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(e.g., cash received from customers, cash received from investment income) and operating cash outflows by use (e.g., cash paid to suppliers, cash paid for interest) in the operating activities section of the cash flow statement. <span>It adjusts each item in the income statement to its cash equivalent. It shows operating cash receipts and payments. More cash flow information can be obtained and it is more easily understood by the average reader. The indirect me

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Definition of Proportional When two quantities always have the same size in relation to each other. In other words they have the same ratio

Home Subjects ▼ Algebra Calculus Data Geometry Measure Money Numbers Physics More ▼ Activities Dictionary Games Puzzles Worksheets ☰ ABCDEFGHIJKLMNOPQRSTUVWXYZ Texto original Sugiere una traducción mejor <span>Definition of Proportional more ... When two quantities always have the same size in relation to each other. In other words they have the same ratio. Example: A rope's length and weight are in proportion. When 20m of rope weighs 1kg, then: • 40m of that rope weighs 2kg • 200m of that rope weighs 10kg etc. Another example: The len

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Deep Gaussian processes (DGPs) are multi-layer hierarchical generalisations of Gaussian pro- cesses (GPs) and are formally equivalent to neural networks with multiple, infinitely wide hidden layers.

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According to A History of Formal Logic (1961) by the distinguished J M Bocheński, the golden periods for logic were the ancient Greek period, the medieval scholastic period, and the mathematical period of the 19th and 20th centuries.

the hands of thinkers such as George Boole, Gottlob Frege, Bertrand Russell, Alfred Tarski and Kurt Gödel, it’s clear that Kant was dead wrong. But he was also wrong in thinking that there had been no progress since Aristotle up to his time. <span>According to A History of Formal Logic (1961) by the distinguished J M Bocheński, the golden periods for logic were the ancient Greek period, the medieval scholastic period, and the mathematical period of the 19th and 20th centuries. (Throughout this piece, the focus is on the logical traditions that emerged against the background of ancient Greek logic. So Indian and Chinese logic are not included, but medieval Ara

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Deep Gaussian processes (DGPs) are multi-layer hierarchical generalisations of Gaussian pro- cesses (GPs) and are formally equivalent to neural networks with multiple, infinitely wide hidden layers.

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The fall of disputational culture wasn’t the only cause for the demise of scholastic logic, however. Scholastic logic was also viewed – rightly or wrongly – as being tied to broadly Aristotelian conceptions of language

s Diafoirus resorts to disputational vocabulary to make a point about love: Distinguo, Mademoiselle; in all that does not concern the possession of the loved one, concedo, I grant it; but in what does regard that possession, nego, I deny it. <span>The fall of disputational culture wasn’t the only cause for the demise of scholastic logic, however. Scholastic logic was also viewed – rightly or wrongly – as being tied to broadly Aristotelian conceptions of language and metaphysics, which themselves fell out of favour in the dawn of the modern era with the rise of a new scientific paradigm. Despite all this, disputations continued to be practised in certain university contexts for some time – indeed, they live on in the ceremonial character of PhD defences. The point, thou

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The fall of disputational culture wasn’t the only cause for the demise of scholastic logic, however. Scholastic logic was also viewed – rightly or wrongly – as being tied to broadly Aristotelian conceptions of language and metaphysics, which themselves fell out of favour in the dawn of the modern era with the rise of a new scientific paradigm. <body><html>

s Diafoirus resorts to disputational vocabulary to make a point about love: Distinguo, Mademoiselle; in all that does not concern the possession of the loved one, concedo, I grant it; but in what does regard that possession, nego, I deny it. <span>The fall of disputational culture wasn’t the only cause for the demise of scholastic logic, however. Scholastic logic was also viewed – rightly or wrongly – as being tied to broadly Aristotelian conceptions of language and metaphysics, which themselves fell out of favour in the dawn of the modern era with the rise of a new scientific paradigm. Despite all this, disputations continued to be practised in certain university contexts for some time – indeed, they live on in the ceremonial character of PhD defences. The point, thou

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It is also not happenstance that the downfall of the disputational culture roughly coincided with the introduction of new printing techniques in Europe by Johannes Gutenberg, around 1440.

