doob martingale
within the
usual
terminology every doob martingale is a martingale but not vice versa. the doob martingales are exactly the uniformly integrable martingales .
doob’s martingale inequalities are a consequence of the following inequalities applied to the submartingale .
@ tony clayton: would please give us of your definition of a doob martingale. best regards
every sequence that was termed as a doob' s martingale, i was able to deduce that it was also a martingale. so here are few of my questions :
theorem (doob’s maximal inequalities) let be a filtration on probability space and let be a continuous martingale with respect to the filtration .
i have to apply azouma inequality to a doob martingale. to do this i have to prove that my doob martingale has bounded increments. wikipedia explains a tecnique which could be very useful to me in order to bound increments (see doob martingale - wikipedia, the free encyclopedia
mcdiarmid' s inequality
paragraph), but i can' t understand a passage: why if the function f has bounded increments, then the martingale b has bounded increments? could someone help me to write down a proof ?
filed under: filtrations and processes, stochastic calculus notes — george lowther @ 12: 00 pm tags: martingale, martingale inequalities, math. pr, stochastic calculus, submartingale
it is correct that doob' s inequality doesn' t apply for $ p = 1 $. in particular a non - negative martingale is bounded in $ l ^ 1 $ but it is not necessarily uniformly integrable (and hence doob' s inequality cannot be true for $ p = 1 $). the most common technique used in proving a. s. convergence (and therefore also $ l ^ 1 $ convergence) of a martingale is doob' s upcrossing inequality .
example of adapted process that is a martingale w. r. t to one filtration but not another
then, doob’s inequalities bound the distribution of the maximum of a martingale by the - norm of its terminal value, and bound the - norm of its maximum by the - norm of its terminal value for all .
doob decomposition of $ | s _ n | $ where $ s _ n $ is simple random walk .
martingale inequalities are an important subject in the study of stochastic processes. the subject of this post is doob’s inequalities which bound the distribution of the maximum value of a martingale in terms of its terminal distribution, and is a consequence of the optional sampling theorem. we work with respect to a filtered probability space. the absolute maximum process of a martingale is denoted by. for any real number, the - norm of a random variable is
now, if, we consider for, the stopping time. by using the above result to the martingale, we obtain
with the usual convention that. it is seen that is an almost surely bounded stopping time. therefore, from the doob’s stopping theorem
sorry, i thought of the inequality of moments which only holds for $ p > 1 $. just disregard my comment. also you could type up these definitions of doob' s inequality and doob' s decomposition theorem in your question. it makes it much easier to read: )
maybe it is not clear enough. i know that to prove the $ l ^ p $, $ p > 1 $, convergence of a martingale, we can use doob' s inequality in $ l ^ p $ form. however, it seems in the proof of convergence in $ l ^ 1 $, that doob' s inequality is not used (as far as i understand, it is not enough - we need uniformly integrability) .
lecture 18: examples of martingales. doob' s martingale. method of bounded differences (mcdiarmid' s inequality). applications to chernoff bounds, and balls - bins. vertex exposure and edge exposure martingales. concentration of chromatic number of g (n, p) (shamir - spencer theorem) .
there are several formly analogous inequalities in probability theory, for example, komolgorov' s inequality, doob' s inequality, and an analogous doob' s inequality in ergodic theory. they all estimate the probability or expectation of a random variable, $ | x _ n | > m $ where $ m $ is a constant. why are they useful ?
here for an integrable r. v. $ x $ and a filtration $ (\ mathcal { f } _ t) _ { t \ in t } $ the corresponding doob martingale $ d _ t $ is defined via $ d _ t: = e (x | \ mathcal { f } _ t), t \ in t $ .
i' m working on my first ever proof that a stochastic process is a martingale, and i' m a bit confused. is there a
standard machine
for these proofs ?
