most recent 30 from http://mathoverflow.net 2013-05-26T08:14:42Z http://mathoverflow.net/feeds/user/5709 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/76791/quanitative-de-moivrelaplace-theorem-reference-request/76826#76826 Yvan Velenik 2011-09-30T06:20:16Z 2011-09-30T06:20:16Z

<p>You just want a local limit theorem for a sum of i.i.d. Bernoulli random variables. A standard reference (not just for Bernoulli r.v.!) is "Sums of Independent Random Variables" by Petrov, in particular Chapter VII, §3. </p>

http://mathoverflow.net/questions/73647/do-the-converses-of-weak-law-of-large-numbers-central-limit-theorem-hold/73828#73828 Yvan Velenik 2011-08-27T08:06:36Z 2011-08-27T08:06:36Z

<p>(As suggested, I promote my comment to an answer, with pgassiat's complement.)</p> <p>Necessary and sufficient conditions (in terms close to those you want) for the WLLN and the CLT can be found, e.g., in "Foundations of modern probability" by Kallenberg (Theorems 4.16 and 4.17 in the first edition, Theorems 5.16 and 5.17 in the second edition). </p>

http://mathoverflow.net/questions/73388/analytic-implicit-function-theorem/73390#73390 Yvan Velenik 2011-08-22T09:35:20Z 2011-08-22T09:50:26Z

<p>One possible reference is <a href="http://books.google.com/books?id=nDgBsOurnAIC&amp;lpg=PA72&amp;vq=implicit&amp;hl=fr&amp;pg=PP1#v=onepage&amp;q&amp;f=false" rel="nofollow">"Holomorphic functions of several variables: an introduction to the fundamental theory"</a> by Ludger Kaup and Burchard Kaup (section 8 of chapter 0).</p>

http://mathoverflow.net/questions/47302/is-there-an-example-of-gibbs-measure-that-is-not-a-weak-limit-of-finite-volume-gi/47894#47894 Yvan Velenik 2010-12-01T12:08:13Z 2010-12-01T12:08:13Z

<p>I don't think so. Just consider Dobrushin boundary conditions (positive spins at vertices with nonnegative second coordinate, negative elsewhere), and a box of the form $$ \Lambda_n=\{-n,\ldots,n\}\times\{-n-[a\sqrt{n}],\ldots,n-[a\sqrt{n}]\}. $$ Then the mixture you'll get in the limit will have $\lambda$ equal to the probability that the open contour passes below $0$, which should go continuously from $1$ to $0$ as $a$ goes from $-\infty$ to $+\infty$ (it is known that the interface converges weakly to a Brownian bridge under diffusive scaling).</p> <p>Note that this is very much a two-dimensional phenomenon. In 3d, at low enough temperature, I strongly doubt that you can find boundary conditions such that the limiting state is a nontrivial mixture of, say, Dobrushin states. (Of course, it is always true that you can reach <em>extremal</em> states in this way.)</p>

http://mathoverflow.net/questions/44417/on-generalisation-of-aizenman-higuchi-theorem/44599#44599 Yvan Velenik 2010-11-02T18:26:00Z 2010-11-02T18:26:00Z

<p>Concerning questions 1 and 2, if I understand it correctly (i.e., you're simply looking at the finite-range ferromagnetic Ising model with free, resp + or -, b.c.), then the sequences actually converge and the limits are translation invariant. This follows from monotonicity in the volume (GKS for free, GKS of FKG for + or -).</p> <p>(In all these cases, just fix a local function and take two different increasing sequences of boxes. Show using correlation inequalities that the expectation in both cases necessarily converge to the same limit by sandwiching those in one sequence by those in the other. You can find a version of this classical argument in my lecture notes (in french), see my homepage.)</p> <p>Concerning 3, the answer is certainly yes, but no proof is known (both the original proofs and the improved one we devised recently with Loren Coquille rely in an essential way on the n.n. nature of the interaction). (There might be some results at very large beta, of course.)</p>

http://mathoverflow.net/questions/37151/what-are-the-big-problems-in-probability-theory/37224#37224 Yvan Velenik 2010-08-31T06:48:42Z 2010-08-31T06:48:42Z

<p>You can also have a look at the list of open problems on Michael Aizenman's homepage:</p> <p><a href="http://www.math.princeton.edu/~aizenman/OpenProblems.iamp/" rel="nofollow">http://www.math.princeton.edu/~aizenman/OpenProblems.iamp/</a></p> <p>These are very important for (mathematical) physics, and several fall in the realm of probability theory (in particular: Soft phases in 2D O(N) models, and Spin glass).</p>

http://mathoverflow.net/questions/30886/applications-of-classical-field-theory/30892#30892 Yvan Velenik 2010-07-07T13:58:56Z 2010-07-07T13:58:56Z