ich is thoroughly disputational, with Meditations on First Philosophy (1641) by Descartes, a book argued through long paragraphs driven by the first-person singular. The nature of intellectual enquiry shifted with the downfall of disputation. <span>It is also not happenstance that the downfall of the disputational culture roughly coincided with the introduction of new printing techniques in Europe by Johannes Gutenberg, around 1440. Before that, books were a rare commodity, and education was conducted almost exclusively by means of oral contact between masters and pupils in the form of expository lectures in which

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It is also not happenstance that the downfall of the disputational culture roughly coincided with the introduction of new printing techniques in Europe by Johannes Gutenberg, around 1440.

ich is thoroughly disputational, with Meditations on First Philosophy (1641) by Descartes, a book argued through long paragraphs driven by the first-person singular. The nature of intellectual enquiry shifted with the downfall of disputation. <span>It is also not happenstance that the downfall of the disputational culture roughly coincided with the introduction of new printing techniques in Europe by Johannes Gutenberg, around 1440. Before that, books were a rare commodity, and education was conducted almost exclusively by means of oral contact between masters and pupils in the form of expository lectures in which

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tually unthinkable before the wide availability of printed books) was well-established. Moreover, as indicated by the passage from Descartes above, the very term ‘logic’ came to be used for something other than what the scholastics had meant. <span>Instead, early modern authors emphasise the role of novelty and individual discovery, as exemplified by the influential textbook Port-Royal Logic (1662), essentially, the logical version of Cartesianism, based on Descartes’s conception of mental operations and the prima

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Instead, early modern authors emphasise the role of novelty and individual discovery

tually unthinkable before the wide availability of printed books) was well-established. Moreover, as indicated by the passage from Descartes above, the very term ‘logic’ came to be used for something other than what the scholastics had meant. <span>Instead, early modern authors emphasise the role of novelty and individual discovery, as exemplified by the influential textbook Port-Royal Logic (1662), essentially, the logical version of Cartesianism, based on Descartes’s conception of mental operations and the prima

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without judgment about things one does not know. Such logic corrupts good sense rather than increasing it. I mean instead the kind of logic which teaches us to direct our reason with a view to discovering the truths of which we are ignorant. <span>Descartes hits the nail on the head when he claims that the logic of the Schools (scholastic logic) is not really a logic of discovery. Its chief purpose is justification and exposition, which makes sense particularly against the background of dialectical practices, where interlocutors explain and debate what they themselves already know. Indeed, for much of the history of logic, both in ancient Greece and in the Latin medieval tradition, ‘dialectic’ and ‘logic’ were taken to be synonymous. Up to Descartes’s time, the ch

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Descartes hits the nail on the head when he claims that the logic of the Schools (scholastic logic) is not really a logic of discovery. Its chief purpose is justification and exposition , which makes sense particularly against the background of dialectical practices, where interlocutors explain and debate what they themselves already know.

without judgment about things one does not know. Such logic corrupts good sense rather than increasing it. I mean instead the kind of logic which teaches us to direct our reason with a view to discovering the truths of which we are ignorant. <span>Descartes hits the nail on the head when he claims that the logic of the Schools (scholastic logic) is not really a logic of discovery. Its chief purpose is justification and exposition, which makes sense particularly against the background of dialectical practices, where interlocutors explain and debate what they themselves already know. Indeed, for much of the history of logic, both in ancient Greece and in the Latin medieval tradition, ‘dialectic’ and ‘logic’ were taken to be synonymous. Up to Descartes’s time, the ch

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are considered the most important and central in the theory of stochastic processes, [1] [4] [23] and were discovered repeatedly and independently, both before and after Bachelier and Erlang, in different settings and countries. [21] [24] <span>The term random function is also used to refer to a stochastic or random process, [25] [26] because a stochastic process can also be interpreted as a random element in a function space. [27] [28] The terms stochastic process and random process are used interchangeably, often with no specific mathematical space for the set that indexes the random variables. [27] [29]

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hangeably, often with no specific mathematical space for the set that indexes the random variables. [27] [29] But often these two terms are used when the random variables are indexed by the integers or an interval of the real line. [5] [29] <span>If the random variables are indexed by the Cartesian plane or some higher-dimensional Euclidean space, then the collection of random variables is usually called a random field instead. [5] [30] The values of a stochastic process are not always numbers and can be vectors or other mathematical objects. [5] [28] Based on their properties, stochastic processes can be d

