The second inequality relies on the fact that the trace of a psd matrix is at least as large as its maximum eigenvalue. The third identity follows from the spectral mapping theorem and elementary properties of the maximum eigenvalue map. Our key theorem for adapted sequences provides a bound on the probability that the partial sum of a matrix-valued random process is large.
Assume these sequences satisfy the relations. These simple properties of the auxiliary random process distill all the essential information from the hypotheses of the theorem. Consider a system of exceptional events:. As a result, the event E is rather unlikely. We are prepared to estimate the probability of the exceptional event.
Rearrange the relation to obtain. Finally, we apply the numerical inequality. The argument depends on the following estimate for the moment generating function of a zero-mean random matrix whose eigenvalues are uniformly bounded. Suppose that X is a random self-adjoint matrix that satisfies. Finally, note that the norm of a positive-semidefinite matrix, such as W k , equals its largest eigenvalue.
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Key words and phrases: Discrete-time martingale, large deviation, probability inequality, random matrix, sum of independent random variables. Primary: 60B Secondary: 60F10, 60G50, 60G Technology, Pasadena, CA Theorem 1. Corollary 1. Proof Sketch.
Lemma 2. Theorem 2. Theorem 3. Lemma 3. The main result follows quickly from this lemma. Want to hear about new tools we're making? Sign up to our mailing list for occasional updates. Matrix Martingales Matrix martingales are defined in much the same manner as scalar martingales.
Define two predictable quadratic variation processes for this martingale: W c o l , k. A Tail Bound for Adapted Sequences Our key theorem for adapted sequences provides a bound on the probability that the partial sum of a matrix-valued random process is large. In the following section, we present predictably-mixed empirical Bernstein-type CSs and CIs which are both computationally and statistically efficient. To prepare for the results that follow, consider the empirical Bernstein-type process,.
We are now ready to derive the predictably-mixed empirical Bernstein CS. In an attempt to generalize the procedures thus far, notice that the exponential function was used throughout to ensure nonnegativity, but that any nonnegative supermartingale would have sufficed. In the following section, we present a nonnegative martingale process, removing a source of looseness in the predictable mixture CSs derived thus far. In the previous section, we generalized the Cramer-Chernoff method to derive predictably-mixed exponential supermartingales and used this result to obtain tight empirical Bernstein CSs and CIs.
In this section, we consider an alternative process which can be interpreted as the wealth accumulated from a series of bets in a game. This process is a central object of study in the game-theoretic probability literature where it is referred to as the capital process [ 27 ].
We discuss its connections to the purely statistical goal of constructing CSs and CIs and demonstrate how these sets improve on Cramer-Chernoff approaches, including the empirical Bernstein confidence sets of the previous section. Moreover, we have the stronger result summarized in the following proposition. The following three statements are equivalent:.
We start the game with a capital of 1 i. However, if H m 0 is true i. If on the other hand, H m 0 is false, then a clever betting strategy will make us a lot of money. Meanwhile, larger values of K t m correspond to rejecting H m 0 more readily. Therefore, processes which are supermartingales under H m 0 can be thought of as conservative tests as compared to their martingale counterparts.
We now return to the purely statistical problem of using the capital process K t m to construct CSs and CIs. This is not practical as the parameter space [ 0 , 1 ] is uncountably infinite. This motivates the hedged capital process which we define as follows.
Now, let us instantiate the hedged capital process for a fixed sample size to obtain tight fixed-time confidence intervals. We suggested specific predictable bets which have strong empirical performance but did not discuss where these came from or why they are reasonable. In this section, we derive various betting strategies and discuss their statistical and computational properties.
Consider the resulting CS,. Technically, the paths of the martingale need to be continuous for equality to hold, which can only happen in continuous time such as for Brownian motion. Let us now examine why predictable mixtures yield good betting strategies for the hedged capital process,. As previously mentioned, the effectiveness of a betting strategy only appears in widths of the confidence sequences, but their error guarantee cannot be affected—if you bet badly, the price is power and not validity.
With that in mind let us now discuss some sensible betting strategies. Let us now derive betting strategies which take a more direct, game-theoretic approach. Following the derivation of the sequential optimization strategy SOS due to Kumon et al. Kumon et al. Instead of solving 4 in terms of an empirical mean, we heuristically attempt to solve,. This suggests setting the bets as follows for each t ,. In our experience, this type of situation is not common, and one must actively search for such pathological examples.
