User mike - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-24T04:35:26Z http://mathoverflow.net/feeds/user/22650 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/118562/tails-of-sums-of-weibull-random-variables/119289#119289 Answer by mike for Tails of sums of Weibull random variables mike 2013-01-18T20:45:24Z 2013-01-18T20:51:37Z <ol> <li>$f(x) = cosh(x^{\alpha})$ is convex for $\alpha > \frac 12$ so 2. $\mathbb E f(\frac {S_n} n) \le \mathbb E f(X)$ which, with chebyshev, implies your claim for any $\frac 12 \le \alpha \le \epsilon$. 3. For general case, replace $cosh$ with $\sum \frac {x^{nk}}{(nk)!}$ where K satisfies $\alpha k > 1$. You have to know that this thing is about $e^x$, which can be gotten from the fact that it is $\sum_j e^{xe^{(2 \pi i j)/k}}/k$</li> </ol> http://mathoverflow.net/questions/102047/arbitrage-free-price-of-a-derivative-when-the-price-is-collected-over-the-lifetim/102081#102081 Answer by mike for Arbitrage free price of a derivative when the price is collected over the lifetime of the derivative mike 2012-07-12T23:54:55Z 2012-07-12T23:54:55Z <p>good <em>MATH</em> question ! so closely related to what people in the <em>MATH</em> departments at Columbia , Chicago , Rutgers , Carnegie Mellon etc etc are doing. Can't help you with the answer to this <em>MATH</em> question though, but I wonder , are you paying in shares , receiving $X_T$ shares ? or what is that factor of $B_T$ doing in your <em>MATH</em> equations ?</p> http://mathoverflow.net/questions/98300/estimating-joint-and-conditional-probabilities-with-incomplete-information/98311#98311 Answer by mike for Estimating joint and conditional probabilities with incomplete information mike 2012-05-29T21:20:53Z 2012-05-29T21:20:53Z <p>No, and to see that there is not, you might produce a collection of rademacher ($\pm 1$) random variables that are pairwise independent but not independent. No statement about 2 at a time, etc., would be different that for independent rademacher , but statements about the 3 way etc would be. You can do this, and generalise it to an N-K setting , by taking $X_1,X_2,X_3$ i.i.d. and $= \pm 1$ with prob $\frac 12$, and $Z_1 = X_2X_3, Z_2 = X_1X_3, Z_3 = X_1X_2$ .</p> http://mathoverflow.net/questions/97802/span-of-symmetrically-truncated-symmetric-random-variables span of symmetrically truncated symmetric random variables mike 2012-05-24T00:03:34Z 2012-05-24T20:57:28Z <p>If $X_i$ are symmetric independent random variables, is $\vert \sum X_i I_{\vert X_i \vert &lt; N_i}\vert$ stochastically smaller than $\vert \sum X_i \vert$ ? Is it comparable in any way which would guarantee the former was stochastically bounded if the latter was ? The motivating example is i.i.d. cauchy, when the convex hull of the r.v.s consists entirely of Cauchys, but can the convex hull of the truncated guys contain stochastically unbounded guys ? (The $X_i$ are stochastically bounded if there is $S(x), S(x) \rightarrow 0$ as $x \rightarrow \infty$ with $\mathbb P(\vert X_i \vert > x ) &lt; S(x) \; \forall i$)</p> http://mathoverflow.net/questions/96007/gaussian-copula-and-the-addition-of-an-identity-matrix/96009#96009 Answer by mike for Gaussian Copula and the addition of an Identity matrix mike 2012-05-04T18:55:43Z 2012-05-04T18:55:43Z <p>It's the jacobian.</p> http://mathoverflow.net/questions/94516/limit-of-a-rescaled-random-sum-of-i-i-d-random-variables/94538#94538 Answer by mike for Limit of a rescaled random sum of i.i.d. random variables mike 2012-04-19T15:01:01Z 2012-04-19T15:01:01Z <p>Do you want $E(N(\alpha))$ in the denominator ?