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arkets have motivated the extensive use of stochastic processes in finance. [16] [17] [18] Applications and the study of phenomena have in turn inspired the proposal of new stochastic processes. Examples of such stochastic processes include <span>the Wiener process or Brownian motion process, [a] used by Louis Bachelier to study price changes on the Paris Bourse, [21] and the Poisson process, used by A. K. Erlang to study the number of phone calls occurring in a certain period of time. [22] These two stochastic processes are considered the mo

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element in the set. [4] [5] The set used to index the random variables is called the index set. Historically, the index set was some subset of the real line, such as the natural numbers, giving the index set the interpretation of time. [1] <span>Each random variable in the collection takes values from the same mathematical space known as the state space. This state space can be, for example, the integers, the real line or n {\displaystyle n} -dimensional Euclidean space. [1] [5] An increment i

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stochastic or random process can be defined as a collection of random variables that is indexed by some mathematical set, meaning that each random variable of the stochastic process is uniquely associated with an element in the set. [4] [5] <span>The set used to index the random variables is called the index set. Historically, the index set was some subset of the real line, such as the natural numbers, giving the index set the interpretation of time. [1] Each random variable in the collection t

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r n {\displaystyle n} -dimensional Euclidean space. [1] [5] An increment is the amount that a stochastic process changes between two index values, often interpreted as two points in time. [48] [49] <span>A stochastic process can have many outcomes, due to its randomness, and a single outcome of a stochastic process is called, among other names, a sample function or realization. [28] [50] [imagelink] A single computer-simulated sample function or realization, among other terms, of a three-dimensional Wiener or Brownian motion process for time 0 ≤ t ≤ 2.

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f a three-dimensional Wiener or Brownian motion process for time 0 ≤ t ≤ 2. The index set of this stochastic process is the non-negative numbers, while its state space is three-dimensional Euclidean space. Classifications[edit source] <span>A stochastic process can be classified in different ways, for example, by its state space, its index set, or the dependence among the random variables. One common way of classification is by the cardinality of the index set and the state space. [51] [52] [53] When interpreted as time, if the index set of a stochastic process has a fi

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} -dimensional vector process or n {\displaystyle n} -vector process. [51] [52] Examples of stochastic processes[edit source] Bernoulli process[edit source] Main article: Bernoulli process <span>One of the simplest stochastic processes is the Bernoulli process, [60] which is a sequence of independent and identically distributed (iid) random variables, where each random variable takes either the value one or zero, say one with probability p {\displaystyle p} and zero with probability 1 − p {\displaystyle 1-p} . This process can be likened to somebody flipping a coin, where the probability of obtaining a head is p {\displaystyle p} and its value is on

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one, while the value of a tail is zero. [61] In other words, a Bernoulli process is a sequence of iid Bernoulli random variables, [62] where each coin flip is a Bernoulli trial. [63] Random walk[edit source] Main article: Random walk <span>Random walks are stochastic processes that are usually defined as sums of iid random variables or random vectors in Euclidean space, so they are processes that change in discrete time. [64] [65] [66] [67] [68] But some also use the term to refer to processes that change in continuous time, [69] particularly the Wiener process used in finance, which has led to some c

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ere are other various types of random walks, defined so their state spaces can be other mathematical objects, such as lattices and groups, and in general they are highly studied and have many applications in different disciplines. [69] [71] <span>A classic example of a random walk is known as the simple random walk, which is a stochastic process in discrete time with the integers as the state space, and is based on a Bernoulli process, where each iid Bernoulli variable takes either the value positive one or negative one. In other words, the simple random walk takes place on the integers, and its value increases by one with probability, say, p {\displaystyle p}

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wnian motion due to its historical connection as a model for Brownian movement in liquids. [75] [76] [76] [77] [imagelink] Realizations of Wiener processes (or Brownian motion processes) with drift (blue) and without drift (red). <span>Playing a central role in the theory of probability, the Wiener process is often considered the most important and studied stochastic process, with connections to other stochastic processes. [1] [2] [3] [78] [79] [80] [81] Its index set and state space are the non-negative numbers and real numbers, respectively, so it has both continuous index set and states space. [82] But the process can be defined more generally so its state space can be n {\displaystyle n} -dimensional Euclidean space. [71] [79] [83]