a doob martingale is a generic construction that is always a martingale. specifically, consider any set of random variables $ $ \ vec { x } = x _ 1, x _ 2, ..., x _ n $ $ taking values in a set $ a $ for which we are interested in the function $ f: a ^ n \ to \ bbb { r } $ and define: $ $ b _ i = e _ { x _ { i + 1 }, x _ { i + 2 }, ..., x _ { n } } [ f (\ vec { x }) | x _ { 1 }, x _ { 2 }, ... x _ { i } ] $ $ it is possible to show that $ b _ i $ is always a martingale regardless of the properties of $ x _ i $. the sequence $ \ { b _ i \ } $ is the doob martigale for $ f $ .
for the second question, in durrett' s book, it says that doobs decomposition is usefull in reducing questions about submartingales to questions about martingales. as far as i understand, if there is a question about submartingale, we can consider to use decomposition to transfer it to a problem on martingale .
about 3, well for countable extensions you need at least an infinite dimensional function $ f $ of course (or a fonctional for the uncountable case), this is also motivated by my answer to point 1. at the moment i can' t see a sufficient condition to give a real meaning to doob' s martingale in that context, but i feel that uniform integrability condition over the $ x _ i $ would probably be usefull to get limits more easily, but this by no way mandatory i think. if i can figure out an extension formally satisfactory i' ll edit the answer accordingly .
to be more specific, i am trying to show that if $ y $ is a random variable with $ e (y) < \ infty $, the process $ x _ n = e (y | i _ n) $ is a martingale (where $ i _ n $ is an arbitrary filtration) .
doob' s inequality: let $ x $ be a submartingale taking on non - negative real values, that is, for all times $ s $ and $ t $ with $ s < t $, $ e [ x _ t \ mid \ mathcal f _ s ] \ geq x _ s $. then, for any constant $ c > 0 $ and $ t > 0 $ we have
what i mean doob' s inequality in l1 is following: let x be a submartingale taking non - negative real values, that is, for all times s and t with s < t, $ \ mathbf { e } \ big [ x _ { t } \ big | \ mathcal { f } _ { s } \ big ] \ geq x _ { s }. $ then, for any constant c > 0, $ \ mathbf { p } \ left [ \ sup _ { 0 \ leq t \ leq t } x _ { t } \ geq c \ right ] \ leq \ frac { \ mathbf { e } \ big [ x _ { t } \ big ] } { c } $ .
@ tim: the notation $ e _ { (x _ { i + 1, ..., x _ n) } } $ is really misleading or confusing in my opinion (for example consider such notation in the context of a brownian motion it is really disturbing isn' t it). the proof of the fact that $ b _ i $ is martingale with respect to $ \ mathcal { f } _ i = \ sigma (x _ 1, ..., x _ i) $ is clear from the proof of lemma 2 (which just the very well known tower property of iterated conditional expectations) in the document of anupam gupta given as reference on wiki' s page. i shall try to answer your question later on .
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$ e (x _ n | i _ m) = x _ m $ for $ m < n $ .
i can do condition 2 easily enough, by substituting $ x _ n = e (y | i _ n) $ and using the tower property .
$ $ e (x _ n | i _ m) = e (e (y | i _ n) | i _ m) \ stackrel {? } { = } e (y | i _ m) = x _ m. $ $
$ e (e (y | i _ n) | i _ m) = e (y | i _ m) $ holds because of the law of iterated expectations, for which $ e (e (y | i _ n) ) = e (y) $ is just a special case .
it might also help to specify what exactly is $ i _ n $ in your question .
here' s a somewhat expanded answer, since this is a fairly straightforward question if you' ve covered the relevant definitions .
let $ m < n $, then we have that $ \ mathbb { e } (x _ n | i _ m) = \ mathbb { e } (\ mathbb { e } (y | i _ n) | i _ m) $. but from the definition of a filtration, we know that $ i _ m \ subseteq i _ n $, so the law of iterated expectations applies, and it follows that $ \ mathbb { e } (\ mathbb { e } (y | i _ n) | i _ m) = \ mathbb { e } (y | i _ m) = x _ m $, which proves 3 .