</p> http://mathoverflow.net/questions/94226/a-random-walk-with-uniformly-distributed-steps/94238#94238 Answer by mike for A random walk with uniformly distributed steps mike 2012-04-16T18:24:35Z 2012-04-16T18:24:35Z <p>Me too, or the first hitting time for $(-\infty,0)$, and I think what he says is right &amp; the distribution is the same for all symmetric continuous r.v.s. I'd look it up if I were home, the ideas are circle of wiener hopf factorization &amp; I think are probably in Feller vol 2., the renewal theory chapter.</p> http://mathoverflow.net/questions/90766/exchangeable-normal-distribution-mixing-measure/93155#93155 Answer by mike for Exchangeable normal distribution mixing measure mike 2012-04-04T18:40:43Z 2012-04-04T18:40:43Z <p>The mixing distribution is the same as the long term average . Since your gaussians can be repesented as $Z + X_i$ where $X_i$ are i.i.d., and also independent of $Z$, the long term average for the events I think you are looking at is $\frac 1 n \sum^n 1_{X_i + Z > c}$. By conditioning on $Z$ this is seen to have distribution $\Phi(c - Z)$ and that should be your mixer.</p> http://mathoverflow.net/questions/57407/testing-for-a-change-in-mean-in-a-time-series/93152#93152 Answer by mike for Testing for a change in mean in a time series mike 2012-04-04T18:15:02Z 2012-04-04T18:15:02Z <p>The situation you describe, a change in seat belt laws in UK, is discussed in Brockwell &amp; Davis, Introduction to Time Series..., example 6.63</p> http://mathoverflow.net/questions/78925/brownian-particle-with-jump-boundary-condition/93115#93115 Answer by mike for Brownian particle with jump boundary condition mike 2012-04-04T13:25:41Z 2012-04-04T13:25:41Z <p>the transition for the jump process started from x satisfies a renewal equation where the lifetime distribution is the hitting time for the boundary. You can write a formal solution in the usual manner of solving renewal equations. I have seen this as problem somewhere, but a quick search of Karlin &amp; Taylor did not turn it up.</p> http://mathoverflow.net/questions/129999/probability-distribution-for-two-state-system-that-depends-on-residence-time Comment by mike mike 2013-05-07T20:56:40Z 2013-05-07T20:56:40Z for example, we are in state 2 at time 5, the process last entered state 2 at time 3.7, $X \sim p(x,1.3)$ ? http://mathoverflow.net/questions/110573/covariance-of-inid-order-statistics/110603#110603 Comment by mike mike 2013-03-14T17:28:50Z 2013-03-14T17:28:50Z Try the FKG inequality, I think product measures are fine even if not i.i.d. and (I think) the order stats are increasing fctns of the data. http://mathoverflow.net/questions/123818/almost-surely-finite Comment by mike mike 2013-03-08T16:35:04Z 2013-03-08T16:35:04Z you know you have a wright-fisher model? I have seen boundary classification for this model, so I think the standard method, in terms of the scale function, must be amenable. However, since you have positive drift at 0, 0 cannot cause you any problem and I am sure that what you say is true. http://mathoverflow.net/questions/121495/do-there-exist-almost-surely-c-infty-smooth-gaussian-random-fields Comment by mike mike 2013-02-11T21:34:36Z 2013-02-11T21:34:36Z do you have anything agains taking you ananlytic field and multiplying it by a deterministic $C^{\infty}$ non-analytic function ? http://mathoverflow.net/questions/112128/integral-inequality Comment by mike mike 2012-11-14T13:41:53Z 2012-11-14T13:41:53Z you may be looking for something like <a href="http://en.wikipedia.org/wiki/FKG_inequality" rel="nofollow">en.wikipedia.org/wiki/FKG_inequality</a> which 1) would