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, so it has both continuous index set and states space. [82] But the process can be defined more generally so its state space can be n {\displaystyle n} -dimensional Euclidean space. [71] [79] [83] <span>If the mean of any increment is zero, then the resulting Wiener or Brownian motion process is said to have zero drift. If the mean of the increment for any two points in time is equal to the time difference multiplied by some constant μ {\displaystyle \mu } , w

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stant μ {\displaystyle \mu } , which is a real number, then the resulting stochastic process is said to have drift μ {\displaystyle \mu } . [84] [85] [86] <span>Almost surely, a sample path of a Wiener process is continuous everywhere but nowhere differentiable. It can be considered a continuous version of the simple random walk. [49] [85] The process arises as the mathematical limit of other stochastic processes such as certain random walks rescaled, [87] [88] which is the subject of Donsker's theorem or inva

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e space can be n {\displaystyle n} -dimensional Euclidean space. [71] [79] [83] If the mean of any increment is zero, then the resulting Wiener or Brownian motion process is said to have zero drift. <span>If the mean of the increment for any two points in time is equal to the time difference multiplied by some constant μ {\displaystyle \mu } , which is a real number, then the resulting stochastic process is said to have drift μ {\displaystyle \mu } . [84] [85] [86] Almost surely, a sample path of a Wiener process is continuous everywhere but nowhere differentiable. It can be considered a continuous version of the simple rando

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arameter. This process has the natural numbers as its state space and the non-negative numbers as its index set. This process is also called the Poisson counting process, since it can be interpreted as an example of a counting process. [99] <span>If a Poisson process is defined with a single positive constant, then the process is called a homogeneous Poisson process. [99] [101] The homogeneous Poisson process (in continuous time) is a member of important classes of stochastic processes such as Markov processes and Lévy processes. [49] The homogen

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constant, then the process is called a homogeneous Poisson process. [99] [101] The homogeneous Poisson process (in continuous time) is a member of important classes of stochastic processes such as Markov processes and Lévy processes. [49] <span>The homogeneous Poisson process can be defined and generalized in different ways. It can be defined such that its index set is the real line, and this stochastic process is also called the stationary Poisson process. [102] [103] If the parameter constant of the Poisson process is replaced with some non-negative integrable function of t {\displaystyle t} ,

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sses. [49] The homogeneous Poisson process can be defined and generalized in different ways. It can be defined such that its index set is the real line, and this stochastic process is also called the stationary Poisson process. [102] [103] <span>If the parameter constant of the Poisson process is replaced with some non-negative integrable function of t {\displaystyle t} , the resulting process is called an inhomogeneous or nonhomogeneous Poisson process, where the average density of points of the process is no longer constant. [104] Serving as a fundamental process in queueing theory, the Poisson process is an important process for mathematical models, where it finds applications for models of events randoml

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the Wiener process or Brownian motion process, [a] used by Louis Bachelier to study price changes on the Paris Bourse

arkets have motivated the extensive use of stochastic processes in finance. [16] [17] [18] Applications and the study of phenomena have in turn inspired the proposal of new stochastic processes. Examples of such stochastic processes include <span>the Wiener process or Brownian motion process, [a] used by Louis Bachelier to study price changes on the Paris Bourse, [21] and the Poisson process, used by A. K. Erlang to study the number of phone calls occurring in a certain period of time. [22] These two stochastic processes are considered the mo

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The term random function is also used to refer to a stochastic or random process, [25] [26] because a stochastic process can also be interpreted as a random element in a function space. <

are considered the most important and central in the theory of stochastic processes, [1] [4] [23] and were discovered repeatedly and independently, both before and after Bachelier and Erlang, in different settings and countries. [21] [24] <span>The term random function is also used to refer to a stochastic or random process, [25] [26] because a stochastic process can also be interpreted as a random element in a function space. [27] [28] The terms stochastic process and random process are used interchangeably, often with no specific mathematical space for the set that indexes the random variables. [27] [29]

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The term random function is also used to refer to a stochastic or random process, [25] [26] because a stochastic process can also be interpreted as a random element in a function space.

are considered the most important and central in the theory of stochastic processes, [1] [4] [23] and were discovered repeatedly and independently, both before and after Bachelier and Erlang, in different settings and countries. [21] [24] <span>The term random function is also used to refer to a stochastic or random process, [25] [26] because a stochastic process can also be interpreted as a random element in a function space. [27] [28] The terms stochastic process and random process are used interchangeably, often with no specific mathematical space for the set that indexes the random variables. [27] [29]

