Course : Mathematical Economics

We have shown that pointwise convergence is strictly weaker than uniform, and we have provided a theorem that complements pointwise convergence with a joint Lipschitz continuity property for the elements of the sequence, and total boundedness of the common domain in order to obtain uniform convergence.

We have examined the usefulness of the strictness of uniform convergence in a general application that concerns the issue approximation of optimization problems (variational approximation). We h

We formulated an example of application of the ULLN using an iid standard uniform process along with a function space defined on the relevant support that we have already established to be totally bounded wrt the uniform metric. We claimed that the above could be useful in facilitating the convergence in probability of a stochastic minimizer to a limiting deterministic analogue, thereby providing a mathematical structure relevant for the establishment of weak consistency of econometric estimat

Given our parametric example, we have proven that a function space endowed with the uniform metric, consisting of functions that are bijectively parameterized by the elements of a totally bounded metric space, and if the parameterization obeys a Lipschitz continuity property, is totally bounded. 

We have continued with our application of the notion of total boundedness in asymptotic analysis: the derivation of a Uniform Law of Large Numbers in a simple framework using a chaining argument that

We have initiated our examination of the notion of total boundness. We have given properties, e.g. that the centers of the covering balls can be chosen inside the totally bounded set, as well as (trivial) examples-counterexamples (the analytical complexity of the notion clearly manifested itself on that counter examples were easier to come up to) using among others the property of sequential disconnectedness.

Given the analytical complexity of the verification of total boundness, we will use th

Dear Students,

In the first tutorial, we stated the definitions of a metric and a pseudo-metric and we solved exercises 4.8 and 5 from Problem Set 1 (it can be found in the folder TA Sessions 2023-2024 -> TA Session 1). We also presented an example of a metric reflecting the structure of a graph. Specifically, if the graph is connected, the metric is well-defined as the minimum distance between any two vertices of the graph. The relevant notes are here. 

In the second tutorial we discussed t

We continued studying properties of metric induced balls. We have shown that these obey some monotonicity property. The examples examined including the ones of the real line endowed with the usual metric, the real line endowed with the "exponential metric", the real line endowed with the discrete one, showed that the “geometry” of the open (and/or the closed) balls crucially depends on the metric.

We have provided with the geometric realizations of open and closed balls in  endowed with an

Given relations we have established in our real vector spaces examples, we have defined the notions of dominance and equivalence between metrics definable on the same carrier set. Two such metrics can be considered equivalent-and we suspected that the notion is representing "similar properties" on the same carrier-without being necessarily equal as functions.

We examined the important-to the upcoming lectures-example of the space of bounded real functions on a non-empty domain endowed with th

We continued the investigation of examples involving metrics on sets of finite dimensional real vectors. (Some of) The examples have shown that it is possible that different metrics on the same carrier set can obey relations, e.g. in the form of functional inequalities. We suspected that such relations might imply analogous ones between the relevant properties that each metric endows the space with, and that provides as with a motivation of further examination of such relations.

Notes on the

After a presentation of the course's scope and aims, and using the overview of the familiar case of the real numbers, we begun with the definition of a distance function (metric) over a non empty set of reference as a real function defined on the product of this set with itself that satisfies positivity, separation, symmetry and triangle inequality. The example of the discrete metric showed that any such set bears at least one such function, and further examples implied that more than one metri