want you to restrict to increasing sets A,B and 2) would show that the inequality can go either way http://mathoverflow.net/questions/103755/is-this-process-strictly-positive Comment by mike mike 2012-08-02T16:00:23Z 2012-08-02T16:00:23Z should have read $\sigma(x_i) &gt; 0$, http://mathoverflow.net/questions/103755/is-this-process-strictly-positive Comment by mike mike 2012-08-02T15:59:04Z 2012-08-02T15:59:04Z Suppose you divide it into 2 cases, one where $\sigma = 0$ in an interval near $0$, and the other where $\exists x_i \rightarrow 0$ with $\sigma(x_i) = 0$. If $\ell_i = sup \lbrace x &lt; x_i : \sigma(x) = 0$ then it seems to me that you can show that a process started from $x_i$ never hits $\ell_i$. Then, if the process is positive with probability 1 at time t it would postive with probability 1 for all t. http://mathoverflow.net/questions/103755/is-this-process-strictly-positive Comment by mike mike 2012-08-02T11:46:56Z 2012-08-02T11:46:56Z why don't you use the scale function ? http://mathoverflow.net/questions/103284/recent-impressive-combinatorial-developments-in-probability-theory/103513#103513 Comment by mike mike 2012-07-30T15:37:16Z 2012-07-30T15:37:16Z I had lunch with Stoock when he spoke at my instituion and asked him what he thought the areas of tomorrow in probability were. I had expected him to answer 'random matrices. sle', the usual suspects. What he said was that when he sees this guy yuval peres' work he always thinks it is interesting. Someone asked him the random matrix/sle question. he said he felt like progress in these areas would depend on importing analytical techniques into probability, whereas new probabilistic techniques would be driven by etc. http://mathoverflow.net/questions/102590/does-a-certain-theorem-on-boltzmann-distributions-exist Comment by mike mike 2012-07-23T21:08:43Z 2012-07-23T21:08:43Z the characteristic function of $\frac {X_n - \mathbb E(X_n)}{\sqrt{a_n}}$ is easily expressed in terms of $F_n$ and it seems like you have enough hypotheses to guarantee it converges to $e^{const. \theta^2}$ http://mathoverflow.net/questions/98724/continuity-of-hitting-distributions Comment by mike mike 2012-06-03T20:27:47Z 2012-06-03T20:27:47Z The way I understand your question as you move along the absorbing boundary from $A$ to $A^c \cap U_{\alpha}$ the function changes (abruptly) from $0$ to 1. http://mathoverflow.net/questions/98405/fourier-decay-rate-of-cantor-measures Comment by mike mike 2012-05-31T00:10:38Z 2012-05-31T00:10:38Z there's a huge literature (with which I'm not faniliar), but you must know erdos, On a family of symmetric Bernoulli convolutions (1939), and the papers that refer to it. http://mathoverflow.net/questions/98228/is-it-known-that-every-pdf-continuous-in-all-rn-has-a-maximum Comment by mike mike 2012-05-28T23:40:34Z 2012-05-28T23:40:34Z I think that if $f$ is a continuous density with compact support $\sum p_i \lambda_i f(\frac x {\lambda_i} + v_i$ is a continuous and has no global maximum, choosing $\sum p_i = 1, \lambda_i \rightarrow \infty, \v_i \rightarrow \infty$, http://mathoverflow.net/questions/98230/distribution-built-up-from-powers-of-a-log-normal-r-v Comment by mike mike 2012-05-28T23:35:21Z 2012-05-28T23:35:21Z you can do that integral somewhat explicitly by completing the square, and express is in terms of $\Phi$, the normal distribution function http://mathoverflow.net/questions/97802/span-of-symmetrically-truncated-symmetric-random-variables Comment by mike mike 2012-05-25T21:06:11Z 2012-05-25T21:06:11Z I think second order stochastic dominance might be what I'm looking for.