Wiener Processes
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Levy Processes
LÉVY PROCESSES, STABLE PROCESSES, AND SUBORDINATORS STEVEN P.LALLEY 1. DEFINITIONSAND EXAMPLES d Definition 1.1. A continuous–time process Xt = X(t ) t 0 with values in R (or, more generally, in an abelian topological groupG ) isf called a Lévyg ≥ process if (1) its sample paths are right-continuous and have left limits at every time point t , and (2) it has stationary, independent increments, that is: (a) For all 0 = t0 < t1 < < tk , the increments X(ti ) X(ti 1) are independent. − (b) For all 0 s t the··· random variables X(t ) X−(s ) and X(t s ) X(0) have the same distribution.≤ ≤ − − − The default initial condition is X0 = 0. A subordinator is a real-valued Lévy process with nondecreasing sample paths. A stable process is a real-valued Lévy process Xt t 0 with ≥ initial value X0 = 0 that satisfies the self-similarity property f g 1/α (1.1) Xt =t =D X1 t > 0. 8 The parameter α is called the exponent of the process. Example 1.1. The most fundamental Lévy processes are the Wiener process and the Poisson process. The Poisson process is a subordinator, but is not stable; the Wiener process is stable, with exponent α = 2. Any linear combination of independent Lévy processes is again a Lévy process, so, for instance, if the Wiener process Wt and the Poisson process Nt are independent then Wt Nt is a Lévy process. More important, linear combinations of independent Poisson− processes are Lévy processes: these are special cases of what are called compound Poisson processes: see sec. -
Levy Processes, Stable Processes, and Subordinators
LÉVY PROCESSES, STABLE PROCESSES, AND SUBORDINATORS STEVEN P.LALLEY 1. DEFINITIONS AND EXAMPLES d Definition 1.1. A continuous–time process X t = X (t ) t 0 with values in R (or, more generally, in an abelian topological group G{) is called} a≥Lévy process if (1) its sample paths are right-continuous and have left limits at every time point t , and (2) it has stationary, independent increments, that is: (a) For all 0 = t0 < t1 < < tk , the increments X (ti ) X (ti 1) are independent. (b) For all 0 s t the··· random variables X (t ) X (−s ) and−X (t s ) X (0) have the same distribution. − − − The default initial condition is X0 = 0. It can be shown that every process with sta- tionary, independent increments has a version with paths that are right-continuous with left limits, but this fact won’t be needed in this course. See BREIMAN, Probability, chs. 9, 14 for the proof. Definition 1.2. A subordinator is a real-valued Lévy process with nondecreasing sam- ple paths. A stable process is a real-valued Lévy process X t t 0 with initial value X0 = 0 that satisfies the self-similarity property { } ≥ 1/↵ (1.1) X t /t =D X1 t > 0. 8 The parameter ↵ is called the exponent of the process. Example 1.1. The fundamental Lévy processes are the Wiener process and the Poisson process. The Poisson process is a subordinator, but is not stable; the Wiener process is stable, with exponent ↵ = 2. Any linear combination of independent Lévy processes is again a Lévy process, so, for instance, if the Wiener process Wt and the Poisson process Nt are independent then Wt Nt is a Lévy process. -
Notes an Infinitesimally Small Change in T ≈ ≤ T ≤ X
Intro to Stochastic Systems (Spring 16) Lecture 3 3 Random signals and probabilistic systems 1 We have defined stochastic signals in an earlier lecture. A stochastic signal X is a collection (Xt)t T of random variables indexed by a time variable t T . Depending on the structure of the set T , we2 2 speak of discrete- or continuous-time signals: • If T is a subset of the integers Z, then X is a discrete-time stochastic signal. • If T is a finite or an infinite interval of the real line R, then X is a continous-time stochastic signal. The set X of possible values each Xt can take is called the state space of the process. If the elements of X can be put in a one-to-one correspondence with a subset of Z, then we say that X is a discrete-state process; otherwise, if X is a finite or an infinite interval of R, we say that X is a continuous-state process. It is important to point out that, even though we are using the word “state,” it does not mean that Xt is a state variable in the sense that it contains all relevant information about the signal up to time t. Another way to view stochastic signals is as follows. We have a probability space (Ω; ;P), such F that each X is a random variable with values in X, i.e., a function X : Ω X. This specifies a t t ! random variable for each t T . On the other hand, fixing ! Ω specifies a function from T into X 2 2 — this function simply assigns the value X (!) to each t T .