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If the random variables are indexed by the Cartesian plane or some higher-dimensional Euclidean space, then the collection of random variables is usually called a random field instead.

hangeably, often with no specific mathematical space for the set that indexes the random variables. [27] [29] But often these two terms are used when the random variables are indexed by the integers or an interval of the real line. [5] [29] <span>If the random variables are indexed by the Cartesian plane or some higher-dimensional Euclidean space, then the collection of random variables is usually called a random field instead. [5] [30] The values of a stochastic process are not always numbers and can be vectors or other mathematical objects. [5] [28] Based on their properties, stochastic processes can be d

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The set used to index the random variables is called the index set.

stochastic or random process can be defined as a collection of random variables that is indexed by some mathematical set, meaning that each random variable of the stochastic process is uniquely associated with an element in the set. [4] [5] <span>The set used to index the random variables is called the index set. Historically, the index set was some subset of the real line, such as the natural numbers, giving the index set the interpretation of time. [1] Each random variable in the collection t

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Each random variable in the collection takes values from the same mathematical space known as the state space.

element in the set. [4] [5] The set used to index the random variables is called the index set. Historically, the index set was some subset of the real line, such as the natural numbers, giving the index set the interpretation of time. [1] <span>Each random variable in the collection takes values from the same mathematical space known as the state space. This state space can be, for example, the integers, the real line or n {\displaystyle n} -dimensional Euclidean space. [1] [5] An increment i

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A stochastic process can have many outcomes, due to its randomness, and a single outcome of a stochastic process is called, among other names, a sample function or realization

r n {\displaystyle n} -dimensional Euclidean space. [1] [5] An increment is the amount that a stochastic process changes between two index values, often interpreted as two points in time. [48] [49] <span>A stochastic process can have many outcomes, due to its randomness, and a single outcome of a stochastic process is called, among other names, a sample function or realization. [28] [50] [imagelink] A single computer-simulated sample function or realization, among other terms, of a three-dimensional Wiener or Brownian motion process for time 0 ≤ t ≤ 2.

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A stochastic process can be classified in different ways, for example, by its state space, its index set, or the dependence among the random variables.

f a three-dimensional Wiener or Brownian motion process for time 0 ≤ t ≤ 2. The index set of this stochastic process is the non-negative numbers, while its state space is three-dimensional Euclidean space. Classifications[edit source] <span>A stochastic process can be classified in different ways, for example, by its state space, its index set, or the dependence among the random variables. One common way of classification is by the cardinality of the index set and the state space. [51] [52] [53] When interpreted as time, if the index set of a stochastic process has a fi

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A stochastic process can be classified in different ways, for example, by its state space, its index set, or the dependence among the random variables.

f a three-dimensional Wiener or Brownian motion process for time 0 ≤ t ≤ 2. The index set of this stochastic process is the non-negative numbers, while its state space is three-dimensional Euclidean space. Classifications[edit source] <span>A stochastic process can be classified in different ways, for example, by its state space, its index set, or the dependence among the random variables. One common way of classification is by the cardinality of the index set and the state space. [51] [52] [53] When interpreted as time, if the index set of a stochastic process has a fi

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One of the simplest stochastic processes is the Bernoulli process, [60] which is a sequence of independent and identically distributed (iid) Bernoulli variables.

} -dimensional vector process or n {\displaystyle n} -vector process. [51] [52] Examples of stochastic processes[edit source] Bernoulli process[edit source] Main article: Bernoulli process <span>One of the simplest stochastic processes is the Bernoulli process, [60] which is a sequence of independent and identically distributed (iid) random variables, where each random variable takes either the value one or zero, say one with probability p {\displaystyle p} and zero with probability 1 − p {\displaystyle 1-p} . This process can be likened to somebody flipping a coin, where the probability of obtaining a head is p {\displaystyle p} and its value is on

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Random walks are stochastic processes that are usually defined as sums of iid random variables or random vectors in Euclidean space, so they are processes that change in discrete time.

one, while the value of a tail is zero. [61] In other words, a Bernoulli process is a sequence of iid Bernoulli random variables, [62] where each coin flip is a Bernoulli trial. [63] Random walk[edit source] Main article: Random walk <span>Random walks are stochastic processes that are usually defined as sums of iid random variables or random vectors in Euclidean space, so they are processes that change in discrete time. [64] [65] [66] [67] [68] But some also use the term to refer to processes that change in continuous time, [69] particularly the Wiener process used in finance, which has led to some c

