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In the previous set of notes we established the central limit theorem, which we formulate here as follows:

Theorem 1 (Central limit theorem) Let ${X_1,X_2,X_3,\dots}$ be iid copies of a real random variable ${X}$ of mean ${\mu}$ and variance ${0 < \sigma^2 < \infty}$, and write ${S_n := X_1 + \dots + X_n}$. Then, for any fixed ${a < b}$, we have

$\displaystyle {\bf P}( a \leq \frac{S_n - n \mu}{\sqrt{n} \sigma} \leq b ) \rightarrow \frac{1}{\sqrt{2\pi}} \int_a^b e^{-t^2/2}\ dt \ \ \ \ \ (1)$

as ${n \rightarrow \infty}$.

This is however not the end of the matter; there are many variants, refinements, and generalisations of the central limit theorem, and the purpose of this set of notes is to present a small sample of these variants.

First of all, the above theorem does not quantify the rate of convergence in (1). We have already addressed this issue to some extent with the Berry-Esséen theorem, which roughly speaking gives a convergence rate of ${O(1/\sqrt{n})}$ uniformly in ${a,b}$ if we assume that ${X}$ has finite third moment. However there are still some quantitative versions of (1) which are not addressed by the Berry-Esséen theorem. For instance one may be interested in bounding the large deviation probabilities

$\displaystyle {\bf P}( |\frac{S_n - n \mu}{\sqrt{n} \sigma}| \geq \lambda ) \ \ \ \ \ (2)$

in the setting where ${\lambda}$ grows with ${n}$. Chebyshev’s inequality gives an upper bound of ${1/\lambda^2}$ for this quantity, but one can often do much better than this in practice. For instance, the central limit theorem (1) suggests that this probability should be bounded by something like ${O( e^{-\lambda^2/2})}$; however, this theorem only kicks in when ${n}$ is very large compared with ${\lambda}$. For instance, if one uses the Berry-Esséen theorem, one would need ${n}$ as large as ${e^{\lambda^2}}$ or so to reach the desired bound of ${O( e^{-\lambda^2/2})}$, even under the assumption of finite third moment. Basically, the issue is that convergence-in-distribution results, such as the central limit theorem, only really control the typical behaviour of statistics in ${\frac{S_n-n \mu}{\sqrt{n} \sigma}}$; they are much less effective at controlling the very rare outlier events in which the statistic strays far from its typical behaviour. Fortunately, there are large deviation inequalities (or concentration of measure inequalities) that do provide exponential type bounds for quantities such as (2), which are valid for both small and large values of ${n}$. A basic example of this is the Chernoff bound that made an appearance in Exercise 47 of Notes 4; here we give some further basic inequalities of this type, including versions of the Bennett and Hoeffding inequalities.

In the other direction, we can also look at the fine scale behaviour of the sums ${S_n}$ by trying to control probabilities such as

$\displaystyle {\bf P}( a \leq S_n \leq a+h ) \ \ \ \ \ (3)$

where ${h}$ is now bounded (but ${a}$ can grow with ${n}$). The central limit theorem predicts that this quantity should be roughly ${\frac{h}{\sqrt{2\pi n} \sigma} e^{-(a-n\mu)^2 / 2n \sigma^2}}$, but even if one is able to invoke the Berry-Esséen theorem, one cannot quite see this main term because it is dominated by the error term ${O(1/n^{1/2})}$ in Berry-Esséen. There is good reason for this: if for instance ${X}$ takes integer values, then ${S_n}$ also takes integer values, and ${{\bf P}( a \leq S_n \leq a+h )}$ can vanish when ${h}$ is less than ${1}$ and ${a}$ is slightly larger than an integer. However, this turns out to essentially be the only obstruction; if ${X}$ does not lie in a lattice such as ${{\bf Z}}$, then we can establish a local limit theorem controlling (3), and when ${X}$ does take values in a lattice like ${{\bf Z}}$, there is a discrete local limit theorem that controls probabilities such as ${{\bf P}(S_n = m)}$. Both of these limit theorems will be proven by the Fourier-analytic method used in the previous set of notes.

