Arzela-Ascoli theorem. I am using the first edition of this book.</p> <blockquote> <p>A set $A$ has compact closure in the Skorohod topology if and only if $$\sup_{x\in A}\ \sup_t |x(t)|&lt;\infty\ \mbox{ and }\ \lim_{\delta\to 0}\ \sup_{x\in A}\ w^\prime_x(\delta)=0.$$</p> </blockquote> <p>Here $$w^\prime_x(\delta)=\inf_{t_i}\ \max_{0 &lt; i\leq r}\ \sup\{ |x(s)-x(t)|: s,t\in [ t_{i-1} , t_i) \} ,$$ where the infimum extends over finite sets of points $\{ t_i\}$ such that $0=t_0 &lt; t_1 &lt; \cdots &lt; t_r=1$ and $t_i-t_{i-1}>\delta\$ for $1\leq i \leq r$. </p> <p>Billingsley also gives a second characterization of compactness in Theorem 14.4 that he says is sometimes more convenient to work with. </p> <p>I presume that this is the same as the solutions already given, but I thought another reference wouldn't hurt. </p> <p>Here is another reference. In section 6 of chapter 3 of Ethier and Kurtz's <em>Markov processes: Characterization and Convergence</em> you will find similar criteria for relative compactness of subsets of $D_E[0,\infty)$, that is, $E$-valued Skorohod space, where $E$ is a metric space. </p> http://mathoverflow.net/questions/44326/most-memorable-titles/44346#44346 Answer by Byron Schmuland for Most memorable titles Byron Schmuland 2010-10-31T16:34:17Z 2010-10-31T16:34:17Z <p><a href="http://www.dms.umontreal.ca/~andrew/PDF/beeb.pdf" rel="nofollow">Zaphod Beeblebrox's Brain and the Fifty-ninth Row of Pascal's Triangle</a></p> http://mathoverflow.net/questions/43662/expectation-of-first-positive-value-in-random-walk/43811#43811 Answer by Byron Schmuland for Expectation of first positive value in random walk Byron Schmuland 2010-10-27T15:13:42Z 2010-10-28T06:38:41Z <p>You are looking for the "mean ladder height" of a random walk. There is a (not very tractable) formula due to Spitzer that gives the answer:</p> <p>$$E(S_N)={\sigma\over\sqrt{2}} \exp\left(\sum_{n=1}^\infty {1\over n}(P(S_n&lt;0)-1/2)\right)$$</p> <p>Here $\sigma^2$ is the variance of the jump distribution. Maybe it would be possible to work this out in your special case. </p> <p>[1] Chow, Yuan S. <em>On Spitzer's formula for the moment of ladder variables.</em> Statist. Sinica <strong>7</strong> no. 1, 1997, 149–156.</p> <p>[2] Spitzer, Frank <em>A Tauberian theorem and its probability interpretation.</em> Trans. Amer. Math. Soc. <strong>94</strong>, 1960, 150–169. </p> http://mathoverflow.net/questions/43855/continuity-of-cylindrical-functions/43871#43871 Answer by Byron Schmuland for Continuity of cylindrical functions. Byron Schmuland 2010-10-27T20:37:46Z 2010-10-27T20:37:46Z <p>For each $j$, the inequality $$|\langle x , e_j\rangle| = j \sqrt{\langle x , e_j\rangle\langle e_j , x\rangle\over j^2}\leq j\sqrt{\langle x , x\rangle_\omega}$$ shows that the map $x\mapsto \langle x , e_j\rangle$ is continuous from $(X , \langle\cdot ,\cdot\rangle_\omega)$ to $\mathbb R$. Thus, for fixed $d$ the map $x\mapsto (\langle x , e_1\rangle, \dots,\langle x , e_d\rangle)$ is continuous from $(X , \langle\cdot ,\cdot\rangle_\omega)$ into ${\mathbb R}^d$. Composing with the smooth map $\phi:{\mathbb R}^d\to {\mathbb R}$ gives you a continuous function from $(X , \langle\cdot ,\cdot\rangle_\omega)$ into $\mathbb R$ again. </p> http://mathoverflow.net/questions/43833/a-markov-process-which-is-not-a-strong-markov-process/43838#43838 Answer by Byron Schmuland for A Markov process which is not a strong markov process? Byron Schmuland 2010-10-27T17:17:16Z 2010-10-27T17:17:16Z <p>A standard example is Exercise 6.17 in Sharpe's book <em>The general theory of Markov processes</em>. The process stays at zero for an exponential amount of time, then moves to the right at a uniform speed. </p> http://mathoverflow.net/questions/42310/when-is-l2x-separable/42314#42314 Answer by Byron Schmuland for When is $L^2(X)$ separable? Byron Schmuland 2010-10-15T17:51:01Z 2010-10-15T18:14:56Z <p>In addition to the measure $\mu$ being $\sigma$-finite, I think you also need some conditions on the measurable space $(X,{\cal A})$.</p> <p>Proposition 3.4.5 of Cohn's book <em>Measure Theory</em> says that $L^p(X,{\cal A},\mu)$ ($1\leq p &lt; \infty$) is separable if $\mu$ is $\sigma$-finite and $\cal A$ is countably generated. For example, it holds if $X$ is a complete separable metric space, and $\cal A$ is the Borel $\sigma$-algebra. </p> <p>However, even for a compact group, you can make counterexamples like $[-1/2,1/2]^{[0,1]}$, an uncountable product of a circles. For the product measure, $\mu=\lambda^{[0,1]}$, the coordinate functions are orthogonal in $L^2$ but there are uncountably many. </p> <p>I haven't checked the details, so take my answer with a grain of salt!