Proof:

Let and . Since are pairwise independent, the are uncorrelated and thus

Since

Now using the Chebyshev’s inequality

Hence

For a.s. convergence we have to take subsequences. Let

Then

So

and the Borel-Cantelli lemma implies

which means that

Now pick an s.th. and observe that if then

or

But since

we have that and hence the terms at the right and left ends both .

**References**:

A.N. Shiryaev (1996), *Probability, *Springer

W. Feller (1971), *An Introduction to Probability Theory and Its Applications,* Vol.2. John Wiley & Sons (NY)

R. Durrett (2010). *Probability: Theory and Examples, *Cambridge University Press

O. Kallenberg (2006)**, ***Foundations of modern probability, *Springer

Proof:

First we proove the following useful result

If and then

Note you can find the same lemma on Feller Vol.2 (p. 150) as

Using the result above

So

Now note that if

When , takes the initial value of , so

while when ,

Hence

]]>(1) It is

easy to obtain confirmations, or verifications, for nearly every theory-if we look for confirmations.(2) Confirmations

should count only if they are the result of risky predictions; that is to say, if, unenlightened by the theory in question, we should have expected an event which was incompatible with the theory–an event which would have refuted the theory.(3)

Every ‘good’ scientific theory is a prohibition: it forbids certain things to happen. The more a theory forbids, the better it is.(4)

A theory which is not refutable by any conceivable event is nonscientific. Irrefutability is not a virtue of a theory (as people often think) but a vice.(5)

Every genuine test of a theory is an attempt to falsify it, or to refute it. Testability is falsifiability; but there are degrees of testability: some theories are more testable, more exposed to refutation, than others; they take, as it were, greater risks.(6) Confirming evidence should not count except when it is the result of a

genuine testof the theory; and this means that it can be presented as a serious but unsuccessful attempt to falsify the theory. (I now speak in such cases of ‘corroborating evidence’.)(7) Some genuinely testable theories, when found to be false, are still upheld by their admirers–for example by introducing ad hoc some auxiliary assumption, or by re-interpreting theory ad hoc in such a way that it escapes refutation. Such a procedure is always possible, it rescues the theory from refutation only

at the price of destroying, or at least lowering, scientific status.

—————————-

Excerpt from a lecture given by Karl Popper at Peterhouse, Cambridge, in Summer 1953, as part of a course on Developments and trends in contemporary British philosophy.

]]>**Measurement Equation**

**Transition Equation**

**Future form**

*1.2. Kalman Filter*

Recursive procedure for computing the optimal estimator of the state vector at time t. When the model is Gaussian Kalman filter can be interpreted as updating the mean and covariance matrix of the conditional distribution of the state vector as new observations become available.

then

where

**Predictive distribution of **

**Updating equations**

and

**Contemporaneous filter:**

**Predictive filter: **

In the latter case

or

where the gain matrix is given by

and

**Initialization**

Start Kalman at with diffuse prior

**Prediction**

Taking conditional expectations in the measurement equation for

with MSE matrix

**MLE and prediction error decomposition**

**Prediction errors or innovations**

*Prediction error decomposition*

*Diagnostic tests* can be based on the standardized innovations

which are serially independent if is known

* is maximized w.r.t. numerically. Diffuse prior exact likelihood.

* and run independently on .

—————————————————————————————————————–

Further Reading:

Time Series: Theory and Methods (Springer Series in Statistics)

An Introduction to State Space Time Series Analysis (PRACTICAL ECONOMETRICS SERIES)

Forecasting, Structural Time Series Models and the Kalman Filter

]]>If is a random variable on . The expected value of is defined as

**Inequalities**

*Jensen’s inequality*. If is convex and

*Holder’s inequality*. If with then

*Cauchy-Schwarz Inequality*: For

Let with .

hence

*Markov’s/Chebyshev’s inequality*. Suppose has , let and let

where

Alternatively suppose and that is -measurable and non-decreasing (note ). Then

Example:

**Integration to the limit**

is uniformly integrable if s.th.

whenever and

**Monotone Convergence Theorem**: If then .

**Lemma**: A measurable function is integrable iff implies

**Fatou’s lemma**. If then

**Dominated Convergence Theorem**. If , , and , then

Note when is constant **Bounded Convergence Theorem**.

**Theorem**: Suppose and continous functions with i) and when large. ii) as , and iii) . Then

.

**Sums of non-negative Random Variables**: Collection of useful results

- If and then

.

- If , then (see Monotone Convergence Theorem)

- If s.th. , then

**First Borel-Cantelli Lemma**. Suppose sequence of events s.th. . Set . Then

and by previous result

Note .