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Random walks are stochastic processes that are usually defined as sums of iid random variables or random vectors in Euclidean space, so they are processes that change in discrete time.

one, while the value of a tail is zero. [61] In other words, a Bernoulli process is a sequence of iid Bernoulli random variables, [62] where each coin flip is a Bernoulli trial. [63] Random walk[edit source] Main article: Random walk <span>Random walks are stochastic processes that are usually defined as sums of iid random variables or random vectors in Euclidean space, so they are processes that change in discrete time. [64] [65] [66] [67] [68] But some also use the term to refer to processes that change in continuous time, [69] particularly the Wiener process used in finance, which has led to some c

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A classic example of a random walk is known as the simple random walk, which is a stochastic process in discrete time with the integers as the state space, and is based on a Bernoulli process, where each iid Bernoulli variable takes either the value positive one or negative one.

ere are other various types of random walks, defined so their state spaces can be other mathematical objects, such as lattices and groups, and in general they are highly studied and have many applications in different disciplines. [69] [71] <span>A classic example of a random walk is known as the simple random walk, which is a stochastic process in discrete time with the integers as the state space, and is based on a Bernoulli process, where each iid Bernoulli variable takes either the value positive one or negative one. In other words, the simple random walk takes place on the integers, and its value increases by one with probability, say, p {\displaystyle p}

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A classic example of a random walk is known as the simple random walk, which is a stochastic process in discrete time with the integers as the state space, and is based on a Bernoulli process, where each iid Bernoulli variable takes either the value positive one or negative one.

ere are other various types of random walks, defined so their state spaces can be other mathematical objects, such as lattices and groups, and in general they are highly studied and have many applications in different disciplines. [69] [71] <span>A classic example of a random walk is known as the simple random walk, which is a stochastic process in discrete time with the integers as the state space, and is based on a Bernoulli process, where each iid Bernoulli variable takes either the value positive one or negative one. In other words, the simple random walk takes place on the integers, and its value increases by one with probability, say, p {\displaystyle p}

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Playing a central role in the theory of probability, the Wiener process is often considered the most important and studied stochastic process, with connections to other stochastic processes. [1] [2] [3] [78] [79] [80] [81] Its index set and state space are

wnian motion due to its historical connection as a model for Brownian movement in liquids. [75] [76] [76] [77] [imagelink] Realizations of Wiener processes (or Brownian motion processes) with drift (blue) and without drift (red). <span>Playing a central role in the theory of probability, the Wiener process is often considered the most important and studied stochastic process, with connections to other stochastic processes. [1] [2] [3] [78] [79] [80] [81] Its index set and state space are the non-negative numbers and real numbers, respectively, so it has both continuous index set and states space. [82] But the process can be defined more generally so its state space can be n {\displaystyle n} -dimensional Euclidean space. [71] [79] [83]

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Playing a central role in the theory of probability, the Wiener process is often considered the most important and studied stochastic process, with connections to other stochastic processes. [1] [2] [3] [78] [79] [80] [81] Its index set and state space are the non-negative numbers and real numbers, respectively, so it has both continuous index set and states space.

wnian motion due to its historical connection as a model for Brownian movement in liquids. [75] [76] [76] [77] [imagelink] Realizations of Wiener processes (or Brownian motion processes) with drift (blue) and without drift (red). <span>Playing a central role in the theory of probability, the Wiener process is often considered the most important and studied stochastic process, with connections to other stochastic processes. [1] [2] [3] [78] [79] [80] [81] Its index set and state space are the non-negative numbers and real numbers, respectively, so it has both continuous index set and states space. [82] But the process can be defined more generally so its state space can be n {\displaystyle n} -dimensional Euclidean space. [71] [79] [83]

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If the mean of any increment is zero, then the resulting Wiener or Brownian motion process is said to have zero drift.

, so it has both continuous index set and states space. [82] But the process can be defined more generally so its state space can be n {\displaystyle n} -dimensional Euclidean space. [71] [79] [83] <span>If the mean of any increment is zero, then the resulting Wiener or Brownian motion process is said to have zero drift. If the mean of the increment for any two points in time is equal to the time difference multiplied by some constant μ {\displaystyle \mu } , w

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If the mean of any increment is zero, then the resulting Wiener or Brownian motion process is said to have zero drift.