</p> http://mathoverflow.net/questions/39928/wiener-process-related-counterexample/39932#39932 Answer by Byron Schmuland for Wiener process related counterexample Byron Schmuland 2010-09-25T06:08:15Z 2010-09-25T06:08:15Z <p>This is not hard to find such an example. Let $P$ be Wiener measure on the space $\Omega = C([0,\infty))$ of continuous functions $t\mapsto \omega(t)$. Then the process $\omega(t)$ satisfies all three conditions of a Brownian motion.</p> <p>Now let's define a new process $W(t)$ that is "almost" equal to $\omega(t)$, but where we deliberately wreck the sample path continuity. </p> <p>Take any random time $T:\Omega\to [0,\infty)$ that has a continuous distribution on $(\Omega, P)$, and let $W(t,\omega)=\omega(t)$ when $t\not=T(\omega)$, but $W(t,\omega)=\omega(t)+1$ otherwise. The process $W(t)$ still satisfies 1 and 3 but the sample path continuity fails at exactly at the time point $T(\omega)$ for each $\omega$. </p> <p>There are many such random times $T$, for example you could use $T(\omega):=\inf [t>0: \omega(t)=1 ]$, i.e. the hitting time of 1.</p> http://mathoverflow.net/questions/39882/product-of-borel-sigma-algebras/39886#39886 Answer by Byron Schmuland for Product of Borel sigma algebras Byron Schmuland 2010-09-24T18:38:34Z 2010-09-24T18:38:34Z <p>The answer to question 3 is yes. At least according to Lemma 6.4.2 of the second volume of Bogachev's book "Measure Theory".</p> <p>He requires both spaces to be Hausdorff and one of them to have a countable base. They need not be metric spaces. </p> http://mathoverflow.net/questions/39409/on-measurable-functions-of-two-variables/39419#39419 Answer by Byron Schmuland for On measurable functions of two variables Byron Schmuland 2010-09-20T18:50:45Z 2010-09-20T18:50:45Z <ol> <li><p>Yes, by continuity in the $t$ variable $f(t,x)=\lim_n f(\lfloor n t\rfloor/n,x)$, which expresses $f$ as the pointwise limit of a sequence of measurable functions.</p></li> <li><p>Yes, by continuity in the $t$ variable we have $\min_{t\in[0,1]} f(t,x)=\min_{t\in[0,1]\cap {\mathbb Q}} f(t,x)$, where $\mathbb Q$ means the rational numbers. </p></li> </ol> http://mathoverflow.net/questions/39289/estimate-the-error-term-in-clt/39353#39353 Answer by Byron Schmuland for estimate the error term in CLT Byron Schmuland 2010-09-20T01:07:20Z 2010-09-20T01:07:20Z <p>If $X_m$ has cumulative distribution function $F_m$, and $X$ has cumulative distribution function $F$, then (at least formally) integration by parts gives you $$E(f(X_m))-E(f(X))=\int (F_m(x)-F(x)) df(x).$$ Now you can apply the Berry-Esseen bound. </p> http://mathoverflow.net/questions/87265/symmetric-feller-processes-and-dirichlet-forms/87470#87470 Comment by Byron Schmuland Byron Schmuland 2012-02-08T17:38:28Z 2012-02-08T17:38:28Z @Hans In that case, doesn't the Friedrichs extension of $(G,{\cal D})$ equal the $L^2$ generator of the process by definition? http://mathoverflow.net/questions/87265/symmetric-feller-processes-and-dirichlet-forms/87470#87470 Comment by Byron Schmuland Byron Schmuland 2012-02-07T21:51:55Z 2012-02-07T21:51:55Z @Hans I'm still thinking about your problem from time to time, when I have moment to spare. I believe that a complete resolution of your problem can only come after we have clearly defined &quot;the $L^2$ semigroup induced by $X$&quot;. Do you have a specific definition in mind? http://mathoverflow.net/questions/87265/symmetric-feller-processes-and-dirichlet-forms Comment by Byron Schmuland Byron Schmuland 2012-02-02T16:12:11Z 2012-02-02T16:12:11Z I'm thinking about your problem, and hope to post an answer soon. In the meantime, I think you want to drop the condition $G({\cal D})\subset C_K$. This eliminates too many nice processes, and is not needed to make $\int Gf\, g\, dm$ finite. – Byron Schmuland http://mathoverflow.net/questions/84634/help-prove-a-maximal-inequality Comment by Byron Schmuland Byron Schmuland 2011-12-31T19:11:19Z 2011-12-31T19:11:19Z The motivation: <a href="http://math.stackexchange.com/questions/94948/kolmogorovs-maximal-inequality-for-random-number-composition/94970#94970" rel="nofollow" title="kolmogorovs maximal inequality for random number