We also discuss other limit theorems in which the limiting distribution is something other than the normal distribution. Perhaps the most common example of these theorems is the Poisson limit theorems, in which one sums a large number of indicator variables (or approximate indicator variables), each of which is rarely non-zero, but which collectively add up to a random variable of medium-sized mean. In this case, it turns out that the limiting distribution should be a Poisson random variable; this again is an easy application of the Fourier method. Finally, we briefly discuss limit theorems for other stable laws than the normal distribution, which are suitable for summing random variables of infinite variance, such as the Cauchy distribution.

Finally, we mention a very important class of generalisations to the CLT (and to the variants of the CLT discussed in this post), in which the hypothesis of joint independence between the variables ${X_1,\dots,X_n}$ is relaxed, for instance one could assume only that the ${X_1,\dots,X_n}$ form a martingale. Many (though not all) of the proofs of the CLT extend to these more general settings, and this turns out to be important for many applications in which one does not expect joint independence. However, we will not discuss these generalisations in this course, as they are better suited for subsequent courses in this series when the theory of martingales, conditional expectation, and related tools are developed.

[This guest post is authored by Matilde Lalin, an Associate Professor in the Département de mathématiques et de statistique at the Université de Montréal.  I have lightly edited the text, mostly by adding some HTML formatting. -T.]

Mathematicians (and likely other academics!) with small children face some unique challenges when traveling to conferences and workshops. The goal of this post is to reflect on these, and to start a constructive discussion what institutions and event organizers could do to improve the experiences of such participants.

The first necessary step is to recognize that different families have different needs. While it is hard to completely address everybody’s needs, there are some general measures that have a good chance to help most of the people traveling with young children. In this post, I will mostly focus on nursing mothers with infants ($\leq 24$ months old) because that is my personal experience. Many of the suggestions will apply to other cases such as non-nursing babies, children of single parents, children of couples of mathematicians who are interested in attending the same conference, etc..

The mother of a nursing infant that wishes to attend a conference has three options:

1. Bring the infant and a relative/friend to help caring for the infant. The main challenge in this case is to fund the trip expenses of the relative. This involves trip costs, lodging, and food. The family may need a hotel room with some special amenities such as crib, fridge, microwave, etc. Location is also important, with easy access to facilities such as a grocery store, pharmacy, etc. The mother will need to take regular breaks from the conference in order to nurse the baby (this could be as often as every three hours or so). Depending on personal preferences, she may need to nurse privately. It is convenient, thus, to make a private room available, located as close to the conference venue as possible. The relative may need to have a place to stay with the baby near the conference such as a playground or a room with toys, particularly if the hotel room is far.
2. Bring the infant and hire someone local (a nanny) to help caring for the infant. The main challenges in this case are two: finding the caregiver and paying for such services. Finding a caregiver in a place where one does not live is hard, as it is difficult to conduct interviews or get references. There are agencies that can do this for a (quite expensive) fee: they will find a professional caregiver with background checks, CPR certification, many references, etc. It may be worth it, though, as professional caregivers tend to provide high-quality services and peace of mind is priceless for the mother mathematician attending a conference. As in the previous case, the mother may have particular needs regarding the hotel room, location, and all the other facilities mentioned for Option 1.
3. Travel without the infant and pump milk regularly. This can be very challenging for the mother, the baby, and the person that stays behind taking care of the baby, but the costs of this arrangement are much lower than in Option 1 or 2 (I am ignoring the possibility that the family needs to hire help at home, which is necessary in some cases). A nursing mother away from her baby has no option but to pump her milk to prevent her from pain and serious health complications. This mother may have to pump milk very often. Pumping is less efficient than nursing, so she will be gone for longer in each break or she will have more breaks compared to a mother that travels with her baby. For pumping, people need a room which should ideally be private, with a sink, and located as close to the conference venue as possible. It is often impossible for these three conditions to be met at the same time, so different mothers give priority to different features. Some people pump milk in washrooms, to have easy access to water. Other people might prefer to pump in a more comfortable setting, such as an office, and go to the washroom to wash the breast pump accessories after. If the mother expects that the baby will drink breastmilk while she is away, then she will also have to pump milk in advance of her trip. This requires some careful planning.

