K. Azuma. Weighted Sums of Certain Dependent Random Variables. Tohoku Math. Journ. 19 (1967) pp. 357-367.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....k#p 1 . Thus, the sequence Z 1 ,Z 2 , Z m for m =1, t is a Doob martingale (see e.g. 71] with the properties that , 9.4) 1) t for all m # 1, t . 9.5) When (9. 5) the so called Lipschitz condition, is inserted into Azuma s inequality for martingales [19, 48, 71], we obtain the tail bound # 2 2e # . 9.6) The above inequality is valid for #xed i, j and v.AsetU is good if for all but a fraction # 1 of the vertices in V , the above inequality holds for all j. Thus we keep i and j #xed and construct a new family of Bernoulli random ....

Kazuoki Azuma. Weighted sums of certain dependent random variables. Thoku Mathematical Journal, 19:357#367, 1967.

....constraint satisfaction degrees and priorities. This might lead to problems in some situations. 4) To clarify the relationship between prioritised FCSP schemes and weighted FCSP schemes. Traditionally, the concept of weight is used to indicate the importance level of an object among some objects [3,15,36,28,14,13], and so like the concept of priority it can also be used to indicate the importance level of a constraint among some constraints. Giving this, we clarify the difference between them. We organise the study in this paper according to the principles of knowledge engineering [71] when building a ....

.... (50) holds, the ordering of the new global satisfaction degrees of the two compound labels is (51) So, if before adding new constraint R we have old t x vx; 52) as long as inequality (50) holds, the ordering (52) changes to (51) 6 Posterioritised FCSPs of an object among some objects [3,15,36,28,14,13] and, so like the concept of priority, it can also be used to indicate the importance level of a constraint among some constraints. Given this, an obvious question to be asked is: what is the difference between them In this section, we clarify the relationship between the schemes for prioritised ....

K, Azuma. Weighted sums of certain dependent random variables. Tohoku Mathematical Journal, 19:357-367, 1967.

....constraint satisfaction degrees and priorities. This might lead to problems in some situations. 4) To clarify the relationship between prioritised FCSP schemes and weighted FCSP schemes. Traditionally, the concept of weight is used to indicate the importance level of an object among some objects [3,15,36,28,14,13], and so like the concept of priority it can also be used to indicate the importance level of a constraint among some constraints. Giving this, we clarify the di erence between them. We organise the study in this paper according to the principles of knowledge engineering [71] when building a ....

.... holds, the ordering of the new global satisfaction degrees of the two compound labels is (51) So, if before adding new constraint R we have (v X ) X ) 52) as long as inequality (50) holds, the ordering (52) changes to (51) 6 Posterioritised FCSPs of an object among some objects [3,15,36,28,14,13] and, so like the concept of priority, it can also be used to indicate the importance level of a constraint among some constraints. Given this, an obvious question to be asked is: what is the di erence between them In this section, we clarify the relationship between the schemes for prioritised ....

K, Azuma. Weighted sums of certain dependent random variables. Tohoku Mathematical Journal, 19:357-367, 1967.

.... Results Below we will make repeated use of the following stability result for martingale di erences, due to Algoet [2] A general account of such results can be found in [40] For bounded Z t the lemma may be deduced from standard exponential inequalities for martingale di erence sequences [26, 5]. Lemma A Let X 1 ; X 2 ; be any stochastic process, and let Z 1 ; Z 2 ; 2 R be random vectors such that, for each t 1, Z t is a measurable function of X 1 ; X t . If sup t 1 E (jZ t j) 1 where (u) u log (1 u) then Z t E(Z t j X ) 0 wp1: The ....

K. Azuma. Weighted sums of certain dependent random variables, Tohoku Math. Journal, vol.68, pp.357-367, 1967.

....We will now show that the dominating number of G is concentrated. By Lemma 1 it is enough to prove: Lemma 2 A.s. D(F ) 1 o(1) Exp(D(F ) This of course will imply that a.s. D(G) 1 o(1) Exp(D(G) To prove Lemma 2, we will make use of the following theorem of Azuma: Azuma s Inequality [2] Let 0 = X 0 ; X n be a martingale with j X i 1 X i j 1 for all 0 i n. Let 0 be arbitrary. Then P r[j X n j p n] e 2 =2 This yields the following very useful standard corollary. Corollary Let Y 1 ; Y 2 ; Y n be a sequence of random events. Let f = f(Y 1 ; Y 2 ....