, so it has both continuous index set and states space. [82] But the process can be defined more generally so its state space can be n {\displaystyle n} -dimensional Euclidean space. [71] [79] [83] <span>If the mean of any increment is zero, then the resulting Wiener or Brownian motion process is said to have zero drift. If the mean of the increment for any two points in time is equal to the time difference multiplied by some constant μ {\displaystyle \mu } , w

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If the mean of the increment for any two points in time is equal to the time difference multiplied by some constant , then the resulting stochastic process is said to have drift

e space can be n {\displaystyle n} -dimensional Euclidean space. [71] [79] [83] If the mean of any increment is zero, then the resulting Wiener or Brownian motion process is said to have zero drift. <span>If the mean of the increment for any two points in time is equal to the time difference multiplied by some constant μ {\displaystyle \mu } , which is a real number, then the resulting stochastic process is said to have drift μ {\displaystyle \mu } . [84] [85] [86] Almost surely, a sample path of a Wiener process is continuous everywhere but nowhere differentiable. It can be considered a continuous version of the simple rando

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If the mean of the increment for any two points in time is equal to the time difference multiplied by some constant , then the resulting stochastic process is said to have drift

e space can be n {\displaystyle n} -dimensional Euclidean space. [71] [79] [83] If the mean of any increment is zero, then the resulting Wiener or Brownian motion process is said to have zero drift. <span>If the mean of the increment for any two points in time is equal to the time difference multiplied by some constant μ {\displaystyle \mu } , which is a real number, then the resulting stochastic process is said to have drift μ {\displaystyle \mu } . [84] [85] [86] Almost surely, a sample path of a Wiener process is continuous everywhere but nowhere differentiable. It can be considered a continuous version of the simple rando

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If the mean of the increment for any two points in time is equal to the time difference multiplied by some constant , then the resulting stochastic process is said to have drift

e space can be n {\displaystyle n} -dimensional Euclidean space. [71] [79] [83] If the mean of any increment is zero, then the resulting Wiener or Brownian motion process is said to have zero drift. <span>If the mean of the increment for any two points in time is equal to the time difference multiplied by some constant μ {\displaystyle \mu } , which is a real number, then the resulting stochastic process is said to have drift μ {\displaystyle \mu } . [84] [85] [86] Almost surely, a sample path of a Wiener process is continuous everywhere but nowhere differentiable. It can be considered a continuous version of the simple rando

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Almost surely, a sample path of a Wiener process is continuous everywhere but nowhere differentiable. It can be considered a continuous version of the simple random walk.

stant μ {\displaystyle \mu } , which is a real number, then the resulting stochastic process is said to have drift μ {\displaystyle \mu } . [84] [85] [86] <span>Almost surely, a sample path of a Wiener process is continuous everywhere but nowhere differentiable. It can be considered a continuous version of the simple random walk. [49] [85] The process arises as the mathematical limit of other stochastic processes such as certain random walks rescaled, [87] [88] which is the subject of Donsker's theorem or inva

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Almost surely, a sample path of a Wiener process is continuous everywhere but nowhere differentiable. It can be considered a continuous version of the simple random walk.

stant μ {\displaystyle \mu } , which is a real number, then the resulting stochastic process is said to have drift μ {\displaystyle \mu } . [84] [85] [86] <span>Almost surely, a sample path of a Wiener process is continuous everywhere but nowhere differentiable. It can be considered a continuous version of the simple random walk. [49] [85] The process arises as the mathematical limit of other stochastic processes such as certain random walks rescaled, [87] [88] which is the subject of Donsker's theorem or inva

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Almost surely, a sample path of a Wiener process is continuous everywhere but nowhere differentiable. It can be considered a continuous version of the simple random walk.

stant μ {\displaystyle \mu } , which is a real number, then the resulting stochastic process is said to have drift μ {\displaystyle \mu } . [84] [85] [86] <span>Almost surely, a sample path of a Wiener process is continuous everywhere but nowhere differentiable. It can be considered a continuous version of the simple random walk. [49] [85] The process arises as the mathematical limit of other stochastic processes such as certain random walks rescaled, [87] [88] which is the subject of Donsker's theorem or inva

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If a Poisson process is defined with a single positive constant, then the process is called a homogeneous Poisson process.