you already know that $ \ mathbb { e } (x _ n) = \ mathbb { e } (\ mathbb { e } (y | i _ n) ) = \ mathbb { e } (y) < \ infty $; as i remarked, this is a special case of the law of iterated expectations .
it remains to show that the stochastic process $ (x _ n) _ n $ is adapted to the filtration $ (i _ n) _ n $. that means we want to show that for each $ n $, $ x _ n = \ mathbb { e } (y | i _ n) $ is an $ i _ n $ - measurable random variable. depending on what your definition of the conditional expectation is, the fact that $ \ mathbb { e } (y | i _ n) $ is measurable with respect to $ i _ n $ was either part of the definition itself, or should have been one of the first properties you proved once you defined it .
thank you. see, that' s where i was getting stuck. i' m using a mish - mash of texts, and i don' t have a single clear definition of conditional expectations (so it gets confusing when responsibilities
shift
)
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$ $ \ lim _ { k \ to \ infty } \ sup _ { t \ in t } e (| x _ t | \, 1 _ { | x _ t | > k }) = 0. $ $
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let and. if then, from jensen’s inequality the process is a submartingale. let and
hi, professor baudoin. i think in the first line of the proof of the part 2, it should be p > 1 .
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the second statement above does not generalize to and, in fact, there exist martingales such that the maximum has infinite expectation at all positive times .
proof: by completing the filtration if necessary, without loss of generality we assume that it is complete. consider the first time at which the process reaches the positive value ,
which, by the debut theorem, is a stopping time. then whenever and, in particular, . optional sampling gives
bounding the right hand side above by gives the first inequality. multiply by and integrate up to some limit to get ,
finally, cancel from both sides and take the limit to get the second inequality in the statement of the theorem. ⬜
just a small comment: i think one needs to first consider a finite time grid to be able to assume that { x _ t ^ * > = k } = { \ tau < = t } (second line in proof of theorem 2) and the use mct to get the result for countable time and continuous time in the cadlag case, like you mentioned in the planetmath proof .
hi. you can do it that way, but it is not necessary to start by restricting to the finite case. i already proved the debut theorem and optional sampling for continuous - time cadlag processes. as we assume the process is cadlag, these can be applied directly to the continuous time case .
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i am working on a research problem in uncertainty propagation that involves sums of possibly dependent random variables with bounded sets of support. i am attempting to use the method of bounded differences as an amateur. in general, most presentations of this method promise to handle arbitrary dependences between random variables but all worked out examples involve independent random variables .
$ $ x _ 0 = \ left \ { \ begin { array } { cc } 1 & \ mbox { with probability } \ \ frac { 1 } { 2 } \ \ - 1 & \ mbox { with probability } \ \ frac { 1 } { 2 } \ \ \ end { array } \ right. \, , $ $
and $ x _ { i + 1 } = x _ i $ for $ i \ geq 0 $. in other words, $ x _ i $ s for $ i \ geq 1 $ are
very highly correlated
with $ x _ 0 $. : - )
we are looking at $ \ sum _ { i = 0 } ^ n x _ i $ and with probability $ \ frac { 1 } { 2 } $ each it can be either $ \ pm (n + 1) $ .
$ $ y _ i = \ mathbb { e } \ left (\ sum _ { j = 0 } ^ n x _ j \ | \ x _ 0, \ ldots, x _ { i - 1 } \ right) \, . $ $
we can show that $ $ | y _ { i + 1 } - y _ i | \ leq 1 \, . $ $
clearly i am wrong! this is the same as the chernoff hoeffding bound for the iid case! i am wondering where i went wrong. very grateful for your help !