**Computing Expected Values**

*Change of variables formula*. Let with measure , i.e. . If , so that or , then

———————————————–

source:

]]>So different aspects of Big Data — in this case dense vs. tall — are of different value for different things. Dense data promote accurate volatility estimation, and tall data promote accurate trend estimation.

]]>]]>In recent years, the use of randomised controlled trials has spread from labour market and welfare programme evaluation to other areas of economics (and to other social sciences), perhaps most prominently in development and health economics. This column argues that some of the popularity of such trials rests on misunderstandings about what they are capable of accomplishing, and cautions against simple extrapolations from trials to other contexts.

whenever

whenever

**Definition 2. Set functions and measures**. Let be a set and be an algebra on , and let be a non-negative set function

- is
**additive**if and, for , - The map is called
**countably additive**(or -additive) if and whenever is a sequence of disjoint sets in with union in , then - Let be a measurable space, so that is a -algebra on .
- A map is called a
**measure**on if is countable additive. The triple is called a**measure space**. -
The measure is called

**finite**ifand –

**finite**if, () s.th.

- Measure is called a
**probability measure**if and is then called a**probability triple**. - An element of is called -null if .
- A statement about points of is said to hold
**almost everywhere**(a.s.) if

**Definition 3.** A **probability space **is a triple where is the sample space, is the family of events (i.e. an event is an element of ) which we assume to be a -field and is a function that assigns probabilities to events.

**Definition 4.** A non-empty class of subsets of is a **monotone class** if for every monotone serquence .

**Proposition 1.** A -field is a monotone class while, conversely, a monotone algebra is a -field.

**Corollary 1**. Let be the power-set (the class of all subsets of ) and .Then for any -field of subsets of ,

.

**Definition 5.** A real valued function defined on is said to be a **random variable** if for every Borel set we have

A random variable is an element of . Thus

**Definition 6.** A function is said to be a **measurable map** from to if

**Theorem 1.** If and , then is measurable. A -field **generated by a class** of subsets (denoted ) is the smallest -field on that makes a measurable map. That is where is any other subset containing .

**Proposition 2.** If is the collection of all -fields that contain then

**Definition 8.** The Borel -field is the smallest -field containing all open intervals of . is generated by each of the following

- the collection of all closed subsets of ;

- the collection of all subintervals of of the form ;

- the collection of all subintervals of of the form .

**Theorem 2.** If and are measurable maps, then is a measurable map from to .

**Theorem 3. **If are random variables and is measurable, then is a random variable.

**Theorem 4. **If are random variables then so are

**Definition 9.** converges almost surely (a.s.) if where

**Definition 10.** A -system on a set is a family of subsets of stable under finite intersection:

**Lemma 1.** Let . Supposed and are measures on s.th. and on . Then

**Corollary 1.** If two probability measures agree on a -system then they agree on the -algebra generated by that -system. Example: .

**Definition 11. Dynkin Class**/ or system. Let be a set, and let be a collection of subsets of . Then is called a d-system (on ) if

- ,
- if and then
- if and , then

Recall means and .

**Definition 12.** : The intersection of all d-systems which contain the class of subsets of . Subsequently , is a d-system, the smallest d-system which contains . Furthermore

.

**Lemma 2. Dynkin’s lemma**. If is a -system, then

A d-system which contains a -system contains the -algebra generated by that -system.

**Proposition 3.** A collection of subsets of is a -algebra iff it is both a -system and a d-system.

**Theorem 5. Carath éodory’s Theorem**. Let be a set, let be an algebra on , and let

If is a countably additive map s.th.

]]>The phrase “Black Swan” (arising earlier in the different context of Popperian falsification) is here defined as an event characterized [p. xviii] by rarity, extreme impact, and retrospective (though not prospective) predictability, and Taleb’s thesis is that such events have much greater effect, in financial markets and the broader world of human affairs, than we usually suppose. The book is challenging to review because it requires considerable effort to separate the content from the style. The style is rambling and pugnacious—well described by one reviewer as “with few exceptions, the writers and professionals Taleb describes are knaves or fools, mostly fools. His writing is full of irrelevances, asides and colloquialisms, reading like the conversation of a raconteur rather than a tightly argued thesis”. And clearly this is perfectly deliberate. Such a book invites a review that reflects the reviewer’s opinions more than is customary in the Notices. My own overall reaction is that Taleb is sensible (going on prescient) in his discussion of financial markets and in some of his general philosophical thought but tends toward irrelevance or ridiculous exaggeration otherwise.

more here.

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