<p><a href="http://en.wikipedia.org/wiki/Statistical_field_theory" rel="nofollow">Statistical field theory</a>? This plays an essential role in the statistical mechanical analysis of continuous phase transitions. See, e.g., the books by Itzykson and Drouffe.</p> <p>Continuum mechanics or fluid mechanics would also apply.</p>

http://mathoverflow.net/questions/27549/for-which-values-of-n-is-known-the-lieb-simon-inequality-for-z-n-models/27561#27561 Yvan Velenik 2010-06-09T08:21:48Z 2010-06-09T08:21:48Z

<p>As far as I know, it has only been proved for $N=1$ (Lieb) and $N=2$ (Lieb+Rivasseau) in the form you want. With an additional prefactor $\beta/N$ and with the infinite-volume measure in the RHS, it has been extended to $N=3$ and $N=4$ by Aizenman and Simon (see also Spohn and Zwerger). All these proofs rely on various correlation inequalities that, to my knowledge, have not been extended to general $O(N)$ models.</p>

http://mathoverflow.net/questions/23583/self-avoidance-time-of-random-walk/23597#23597 Yvan Velenik 2010-05-05T15:21:22Z 2010-05-06T12:55:03Z

<p>[I've corrected a stupid mistake below and added an upper bound... Please check the numerical values!]</p> <p>Well, I doubt that an explicit expression exists. However, it should be possible to get good bounds. The lower bound is easy: observe that $$ E(T) = \sum_{k\geq 1} P(T> k-1) = 1+\sum_{k\geq 1} c_k 4^{-k}, $$ where $c_k$ is the number of self-avoiding paths of length $k$ (and, of course, $4^k$ is the number of all paths of length $k$), so that we get a lower bound by truncating this series. Using the (known) values for $c_k$, $k=1,\ldots,71$, we get $$ E(T) > 4.58607909 $$ Now, you can get an upper bound by bounding the neglected part of the series using $c_k\leq 4 \cdot 3^{k-1}$. This gives you a very narrow interval containing the right value: if I have made no mistake ;) , we get $$ 4.58607909 &lt; E(T) &lt; 4.58607911 $$</p>

http://mathoverflow.net/questions/18268/discrete-stochastic-process-exponentially-correlated-bernoulli/23441#23441 Yvan Velenik 2010-05-04T15:13:24Z 2010-05-04T15:13:24Z

<p>The above solution is very nice, but relies on the very special structure of the desired process. In a much more general framework, I think that one could use a perfect simulation algorithm as described in:</p> <p><em>Processes with long memory: Regenerative construction and perfect simulation</em>, Francis Comets, Roberto Fernández, and Pablo A. Ferrari, Ann. Appl. Probab. <strong>12</strong>, Number 3 (2002), 921-943.</p>

http://mathoverflow.net/questions/22820/percolation-model-and-complex-probabilities/22956#22956 Yvan Velenik 2010-04-29T09:20:30Z 2010-04-29T09:20:30Z

<p>Not an answer to your original question, but more a reaction to the previous answer. It does make a lot of sense to analyse the behaviour of various quantities as function of complex-valued physical parameters (here p). For example, analysis of a statistical mechanical system (say, an Ising model) as a function of a complex magnetic field or complex temperature provides many important information about the system. To cite some: the Lee-Yang theorem (about possible locations of singularities, and thus possible locations of phase transitions) or Isakov's theorem (existence of an essential singularity at 0 of the free energy of the Ising model as a function of a complex magnetic field, thus showing that stable phases cannot be analytically continued into the metastable phases, contrarily to what mean-field theory suggests). Actually, there is a old version of Isakov theorem's for percolation (by Kunz and Souillard).</p>

http://mathoverflow.net/questions/108718/expected-value-of-max-x1-xn-in-continuous-distribution-on-0 Yvan Velenik 2012-10-03T18:28:48Z 2012-10-03T18:28:48Z

You're at the wrong place: this is not research level. You should rather try <a href="http://stats.stackexchange.com/" rel="nofollow">stats.stackexchange.com</a> .

http://mathoverflow.net/questions/108399/probability-and-events/108413#108413 Yvan Velenik 2012-09-30T13:05:32Z 2012-09-30T13:05:32Z

@Ant, I'd say that it was closed because it's quite far from research level...

http://mathoverflow.net/questions/105584/mathematics-and-marijuana Yvan Velenik 2012-08-27T08:11:12Z 2012-08-27T08:11:12Z

I am puzzled at how this is more off-topic than many other soft questions on this site...

http://mathoverflow.net/questions/103284/recent-impressive-combinatorial-developments-in-probability-theory/103512#103512 Yvan Velenik 2012-07-31T14:38:42Z 2012-07-31T14:38:42Z