K. Azuma. Weighted Sums of Certain Dependent Random Variables. Tokuku Math. Journal 19 (1967), 357 - 367.

....1.3 There are stronger versions of the Chernoff Bound (see e.g. 2] but this one is strong enough for our purposes. To deal with the case a 1 6 np it will suffice to use Pr( jBIN(n; p) Gamma npj a) Pr(jBIN(n; p) Gamma npj 1 6 np) 2e Gamma 1 108 np . Azuma s Inequality [3] Let 0 = X 0 ; X n be a martingale with j X i 1 Gamma X i j 1 for all 0 i n. Let 0 be arbitrary. Then P r[j X n j p n] e Gamma 2 =2 : This yields the following very useful standard corollary. Corollary Let Y = Y 1 ; Y 2 ; Y n be a sequence of random events. ....

K. Azuma, Weighted Sums of Certain Dependent Random Variables. Tokuku Math. Journal 19 (1967), 357 - 367.

....conditions of Theorem 1. If Q(D) 0 and if for some function 0 (n) n 1 8 Gammaffl F has no vertices of degree greater than (n) then F a.s. has no components with more than ff = dR (n) 2 log ne vertices. The following theorem of Azuma will play an important role: Azuma s Inequality [1] Let 0 = X 0 ; X n be a martingale with j X i 1 Gamma X i j 1 10 for all 0 i n. Let 0 be arbitrary. Then Pr[j X n j p n] e Gamma 2 =2 This yields the following very useful standard corollary. Corollary Let Sigma = Sigma 1 ; Sigma 2 ; Sigma n be a sequence ....

K. Azuma. Weighted Sums of Certain Dependent Random Variables. Tokuku Math. Journal 19 (1967), 357 - 367.

.... y t Gamma1 ) g(Y t j Y t Gamma1 = y t Gamma1 ) # : Now it is easy to see that T f GammaT g = T f (y n ) GammaT g (y n ) is a sum of bounded martingale differences, that is, each term Z t has zero conditional mean and range bounded by 2d t (f; g) Then the Hoeffding Azuma inequality [1] implies that, for all 0, E h e (T f GammaT g ) i exp 2 2 d(f; g) 2 : Thus, the family fT f : f 2 Fg is indeed subgaussian. Hence, recalling that R n (F) 2E [sup F T f ] and applying Proposition 4 we obtain the statement of the lemma. 2 Lemma 5 provides a sharp bound on ....

K. Azuma. Weighted sums of certain dependent random variables. Tohoku Mathematical Journal, 68:357--367, 1967.

....W using B1 edges. Define an exposure martingale Z 0 ; Z 1 ; such that Z 0 = E[Y ] Z i = E[Y j N(w 1 ) N(w i ) Zw = Y . Since the degree of all nodes is bounded by C, a node w i can connect to no more than C nodes outside W . Thus, jZ i Z i 1 j C. Using Azuma s inequality [2] we prove that for sufficiently large constant d, P fjY E[Y ]j f 8 p w C C p wg 2e f 2 128C 2 w 1=N 5 : 2 To complete the proof of the theorem, consider two nodes v and u. By applying the above lemma O(log N) times we prove that with probability 1 O( log N N 5 ) for some ....

K. Azuma. Weighted sums of certain dependent random variables. Tohoku Mathematical Journal, 19, 357-367, 1967. 7

....W using B# edges. Define an exposure martingale Z # ;Z # ; such that Z # # E#Y #, Z # # E#Y # N#w # #; N#w # ##, Z# # Y . Since the degree of all nodes is bounded by C,a node w # can connect to no more than C nodes outside W . Thus, #Z # # Z ### # Using Azuma s inequality [2] we prove that for sufficiently large constant d, P ##Y #E#Y ### f # # w C C # w###e # # # #### # # # #=N # : # To complete the proof of the theorem, consider two nodes v and u. By applying the above lemma O#### N# times we prove that with probability ##O# ### # # # #,forsome ....