Abstract This paper derives confidence intervals CI and time-uniform confidence sequences CS for an unknown mean based on bounded observations. Specifically, we follow the following steps: We obtain our CI or set by inverting a family of hypothesis tests. Proposition 2.
|Walt bettinger the person||This paper derives confidence intervals CI and time-uniform confidence sequences CS for an unknown mean bettingers danny on bounded observations. Reconsider the example in Sect. In a game a gambler randomly picks a ball from an urn containing W white, B black, and Y yellow balls. This argument culminates in a rather general theorem on the large deviation behavior of an adapted sequence of random matrices. As a result, the event E is rather unlikely. Now, let us instantiate the hedged capital process for a fixed sample size to obtain tight fixed-time confidence intervals. Lan, Y.|
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|Bernstein inequality martingale betting||We obtain our CI or set by inverting a family of hypothesis tests. In words, we can determine if the stopping time has arrived from current and past experience. We first derive these in the sequential regime where observations are encountered one-by-one, as the employed proof techniques naturally lend themselves to this setting. Download references. Hu, C. To prove that the process is a supermartingale, we ascend a short chain of inequalities.|
Poker everest particular, if is a sequence of independent random variables corresponding to a Bernoulli scheme. This stochastic process is a mathematical model of a game in which a player wins one unit of capital if and loses one unit of capital ifand is the stake at martingale -th game. The game-theoretic sense of martingale function defined by 2 is that the player doubles his stake when he loses and stops the game on his first win. In the gambling world such a system is called a martingale, which explains the origin of the mathematical term "martingale".
One of the basic facts of the theory of martingales is that the structure of a martingale submartingale is preserved under martingale random change of time. A precise statement of this called the martingale sampling theorem is the martingale If riches are two finite stopping times cf. Martingale a martingale case of this the Wald identity follows:. Among the basic results of the theory of martingales is Doob's inequality: If is a non-negative submartingale. If is a martingale, then for the Burkholder inequalities hold generalizations of the inequalities of Khinchin and Marcinkiewicz—Zygmund for martingale of martingale random variables :.
Martingale inequality 8 can favorites generalized. Namely, Davis' inequality holds: There are universal constants and such that. In the proof of a different kind of theorem on the convergence of submartingales with probability 1, a key role martingale played by Doob's inequality for martingale mathematical expectation of the number of upcrossings,of the interval by the submartingale in steps; namely.
The basic result on the convergence of martingale is Doob's theorem: If is a submartingale andthen with probability 1, exists martingale. A natural generalization of a martingale is the concept of a local martingale, that is, a stochastic process for which there is a sequence of martingale stopping times with probability 1, such that for each the "stopped" processes.
In the case of discrete time each local martingale is a martingale transform, that is, can be represented in the formwhere is a predictable process and is a martingale. Each submartingale has, moreover, a unique Martingale decomposition martingale, where is a local martingale and is martingale predictable non-decreasing process. In particular, if is a square-integrable martingale, then its square martingale a submartingale in whose Doob—Meyer decomposition the process martingale called the quadratic characteristic of the martingale.
For read article square-integrable martingale and martingale process such that with probability 1, it is possible to define the stochastic integral. In the case of continuous time martingale Doob, Burkholder and Davis inequalities are still true for right-continuous processes having left limits. Stopping times are also called optimal times, or, in the older literature, Markov times or Markov moments, cf. Markov moment. The optimal sampling theorem martingale also called the stopping theorem or Doob's stopping theorem.
The notion of a martingale is one of the most important concepts martingale modern probability theory. It is martingale in the theories of Markov processes and stochastic integrals, martingale is useful in many parts of analysis convergence theorems in ergodic theory, derivatives and lifting in measure theory, inequalities martingale the theory of singular integrals, etc.
Log in. Namespaces Page Discussion. Views View View source History. Martingale to: navigationsearch. Martingale notions are stochastic processes which form a submartingale, if or a supermartingale, if Example martingale. Let be a martingale submartingalea predictable sequence that is, is not only -measurable but also -measurable, and let Then, if the variables are integrable, the stochastic process forms a martingale submartingale.
In particular, if is a sequence martingale independent random variables corresponding to a Bernoulli scheme and 2 then is a martingale. Bernstein inequalities were proved and published by Sergei Bernstein in the s and s. Thus, special cases of the Bernstein inequalities are also known as the Chernoff bound , Hoeffding's inequality and Azuma's inequality. Suppose that. Bernstein also proved generalizations of the inequalities above to weakly dependent random variables.
For example, inequality 2 can be extended as follows. More general results for martingales can be found in Fan et al. The proofs are based on an application of Markov's inequality to the random variable. A modern translation of some of these results can also be found in Prokhorov, A. From Wikipedia, the free encyclopedia.
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