It can be viewed as an application of traditional induction on the length of that binary representation. If traditional predecessor induction is interpreted computationally as an n -step loop, prefix induction corresponds to a log n -step loop, and thus proofs using prefix induction are "more feasibly constructive" than proofs using predecessor induction. Predecessor induction can trivially simulate prefix induction on the same statement. Prefix induction can simulate predecessor induction, but only at the cost of making the statement more syntactically complex adding a bounded universal quantifier , so the interesting results relating prefix induction to polynomial-time computation depend on excluding unbounded quantifiers entirely, and limiting the alternation of bounded universal and existential quantifiers allowed in the statement.

This form of induction has been used, analogously, to study log-time parallel computation. The name "strong induction" does not mean that this method can prove more than "weak induction", but merely refers to the stronger hypothesis used in the inductive step; in fact the two methods are equivalent, as explained below. In this form of complete induction one still has to prove the base case, P 0 , and it may even be necessary to prove extra base cases such as P 1 before the general argument applies, as in the example below of the Fibonacci number F n.

This is a special case of transfinite induction as described below. Complete induction is equivalent to ordinary mathematical induction as described above, in the sense that a proof by one method can be transformed into a proof by the other. Suppose there is a proof of P n by complete induction. Then Q n holds for all n if and only if P n holds for all n , and our proof of P n is easily transformed into a proof of Q n by ordinary induction.

Complete induction is most useful when several instances of the inductive hypothesis are required for each inductive step. For example, complete induction can be used to show that. Another proof by complete induction uses the hypothesis that the statement holds for all smaller n more thoroughly. Consider the statement that "every natural number greater than 1 is a product of one or more prime numbers ", which is the " existence " part of the fundamental theorem of arithmetic. The induction hypothesis now applies to n 1 and n 2 , so each one is a product of primes.

Thus m is a product of products of primes; therefore itself a product of primes. We shall look to prove the same example as above , this time with a variant called strong induction. The statement remains the same:. However, there will be slight differences with the structure and assumptions of the proof. Let us begin with the base case. However, proving the validity of the statement for no single number suffices to establish the base case; instead, one needs to prove the statement for an infinite subset of the natural numbers.

For example, Augustin Louis Cauchy first used forward regular induction to prove the inequality of arithmetic and geometric means for all powers of 2, and then used backward induction to show it for all natural numbers. The inductive step must be proved for all values of n. To illustrate this, Joel E. Cohen proposed the following argument, which purports to prove by mathematical induction that all horses are of the same color: [16].

In second-order logic , we can write down the " axiom of induction" as follows:. The axiom of induction asserts the validity of inferring that P n holds for any natural number n from the base case and the inductive step.

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The first quantifier in the axiom ranges over predicates rather than over individual numbers. This is a second-order quantifier, which means that this axiom is stated in second-order logic. Axiomatizing arithmetic induction in first-order logic requires an axiom schema containing a separate axiom for each possible predicate. The article Peano axioms contains further discussion of this issue. In first-order ZFC set theory , quantification over predicates is not allowed, but we can still phrase induction by quantification over sets:.

This is not an axiom, but a theorem, given that natural numbers are defined in the language of ZFC set theory by axioms, analogous to Peano's. Any set of cardinal numbers is well-founded, which includes the set of natural numbers. This form of induction, when applied to a set of ordinals which form a well-ordered and hence well-founded class , is called transfinite induction. It is an important proof technique in set theory , topology and other fields. So the special cases are special cases of the general case. The principle of mathematical induction is usually stated as an axiom of the natural numbers; see Peano axioms.

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However, it can be proved from the well-ordering principle. Indeed, suppose the following:. To derive simple induction from these axioms, one must show that if P n is some proposition predicated of n for which:. Let S be the set of all natural numbers for which P m is false.

Let us see what happens if one asserts that S is nonempty. Well-ordering tells us that S has a least element, say n. Moreover, since P 0 is true, n is not 0. Now m is less than n , and n is the least element of S. It follows that m is not in S , and so P m is true. This is a contradiction, since n was in S.

Therefore, S is empty. It can also be proved that induction, given the other axioms, implies the well-ordering principle. Suppose there exists a non-empty set, S , of naturals that has no least element. Let P n be the assertion that n is not in S.

## Smythe : Strong Laws of Large Numbers for $r$-Dimensional Arrays of Random Variables

Then P 0 is true, for if it were false then 0 is the least element of S. Furthermore, suppose P 1 , P 2 , Therefore, by the induction axiom, P n holds for all n , so S is empty, a contradiction. From Wikipedia, the free encyclopedia. This section includes a list of references , related reading or external links , but its sources remain unclear because it lacks inline citations. The cross moment — i.