K. Azuma. Weighted sums of certain dependent random variables. Tohoku Mathematical Journal, 19, 357-367, 1967.

....sum is O ffl (log n) c Gamma2 : Anticipating the proof of step 2, we record Azuma s inequality for discrete martingales. Definition 9. A martingale is a sequence X 1 ; X 2 ; Xn of real valued random variables for which E [X i 1 j X i ] X i . Theorem 11 (Azuma s Inequality, [Hoe63, Azu67]) Let X 1 ; Xn be a martingale with jX i Gamma X i Gamma1 j 1. Then Pr Theta Xn Gamma E [Xn ] p n e Gamma 2 2 : See [AS92, x7] for a general discussion of discrete martingales and a proof of Azuma s inequality. 9 Proof of Step 2. For convenience fix a specific ....

Kazuoki Azuma. Weighted sums of certain dependent random variables. T ohoku Math. J. (2), 19:357--367, 1967. 11

....follows. Below we make repeated use of a special case of the so called martingale law of large numbers. A general account of such results and a proof of the lemma below can be found in [27] The lemma may also be deduced from standard exponential inequalities for martingale di erence sequences [21, 2]. Lemma A Let X 1 ; X 2 ; be an arbitrary process taking values in X and let 1 ; 2 ; 2 IR d be random vectors such that, for each t 1, t is a measurable function of X 1 ; X t . If there is a constant L 1 such that jj t jj L with probability one for each t 1 ....

K. Azuma. Weighted sums of certain dependent random variables. Tohoku Math. Journal, vol.68, pp.357-367, 1967.

....upper and the lower bound for the path coloring problem. Our approach is similar to the one used in [13] see also [17] to calculate the tail bounds of a well known occupancy problem. We exploit the properties of special sequences of random variables called martingales, using Azuma s inequality [2] for their analysis. Similar results in a more general context are presented in [21] Consider the following process. We have a collection of n balls, of which ffn are red and (1 Gamma ff)n are black (0 ff 1) We select without replacement uniformly at random fin balls (0 fi 1) Let Omega ....

K. Azuma. Weighted Sums of Certain Dependent Random Variables. Tohoku Mathematical Journal, 19:357--367, 1967.

....inf m2M # inf f2Fm P #f# I#m# # # o #2 X m2M expf# 2 3 t 2 m g: The proof uses a well known exponential inequality for martingale di#erence sequences #see, e.g. Ledoux and Talagrand #1991#, Lemma 1.5, or Devroye, Gy#or# and Lugosi #1996#, Theorem 9.1#. This inequality is due to Azuma #1967#. Yurinski #1974# suggested a martingale representation of the norms of sums of independent random vectors and opened a way to use this type of inequalities in Probability in Banach spaces. They also found a number of applications in the local theory of Banach spaces #Milman and Schechtman ....

Azuma, K. #1967# Weighted sums of certain dependent random variables. Tokuku Math. J. 19, 357#367.

....are concentration inequalities for empirical and related processes. We are using in the current version of the results a relatively old form of these inequalities based on the extension of the classical Hoe ding type bounds to the martingale di erences. This extension is due, apparently, to Azuma [5] and it was used very successfully by Yurinskii [69] in the problems of Probability in Banach Spaces. Since then, it has been used in many other applications, including functional limit theorems and empirical processes [41] 42] local theory of Banach spaces [47] combinatorial problems on ....

K. Azuma. Weighted sums of certain dependent random variables. Tokuku Math. J., 19:357-367, 1967.

....from heuristics. For each graph G i , call C 0 its minimum cut size. Taking G i from the ensemble G(N, p) C 0 is a random variable. Following derivations now standard in a number of other stochastic combinatorial optimization problems (COPs) it is possible to show, using Azuma s inequality [1], that the distribution of C 0 becomes peaked as N # #. This means that as N becomes large, C 0 #C 0 #) #C 0 #, the relative fluctuations about the mean tend to zero. This property, often referred to as self averaging, is typical of processes to which many terms contribute. For ....