@an12, look for example at these nice lecture notes: arXiv:1109.1549

http://mathoverflow.net/questions/99987/where-to-publish-a-math-textbook-in-creative-commons/100002#100002 Yvan Velenik 2012-06-21T20:29:21Z 2012-06-21T20:29:21Z

Similarly, Flajolet and Sedgewick's very nice &quot;Analytic combinatorics&quot; was published by Cambridge University Press, while being also made freely available [here](<a href="http://algo.inria.fr/flajolet/Publications/book.pdf" rel="nofollow">algo.inria.fr/flajolet/Publications/book.pdf</a>).

http://mathoverflow.net/questions/98181/coupon-collector-problem Yvan Velenik 2012-05-28T11:08:06Z 2012-05-28T11:08:06Z

There's a rather extensive literature on all kinds of extensions of the original coupon collector problem. Concerning your specific question, you might have a look at <a href="http://arxiv.org/abs/math/0304229" rel="nofollow">arxiv.org/abs/math/0304229</a>, for example.

http://mathoverflow.net/questions/91244/renewal-theorems-for-return-times Yvan Velenik 2012-03-15T15:35:26Z 2012-03-15T15:35:26Z

Well, I don't know where you took the approximation for the convolution from, but it certainly does not apply uniformly over all $k\leq n$. Indeed, the expected number of returns to zero by time n will be $\sum_{k=1}^n u_k \approx n^\alpha$, and the probability of having much more than that will decay fast. So, in your last computation, some cut-off at scale $n^\alpha$ must take place, which yields $\sum_{k=1}^{n^\alpha} k f_n \approx n^{\alpha-1}$, which is consistent. You should look at the conditions underlying the derivation of your second property (I guess: fix $k$, let $n\to\infty$).

http://mathoverflow.net/questions/83092/what-is-the-main-goal-of-a-paper-really Yvan Velenik 2011-12-10T11:09:20Z 2011-12-10T11:09:20Z

It also somewhat depends on the field, I guess. As a mathematical physicist, having a correct proof of a statement is far from enough for me (the fact that the statement holds having already been established with reasonable certainty by theoretical physicists). What I very much care about is <i>really</i> understanding <i>why</i> the statement is true, and that's the main purpose of a proof for me. So, a paper which can only be checked line by line, without ever leading to a full picture is useless, as far as I am concerned (as are proofs relying on long, unenlightening computations).

http://mathoverflow.net/questions/78011/multidimensional-berry-esseen-for-probability-density-functions Yvan Velenik 2011-10-13T16:14:47Z 2011-10-13T16:14:47Z

I don't know of a precise reference, but mathscinet returns a large number of papers on this topic (many in russian, though). I don't have access to these journals, so I can't check, but looking at the first page of <a href="http://springerlink.com/content/v7071t24g455837h" rel="nofollow">springerlink.com/content/v7071t24g455837h</a>, it might contain something relevant...

http://mathoverflow.net/questions/77973/normal-distribution-with-positive-semi-definite-covariance-matrix Yvan Velenik 2011-10-13T07:13:03Z 2011-10-13T07:13:03Z

Do people <i>really</i> believe that this is a research-level question???

http://mathoverflow.net/questions/75829/berry-esseen-type-result-for-probability-density-functions Yvan Velenik 2011-09-19T09:24:37Z 2011-09-19T09:24:37Z

Hum. If you search for &quot;Sums of independent random variables&quot; Petrov on google.com, you should find what you need on the first page of results ;) ...

http://mathoverflow.net/questions/75829/berry-esseen-type-result-for-probability-density-functions Yvan Velenik 2011-09-19T09:02:03Z 2011-09-19T09:02:03Z

Sure. Look, e.g., at the supplements to chapter VII in the classical book &quot;Sums of independent random variables&quot; by Petrov.

http://mathoverflow.net/questions/73647/do-the-converses-of-weak-law-of-large-numbers-central-limit-theorem-hold Yvan Velenik 2011-08-26T06:48:01Z 2011-08-26T06:48:01Z

@pgassiat: Yes, thank you. I hadn't realized that there was a newer edition ;) .

http://mathoverflow.net/questions/73647/do-the-converses-of-weak-law-of-large-numbers-central-limit-theorem-hold Yvan Velenik 2011-08-25T14:09:20Z 2011-08-25T14:09:20Z

Necessary and sufficient conditions (in terms close to those you want) for the WLLN and the CLT can be found, e.g., in &quot;Foundations of modern probability&quot; by Kallenberg (Theorems 4.16 and 4.17).

http://mathoverflow.net/questions/71473/how-far-can-a-particle-travel-from-its-origin-if-it-exhibits-self-avoiding-browni Yvan Velenik 2011-08-10T18:24:10Z 2011-08-10T18:24:10Z

Not quite the process you're looking for, but you might be interested in googling for (or checking on mathscinet for) the &quot;myopic&quot; or &quot;true&quot; self-avoiding walk.