[ 1 ] dubhashi and panconesi, concentration of measure for the analysis of randomized algorithms .
resolution: it seems to be resolved thanks to the comments below from @ alpoge. the mistake here is $ y _ 0 = \ mathbb { e } (\ sum x _ i) = 0 $. but $ y _ 1 = y _ 2 = \ cdots = y _ n = (n + 1) x _ 0 $ and therefore, when applying azuma' s theorem, the denominator will be something like
now the bounds obtained are much weaker and consistent with having $ y _ n $ being $ n + 1 $ with probability $ \ frac { 1 } { 2 } $ and $ - (n + 1) $ with probability $ \ frac { 1 } { 2 } $ .
the lesson is then when the method of bounded difference is applied to sums of dependent random variables, care should be taken to calculate $ \ mathbb { e } (x _ j | x _ 0, \ ldots, x _ { j - 1 }) $ carefully when $ x _ { j } $ could be dependent on $ x _ 0, \ ldots, x _ { j - 1 } $. it is tempting to equate this to $ \ mathbb { e } (x _ j) $ .
azuma gives you a bound on the probability that y _ n - y _ 0 is large, no? (here it' s 0. )
not quite, $ y _ 0 = 0 $ and $ y _ n = (n + 1) x _ 0 $ right? but between your previous comment that you just modified, and edgar' s comment below, i think i can see a mistake. $ y _ 1 $ is already $ (n + 1) x _ 0 $. so the difference $ | y _ { i } - y _ { i - 1 } | = 0 $ for $ i \ geq 2 $ but $ | y _ { 1 } - y _ { 0 } | = 2 (n + 1) $. the result will be different. is that the mistake ?
sorry, i keep indexing wrong! yeah it' s as i said in the comment i deleted - - - the bound on the increments doesn' t hold for y _ 1 - y _ 0. the general bound is prob. (y _ k - y _ 0 \ geq t) \ leq \ exp (- t ^ 2 / (2 \ sum _ { i = 1 } ^ k | | y _ i - y _ { i - 1 } | | _ \ infty ^ 2) ). applying this you get a denominator that is order n ^ 2, rather than order n .
( i' m certainly no expert so the bound above is probably not optimal - - - e. g. the constant in the exponential is probably suboptimal. )
if you would like, please add a solution. otherwise, i have added a resolution to my post. thanks once again! the nice thing is that this gives the same answer as another paper i am reading by jansen that uses the chromatic number of the dependence graph. so i am much happier !
what if we work it out? all of $ x _ j $ are equal to $ x _ 0 $, in particular measurable with respect to $ x _ 0 $. so $ $ \ sum _ { i = 0 } ^ n x _ i = (n + 1) x _ 0, \ \ y _ i = (n + 1) x _ 0 \ qquad \ text { for all } i \ \ y _ { i + 1 } - y _ i = 0 $ $ certainly bounded differences ...
thanks! i am actually translating a much more complicated situation into a
toy problem
that i have posted. in general, there are no other hypotheses. i still do not see where i went wrong .
except that $ y _ 0 $ is not $ (n + 1) x _ 0 $, it is the constant 0. now apply azuma' s theorem using all the differences including the very large difference $ | y _ 1 - y _ 0 | $. (this is much the same as alpoge' s comment above. )
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can the set of random variables $ \ { x _ 1, x _ 2, ..., x _ n \ } $ be generalized from finite to infinite (countably or uncountably) ?
about 1, yes $ f $ is clearly part of the definition of $ b _ i $ .
about 2, a requirement could be for $ f (x _ 1, ..., x _ n) $ to be an integrable random variable with respect to the joint law of $ (x _ 1, ..., x _ n) $, alternatively you can suppose that $ f $ is either positive (or negative) or bounded, what is important is that the conditional expectation involved make sense at every steps .
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$ $ \ mathbf { p } \ left [ \ sup _ { 0 \ leq t \ leq t } x _ { t } \ geq c \ right ] \ leq \ frac { \ mathbf { e } \ big [ x _ { t } \ big ] } { c }. $ $
$ $ \ | x _ { t } \ | _ { p } \ leq \ | s _ { t } \ | _ { p } \ leq \ frac { p } { p - 1 } \ | x _ { t } \ | _ { p }. $ $
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lecture 1: upper bound and lower bound for ramsey numbers. probabilistic and deterministic approximation for max - cut .
lecture 2: bollobas' s theorem. sum - free sets. property - b (lower bound). dominating sets in graphs with large min - degree .