X. Azuma, Weighted sums of certain dependent random variables, Tohoku Math. J., 19 (1967), pp. 357--367.

....a lot of senses a general martingale resembles this particular set of examples. There are many inequalities estimating the probability of the deviation of f = f n from f 0 = Ef in terms of the behavior of the sequence fd i g. In the next proposition we gather some of them. i) is due to K. Azuma [5] or[52] p. 238. ii) and (iii) are due to Pisier [42] ii) was first used in [24] iv) is a generalization to the martingale case of Prokhorov s inequality. In a somewhat weaker form it first appears in [27] The form here is from [21] Proposition 5 (i) For all t 0, P (f ; jf( Gamma Ef j ....

K. Azuma, Weighted sums of certain dependent random variables, Tohoku Math. J. 19 (1967) 357--367.

....jY i Gamma Y i Gamma1 j c i : The next theorem says that when the bounded differences condition holds, the process itself does not go far away from its starting point. The theorem is known as Hoeffding Azuma s inequality and has appeared in different forms in papers by Hoeffding [Hoe63] Azuma [Azu67] Steiger [Ste67] Freedman [Fre75] and others. Theorem 11 (Hoeffding Azuma s inequality. Let Y 0 ; Y n be a martingale with respect to X 1 ; X n that satisfies the bounded differences condition with c = c 1 ; c n ) Then, for any t 0, Pr [jY n Gamma Y 0 j t] 2 ....

K. Azuma. Weighted sums of certain dependent random variables. Tokuku Math. Journal, 19:357--367, 1967.

....result implies that restricting a singleprice auction to be stable does not a ect performance by more that a constant factor. As we have seen, restricting a multiple price auction to be stable may a ect its performance by roughly a factor of log h. 4 In this section we use Azuma s Inequality [1] (see also e.g. 6] A sequence of random variables X 0 ; X k is a martingale sequence if for all i 0, E[X i jX i 1 ; X 0 ] X i 1 . Azuma s inequality is as follows. Lemma 6.1 [1] If X 0 ; X k is a martingale sequence with jX i X i 1 j c i then Pr[jX k X 0 j ] ....

....a ect its performance by roughly a factor of log h. 4 In this section we use Azuma s Inequality [1] see also e.g. 6] A sequence of random variables X 0 ; X k is a martingale sequence if for all i 0, E[X i jX i 1 ; X 0 ] X i 1 . Azuma s inequality is as follows. Lemma 6. 1 [1] If X 0 ; X k is a martingale sequence with jX i X i 1 j c i then Pr[jX k X 0 j ] 2e 2 2 P k i=1 c 2 i Recall that instead of analyzing the auction that selects a sample of size n=2, we analyze a potentially worse, but easier to analyze auction that selects each bid to be ....

K. Azuma. Weighted Sums of Certain Dependent Random Variables. Tohoku Mathematical Journal, 19:357-367, 1967.

....c. One can also obtain entirely similar bounds using Y cn more straightforward martingale arguments. In the following, we assume familiarity with basic martingale theory; see, for example, 4, Chapter 7] for more information. We use the following form of the martingale tail inequality due to Azuma [5]: Lemma 3 [Azuma] Let X 0 , X 1 , X m be a martingale sequence such that for each k, X k X k 1 # 1. Then for any # 0, Pr( X m X 0 # # m) 2e # 2 2 . Theorem 4 In the notation of Theorem 1, Pr( Y cn E[Y cn ] # # cn) 2e # 2 2 for any # 0. Proof: ....

K. Azuma, "Weighted sums of certain dependent random variables", Tohoku Mathematical Journal, Vol. 19, 1967, pp. 357-367.

....sum is bounded by (1 o(1) 4e logn : Anticipating the proof of step 2, we record Azuma s inequality for discrete martingales. Definition 9. A martingale is a sequence X 1 ; X 2 ; X n of real valued random variables for which E [X i 1 j X i ] X i . Theorem 10 (Azuma s Inequality, [Hoe63, Azu67]) Let X 1 ; X n be a martingale with jX i Gamma X i Gamma1 j 1. Then Pr Theta X n Gamma E [X n ] l p n e Gamma l 2 2 : See [AS92, x7] for a general discussion of discrete martingales and a proof of Azuma s inequality. Proof of Step 2. For convenience fix a specific ....

