We have observed that the cost of the search is equal to the number of tosses of a coin of bias p that are necessary until we obtain H successes. That is, we flip the coin repeatedly and stop as soon as we observe H successes. The difficulty here is that the random variable we are studying is the sum of geometrically distributed random variables. The distribution of this random variable is call...

We consider Markovian models on graphs with local dynamics. We show that, under suitable conditions, such Markov chains exhibit both rapid convergence to equilibrium and strong concentration of measure in the stationary distribution. We illustrate our results with applications to some known chains from computer science and statistical mechanics.

Suppose there is a bag containing red balls and blue balls. You would like to estimate the fraction of red balls in the bag. However, you are only allowed to sample randomly from the bag with replacement. The straightforward method is to make T random samples and use the fraction of red balls in the random samples as an estimate of the true fraction. The question is: how many samples are enough...

where Y ∈ Rp×n is a rectangular p×n matrix with random centered entries, and both n and p ≤ n tend to infinity: typically p = p(n), and p(n)/n tends to some limit. M can be seen as the empirical covariance matrix of a random vector of dimension p sampled n times, each sample being a column of Y . It is common in applications to have a number of variables with a comparable order of magnitude wit...

Concentration inequalities have been the subject of exciting developments during the last two decades, and they have been intensively studied and used as a powerful tool in various areas. These include convex geometry, functional analysis, statistical physics, statistics, pure and applied probability theory (e.g., concentration of measure phenomena in random graphs, random matrices and percolat...

We introduce here an index, which we call the Rare Event Concentration Coefficient (RECC), that is a measure of the dispersion/concentration of events which have a low frequency but tend to have a high level of concentration, such as the number of crimes suffered by a person. The Rare Event Concentration Coefficient is a metric based on a statistical mixture model, with a value closer to zero m...

2 Agnostic Learning We will proceed without making any assumptions on the distribution PXY . This situation is often termed as Agnostic Learning. The root of the word agnostic literally means not known. The term agnostic learning is used to emphasize the fact that often, perhaps usually, we may have no prior knowledge about PXY and f∗. The question then arises about how we can reasonably select...