lecture 3: property b (upper bound). coloring non - linear hypergraphs. dominating sets in graphs of high min - degree (probabilistic and algorithmic version) .
lecture 4: tournaments with property s _ k. maximum number of hamiltonian paths in a tournament (lower bound). local - vs - global satisfiability. balancing vectors .
lecture 5: sperner' s theorem. turan and ramsey using the deletion method .
lecture 6: gilbert - vershamov bound (greedy and probabilistic constructions). approximate sum of binomial coefficients. turan' s theorem. high girth and high chromatic number .
lecture 7: maximum number of hamiltonian paths in a tournament (upper bound). erdos - ko - rado theorem .
lecture 8: hardy - ramanujan theorem. distinct sums. g (n, p) threshold for isolated vertices .
lecture 9: g (n, p) threshold for connectivity and for appearance of k _ 4 .
lecture 10: solution of home assignments. 2 - point concentration of clique - number of g (n, p) .
lecture 11: arithmetic progressions in quasi - random integer sets. chernoff bound (special case) .
lecture 13: bennett / bernstein inequality. consistent arcs in tournaments. lower bound for relaxed ramsey problem. improved lower bound for property b .
lecture 14: applications of the the lovasz local lemma. property b. cycles in directed graphs. non - repetative strings .
lecture 15: proof of local lemma. improved bound for r (3, k). independent transversals .
lecture 16: four functions (ahlswede - daykin) theorem. kleitman' s lemma. fkg inequality. correlated events in random graphs .
lecture 17: marica - schonheim theorem. chebyshev' s association inequality. azuma' s inequality .
lecture 19: the chromatic number of g (n, 1 / 2) (bollobas' s theorem). an isoperimetric inequality on the hamming cube .
lecture 20: 4 - point concentration for the chromatic number of sparse random graphs. local - vs - global coloring (erdos' s theorem). tsp in the unit square (deterministic upper bound) .
lecture 21: near gaussian tail for stochastic tsp via method of bounded average differences. choice number of g (n, 1 / 2 )
lecture 22: janson' s inequalities. triangles in sparse random graphs. chromatic number of g (n, 1 / 2) via janson .
lecture 23: diameter of random graphs via janson' s inequality. method of non - uniform bounded differences using talagrand' s inequality. gaussian tail for stochastic tsp. improved bound for balls - bins .
lecture 24: concentration around median vs expectation. concentration of certifiable functions using talagrand' s inequality. concentration of longets increasing subsequence.
a doob martingale ( also known as a levy martingale ) is a mathematical construction of a stochastic process which approximates a given random variable and has the martingale property with respect to the given filtration . it may be thought of as the evolving sequence of best approximations to the random variable based on information accumulated up to a certain time . when analyzing sums , random walks , or other additive functions of independent random variables , one can often apply the central limit theorem , law of large numbers , chernoff 's inequality , chebyshev 's inequality or similar tools . when analyzing similar objects where the differences are not independent , the main tools are martingales and azuma 's inequality . 
a doob martingale (also known as a levy martingale) is a mathematical construction of a stochastic process which approximates a given random variable and has the martingale property with respect to the given filtration. it may be thought of as the evolving sequence of best approximations to the random variable based on information accumulated up to a certain time. when analyzing sums, random walks, or other additive functions of independent random variables, one can often apply the central limit theorem, law of large numbers, chernoff's inequality, chebyshev's inequality or similar tools. when analyzing similar objects where the differences are not independent, the main tools are martingales and azuma's inequality.
a doob martingale (also known as a levy martingale) is a mathematical construction of a stochastic process which approximates a given random variable and has the martingale property with respect to the given filtration. it may be thought of as the evolving sequence of best approximations to the random variable based on information accumulated up to a certain time. when analyzing sums, random walks, or other additive functions of independent random variables, one can often apply the central limit theorem, law of large numbers, chernoff's inequality, chebyshev's inequality or similar tools. when analyzing similar objects where the differences are not independent, the main tools are martingales and azuma's inequality.