This paints a specific picture of weakly stationary processes as those with constant mean and variance. Their properties are contrasted nicely with those of their counterparts in Figure 2 below. Confusingly enough, it is also sometimes referred to simply as stationarity , depending on context see [Boshnakov, ] for an example ; in geo-statistical literature, for example, this is the dominant notion of stationarity.

Note: Strong stationarity does not imply weak stationarity, nor does the latter implies the former see example here! An exception are Gaussian processes, for which weak stationarity does imply strong stationarity. The reason strong stationarity does not imply weak stationarity is that it does not mean the process necessarily has a finite second moment; e.

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Indeed, having a finite second moment is a necessary and sufficient condition for the weak stationarity of a strongly stationary process. White Noise Process : A white noise process is a serially uncorrelated stochastic process with a mean of zero and a constant and finite variance. Note that this implies that every white noise process is a weak stationary process. Very close to the definition of strong stationarity, N -th order stationarity demands the shift-invariance in time of the distribution of any n samples of the stochastic process, for all n up to order N.

Thus, the same condition is required:. Naturally, stationarity to a certain order N does not imply stationarity of any higher order but the inverse is true. An interesting thread in mathoverflow showcases both an example of a 1st order stationary process that is not 2nd order stationary, and an example for a 2nd order stationary process that is not 3rd order stationary.

And similarly, having a finite second moment is a sufficient and necessary condition for a 2nd order stationary process to also be a weakly stationary process. The term first-order stationarity is sometimes used to describe a series that has means that never changes with time, but for which any other moment like variance can change. Cyclostationarity is prominent in signal processing. A stochastic process is trend stationary if an underlying trend function solely of time can be removed, leaving a stationary process.

In the presence of a shock a significant and rapid one-off change to the value of the series , trend-stationary processes are mean-reverting; i. Intuitive extensions exist of all of the above types of stationarity for pairs of stochastic processes. Weak stationarity and N -th order stationarity can be extended in the same way the latter to M - N -th order joint stationarity.

A weaker form of weak stationarity, prominent in geostatistical literature see [Myers ] and [Fischer et al. An important class of non-stationary processes are locally stationary LS processes.

Alternatively, [Dahlhaus, ] defines them informally as processes which locally at each time point are close to a stationary process but whose characteristics covariances, parameters, etc. A formal definition can be found in [Vogt, ], and [Dahlhaus, ] provides a rigorous review of the subject. LS processes are of importance because they somewhat bridge the gap between the thoroughly explored sub-class of parametric non-stationary processes see the following section and the uncharted waters of the wider family of non-parametric processes, in that they have received rigorous treatment and a corresponding set of analysis tools akin to those enjoyed by parametric processes.

A great online resource on the topic is the home page of Prof. Guy Nason , who names LS processes as his main research interest. The following typology figure, partial as it may be, can help understand the relations between the different notions of stationarity we just went over:.

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The definitions of stationarity presented so far have been non-parametric; i. The related concept of a difference stationarity and unit root processes, however, requires a brief introduction to stochastic process modeling. The topic of stochastic modeling is also relevant insofar as various simple models can be used to create stochastic processes see figure 5. The forecasting of future values is a common task in the study of time series data.

To make forecasts, some assumptions need to be made regarding the Data Generating Process DGP , the mechanism generating the data. These assumptions often take the form of an explicit model of the process, and are also often used when modeling stochastic processes for other tasks, such as anomaly detection or causal inference.

We will go over the three most common such models. This is a memory-based model, in the sense that each value is correlated with the p preceding values; an AR model with lag p is denoted with AR p. The vector autoregressive VAR model generalizes the univariate case of the AR model to the multivariate case; now each element of the vector x[t] of length k can be modeled as a linear function of all the elements of the past p vectors:. Like for autoregressive models, a vector generalization, VMA, exists. With a basic understanding of common stochastic process models, we can now discuss the related concept of difference stationary processes and unit roots.

This concept relies on the assumption that the stochastic process in question can be written as an autoregressive process of order p, denoted as AR p :. We can write the same process as:. The part inside the parenthesis on the left is called the characteristic equation of the process. We can consider the roots of this equation:.

This means that the process can be transformed into a weakly-stationary process by applying a certain type of transformation to it, called differencing. Difference stationary processes have an order of integration , which is the number of times the differencing operator must be applied to it in order to achieve weak stationarity. A process that has to be differenced r times is said to be integrated of order r, denoted by I r. Maccari, and C. Castellani Phys. B 99 , — Published 28 February Abstract The nature of the superfluid-insulator transition in one dimension has been much debated recently.

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