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== Formal statement == |
== Formal statement == |
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Let <math>(X_n)_{n \geq 1}</math> be a sequence of i.i.d. random variables whose moment generating function <math>M: |
Let <math>(X_n)_{n \geq 1}</math> be a sequence of i.i.d. random variables whose moment generating function <math>M: t \mapsto \mathbb{E}(e^{t X_1})</math> is finite for some <math>\theta > 0</math>, and let <math>S_n = X_1 + \cdots + X_n</math>, with <math>S_0 = 1</math>. Then, the process <math>(W_n)_{n \geq 0}</math> defined by |
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:<math>W_n = \frac{e^{\theta S_n}}{M(\theta)^n}</math> |
:<math>W_n = \frac{e^{\theta S_n}}{M(\theta)^n}</math> |
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is a martingale known as ''Wald's martingale''.<ref>{{cite web |url=https://ocw.mit.edu/courses/15-070j-advanced-stochastic-processes-fall-2013 |title= Advanced Stochastic Processes, Lecture 10 |last=Gamarnik |first=David |date=2013 |website=MIT OpenCourseWare |access-date=24 June 2023}}</ref> In particular, <math>\mathbb{E}(W_n) = 1</math> for all <math>n \geq 0</math>. |
is a martingale known as ''Wald's martingale''.<ref>{{cite web |url=https://ocw.mit.edu/courses/15-070j-advanced-stochastic-processes-fall-2013 |title= Advanced Stochastic Processes, Lecture 10 |last=Gamarnik |first=David |date=2013 |website=MIT OpenCourseWare |access-date=24 June 2023}}</ref> In particular, <math>\mathbb{E}(W_n) = 1</math> for all <math>n \geq 0</math>. |
Revision as of 17:47, 10 December 2023
In probability theory, Wald's martingale is the name sometimes given to a martingale used to study sums of i.i.d. random variables. It is named after the mathematician Abraham Wald, who used these ideas in a series of influential publications.[1][2][3]
Wald's martingale can be seen as discrete-time equivalent of the Doléans-Dade exponential.
Formal statement
Let be a sequence of i.i.d. random variables whose moment generating function is finite for some , and let , with . Then, the process defined by
is a martingale known as Wald's martingale.[4] In particular, for all .
See also
Notes
- ^ Wald, Abraham (1944). "On cumulative sums of random variables". Ann. Math. Stat. 15 (3): 283–296. doi:10.1214/aoms/1177731235.
- ^ Wald, Abraham (1945). "Sequential tests of statistical hypotheses". Ann. Math. Stat. 16 (2): 117–186. doi:10.1214/aoms/1177731118.
- ^ Wald, Abraham (1945). Sequential analysis (1st ed.). John Wiley and Sons.
- ^ Gamarnik, David (2013). "Advanced Stochastic Processes, Lecture 10". MIT OpenCourseWare. Retrieved 24 June 2023.