composition">math.stackexchange.com/questions/94948/&hellip;</a> http://mathoverflow.net/questions/81606/what-is-the-correct-notation-for-stating-that-for-every-element-of-a-set-of-2-tup Comment by Byron Schmuland Byron Schmuland 2011-11-22T13:12:06Z 2011-11-22T13:12:06Z Crossposted: <a href="http://math.stackexchange.com/questions/84552/what-is-the-correct-notation-for-stating-that-for-every-element-of-a-set-of-2-tu" rel="nofollow" title="what is the correct notation for stating that for every element of a set of 2 tu">math.stackexchange.com/questions/84552/&hellip;</a> http://mathoverflow.net/questions/81419/how-to-calculate-this-expectation-where-the-random-variable-is-restricted-on-a-sp Comment by Byron Schmuland Byron Schmuland 2011-11-20T19:50:02Z 2011-11-20T19:50:02Z Crossposted: <a href="http://math.stackexchange.com/questions/83931/how-to-calculate-this-expectation-where-the-random-variable-is-restricted-on-a-s" rel="nofollow" title="how to calculate this expectation where the random variable is restricted on a s">math.stackexchange.com/questions/83931/&hellip;</a> http://mathoverflow.net/questions/78987/deriving-a-closed-form-for-rolling-a-sum-n-with-k-dice-using-stars-and-bars/78990#78990 Comment by Byron Schmuland Byron Schmuland 2011-10-24T18:59:54Z 2011-10-24T18:59:54Z Standard dice have $m=6$ sides, not 18. http://mathoverflow.net/questions/78706/probability-one-event-for-markov-chain/78721#78721 Comment by Byron Schmuland Byron Schmuland 2011-10-21T15:17:35Z 2011-10-21T15:17:35Z In my solution $\varepsilon_K(\omega)&gt;0$ and $K(\omega)$ can be random. Only $S$ must be a non-random set. http://mathoverflow.net/questions/76712/determine-if-circle-is-covered-by-some-set-of-other-circles Comment by Byron Schmuland Byron Schmuland 2011-09-29T02:23:31Z 2011-09-29T02:23:31Z Crossposted: <a href="http://math.stackexchange.com/questions/68395/determine-if-circle-is-covered-by-some-set-of-other-circles" rel="nofollow" title="determine if circle is covered by some set of other circles">math.stackexchange.com/questions/68395/&hellip;</a> http://mathoverflow.net/questions/76347/distribution-of-the-biggest-gap/76361#76361 Comment by Byron Schmuland Byron Schmuland 2011-09-25T20:08:55Z 2011-09-25T20:08:55Z Can someone help me hyperlink to the MSE question? I couldn't make it work. http://mathoverflow.net/questions/76153/probability-of-generating-a-connected-graph Comment by Byron Schmuland Byron Schmuland 2011-09-23T00:56:54Z 2011-09-23T00:56:54Z Crossposted: <a href="http://math.stackexchange.com/questions/66777/probability-of-generating-a-connected-graph-on-the-unit-square" rel="nofollow" title="probability of generating a connected graph on the unit square">math.stackexchange.com/questions/66777/&hellip;</a> http://mathoverflow.net/questions/72978/radon-nikodym-derivative-as-a-measurable-function-in-a-product-space Comment by Byron Schmuland Byron Schmuland 2011-08-30T15:27:55Z 2011-08-30T15:27:55Z Asked and answered at MathSE: <a href="http://math.stackexchange.com/questions/58018/radon-nikodym-derivative-as-a-measurable-function-in-a-product-space" rel="nofollow" title="radon nikodym derivative as a measurable function in a product space">math.stackexchange.com/questions/58018/&hellip;</a> http://mathoverflow.net/questions/63056/an-elementary-problem-in-euclidean-geometry Comment by Byron Schmuland Byron Schmuland 2011-04-26T17:46:48Z 2011-04-26T17:46:48Z This is part (a) of problem 11524 in the October 2010 issue of the American Mathematical Monthly. http://mathoverflow.net/questions/62747/support-of-probability-measures-on-separable-metric-spaces Comment by Byron Schmuland Byron Schmuland 2011-04-23T15:49:21Z 2011-04-23T15:49:21Z This could also be of interest: <a href="http://mathoverflow.net/questions/44408/does-the-support-of-a-borel-probability-measure-always-have-full-measure-in-a-met" rel="nofollow" title="does the support of a borel probability measure always have full measure in a met">mathoverflow.net/questions/44408/&hellip;</a> http://mathoverflow.net/questions/57808/correlated-brownian-motion-and-poisson-process Comment by Byron Schmuland Byron Schmuland 2011-03-09T02:39:16Z 2011-03-09T02:39:16Z @George: Would the negative result also follow from the Martingale Representation Theorem? The martingale jump process $N_t-t$ cannot be represented as a stochastic integral with respect to BM, since it doesn't have continuous sample paths.