arameter. This process has the natural numbers as its state space and the non-negative numbers as its index set. This process is also called the Poisson counting process, since it can be interpreted as an example of a counting process. [99] <span>If a Poisson process is defined with a single positive constant, then the process is called a homogeneous Poisson process. [99] [101] The homogeneous Poisson process (in continuous time) is a member of important classes of stochastic processes such as Markov processes and Lévy processes. [49] The homogen

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If a Poisson process is defined with a single positive constant, then the process is called a homogeneous Poisson process.

arameter. This process has the natural numbers as its state space and the non-negative numbers as its index set. This process is also called the Poisson counting process, since it can be interpreted as an example of a counting process. [99] <span>If a Poisson process is defined with a single positive constant, then the process is called a homogeneous Poisson process. [99] [101] The homogeneous Poisson process (in continuous time) is a member of important classes of stochastic processes such as Markov processes and Lévy processes. [49] The homogen

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The homogeneous Poisson process can be defined and generalized in different ways. It can be defined such that its index set is the real line, and this stochastic process is also called the stationary Poisson process.

constant, then the process is called a homogeneous Poisson process. [99] [101] The homogeneous Poisson process (in continuous time) is a member of important classes of stochastic processes such as Markov processes and Lévy processes. [49] <span>The homogeneous Poisson process can be defined and generalized in different ways. It can be defined such that its index set is the real line, and this stochastic process is also called the stationary Poisson process. [102] [103] If the parameter constant of the Poisson process is replaced with some non-negative integrable function of t {\displaystyle t} ,

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The homogeneous Poisson process can be defined and generalized in different ways. It can be defined such that its index set is the real line, and this stochastic process is also called the stationary Poisson process.

constant, then the process is called a homogeneous Poisson process. [99] [101] The homogeneous Poisson process (in continuous time) is a member of important classes of stochastic processes such as Markov processes and Lévy processes. [49] <span>The homogeneous Poisson process can be defined and generalized in different ways. It can be defined such that its index set is the real line, and this stochastic process is also called the stationary Poisson process. [102] [103] If the parameter constant of the Poisson process is replaced with some non-negative integrable function of t {\displaystyle t} ,

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If the parameter constant of the Poisson process is replaced with some non-negative integrable function of , the resulting process is called an inhomogeneous or nonhomogeneous Poisson process, where the average density of points of the process is no longer constant. <

sses. [49] The homogeneous Poisson process can be defined and generalized in different ways. It can be defined such that its index set is the real line, and this stochastic process is also called the stationary Poisson process. [102] [103] <span>If the parameter constant of the Poisson process is replaced with some non-negative integrable function of t {\displaystyle t} , the resulting process is called an inhomogeneous or nonhomogeneous Poisson process, where the average density of points of the process is no longer constant. [104] Serving as a fundamental process in queueing theory, the Poisson process is an important process for mathematical models, where it finds applications for models of events randoml

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If the parameter constant of the Poisson process is replaced with some non-negative integrable function of , the resulting process is called an inhomogeneous or nonhomogeneous Poisson process, where the average density of points of the process is no longer constant.

sses. [49] The homogeneous Poisson process can be defined and generalized in different ways. It can be defined such that its index set is the real line, and this stochastic process is also called the stationary Poisson process. [102] [103] <span>If the parameter constant of the Poisson process is replaced with some non-negative integrable function of t {\displaystyle t} , the resulting process is called an inhomogeneous or nonhomogeneous Poisson process, where the average density of points of the process is no longer constant. [104] Serving as a fundamental process in queueing theory, the Poisson process is an important process for mathematical models, where it finds applications for models of events randoml

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body> If the parameter constant of the Poisson process is replaced with some non-negative integrable function of , the resulting process is called an inhomogeneous or nonhomogeneous Poisson process, where the average density of points of the process is no longer constant. <body><html>

sses. [49] The homogeneous Poisson process can be defined and generalized in different ways. It can be defined such that its index set is the real line, and this stochastic process is also called the stationary Poisson process. [102] [103] <span>If the parameter constant of the Poisson process is replaced with some non-negative integrable function of t {\displaystyle t} , the resulting process is called an inhomogeneous or nonhomogeneous Poisson process, where the average density of points of the process is no longer constant. [104] Serving as a fundamental process in queueing theory, the Poisson process is an important process for mathematical models, where it finds applications for models of events randoml

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