Many pumping mothers try to store the pumped milk and bring it back home. In this case the mother needs a hotel room with a fridge which (ideally, but hard to find) has a freezer. In a perfect world there would also be a fridge in the place where she pumps/where the conference is held.

It is important to keep in mind that each option has its own set of challenges (even when expenses and facilities are all covered) and that different families may be restricted in their choice of options for a variety of reasons. It is therefore important that all these three options be facilitated.

As for the effect these choices have on the conference experience for the mother, Option 1 means that she has to balance her time between the conference and spending time with her relative/friend. This pressure disappears when we consider Option 2, so this option may lead to more participation in the conferences activities. In Option 3, the mother is in principle free to participate in all the conference activities, but the frequent breaks may limit the type of activity. A mother may choose different options depending on the nature of the conference.

I want to stress, for the three options, that having to make choices about what to miss in the conference is very hard. While talks are important, so are the opportunities to meet people and discuss mathematics that happen during breaks and social events. It is very difficult to balance all of this. This is particularly difficult for the pumping mother in Option 3: because she travels without her baby, she is not perceived to be a in special situation or in need of accommodation. However, this mother is probably choosing between going to the last lecture in the morning or having lunch alone, because if she goes to pump right after the last lecture, by the time she is back, everybody has left for lunch.

Here is the Hall of Fame for those organizations that are already supporting nursing mothers’ travels in mathematics:

• The Natural Sciences and Engineering Research Council of Canada (NSERC) (search for “child care”) allows to reimburse the costs of child care with Option 2 out of the mother’s grants. They will also reimburse the travel expenses of a relative with Option 1 up to the amount that would cost to hire a local caregiver.
• The ENFANT/ELEFANT conference (co-organized by Lillian Pierce and Damaris Schindler) provided a good model to follow regarding accommodation for parents with children during conferences that included funding for covering the travel costs of accompanying caretakers (the funding was provided by the Deutsche Forschungsgemeinschaft, and lactation rooms and play rooms near the conference venue (the facilities were provided by the Hausdorff Center for Mathematics).

Additional information (where to go with kids, etc) was provided on site by the organizers and was made available to all participants all the time, by means of a display board that was left standing during the whole week of the conference.
• The American Institute of Mathematics (AIM) reimburses up to 500 dollars on childcare for visitors and they have some online resources that assist in finding childcare and nannies.

[UPDATED] Added a few more things to the Hall of Fame

In closing, here is a (possibly incomplete) list of resources that institutes, funding agencies, and conferences could consider providing for nursing mother mathematicians:

1. Funding (for cost associated to child care either professional or by an accompanying relative).
2. List of childcare resources (nannies, nanny agencies, drop-in childcare centre, etc).
3. Nursing rooms and playrooms near the conference venue. Nearby fridge.
4. Breaks of at least 20 minutes every 2-3 hours.
5. Information about transportation with infants. More specific, taxi and/or shuttle companies that provide infant car seats. Information regarding the law on infant seats in taxis and other public transportation.
6. Accessibility for strollers.
7. [UPDATED] A nearby playground location. (comment from Peter).

I also find it important that these resources be listed publicly in the institute/conference website. This serves a double purpose: first, it helps those in need of the resources to access them easily, and second, it contributes to make these accommodations normal, setting a good model for future events, and inspiring organizers of future events.

Finally, I am pretty sure that the options and solutions I described do not cover all cases. I would like to finish this note by inviting readers to make suggestions, share experiences, and/or pose questions about this topic.