Kazuoki Azuma. Weighted sums of certain dependent random variables. T ohoku Math. J. (2), 19:357--367, 1967.

....[15] Let( Omega ; F ; Pr) be a probability space, and let F 0 , F 1 , F n be a filter with respect to it. Let X be any random variable over this probability space and define X i = E[XjF i ] Then, the sequence X 0 , X n is a martingale. The next theorem states Azuma s inequality [1]; a powerful tool for the analysis of martingales. Theorem 5 (Azuma s Inequality [1] Let X 0 ; X 1 ; be a martingale sequence such that for each k, jX k Gamma X k Gamma1 j c k ; where c k may depend on k. Then, for all t 0 and any 0, Pr[jX t Gamma X 0 j ] 2 exp Gamma 2 2 ....

....be a filter with respect to it. Let X be any random variable over this probability space and define X i = E[XjF i ] Then, the sequence X 0 , X n is a martingale. The next theorem states Azuma s inequality [1] a powerful tool for the analysis of martingales. Theorem 5 (Azuma s Inequality [1]) Let X 0 ; X 1 ; be a martingale sequence such that for each k, jX k Gamma X k Gamma1 j c k ; where c k may depend on k. Then, for all t 0 and any 0, Pr[jX t Gamma X 0 j ] 2 exp Gamma 2 2 P t k=1 c 2 k : Consider the following process. We have a collection C of n ....

K. Azuma. Weighted Sums of Certain Dependent Random Variables. Tohoku Mathematical Journal, 19:357--367, 1967.

.... W # 2#f(D) V D and for induced sub hypergraph H # on the set V # = V (W # # e#C e) of uncovered vertices, V # # e # V , e (k 1)# D f(e (k 1)# D) # d H # (x) # e (k 1)# D for all x # V # . The proof of Lemma 3. 1 is based on the following Azuma Hoe#uding [5, 11]) type martingale inequality. For more general inequalities and their proofs, one may refer to [3] and or [16] Lemma 3.2 Let X 1 , Xm be independent random variables with Pr[X i = 0] 1 p i and Pr[X 1 = 1] p i . For Y = Y (X 1 , Xm ) suppose Y (X 1 , X i 1 , 1, X i 1 , ....

K. Azuma, Weighted sums of certain dependent random variables, Tokuku Math. J. 19 (1967), 357-367.

....2ffi (9 Gamma (0:23 Delta 81=2) 1 Gamma 2ffl) q n(log n) 3 Gamma0:3 q n(log n) 3 0 ; and (17) follows. 2 10 3 Tools 3. 1 Azuma Hoeffding type martingale inequalities Most applications of the semirandom method involve Azuma Hoeffding type martingale inequalities (from [17] [4]) which are very useful in showing that many events can happen simultaneously. Indeed, Azuma Hoeffding type martingale inequalities, followed by Lov asz s local lemma, have become the most popular way to prove the existence of certain packings, colorings and list colorings mentioned in Section 1. ....

K. Azuma. Weighted sums of certain dependent random variables. Tokuku Math. J., 19:357--367, 1967.

.... ln f(Y t j Y t 1 = y t 1 ) g(Y t j Y t 1 = y t 1 ) # : Now it is easy to see that T f T g = T f (y n ) T g (y n ) is a sum of bounded martingale di erences, that is, each term Z t has zero conditional mean and range bounded by 2d t (f; g) Then the Hoe ding Azuma inequality [1] implies that, for all 0, E h e (T f Tg ) i exp 2 2 d(f; g) 2 : Thus, the family fT f : f 2 Fg is indeed subgaussian. Hence, recalling that R n (F) 2E [sup F T f ] and applying Proposition 4 we obtain the statement of the lemma. 2 Lemma 5 provides a sharp bound on the ....

K. Azuma. Weighted sums of certain dependent random variables. Tohoku Mathematical Journal, 68:357-367, 1967.

....1 Introduction Suppose fXm ; Fm g 1 m=0 is a discrete parameter real valued martingale with bounded jumps jXm Gamma Xm Gamma1 j a, m 2 IN, filtration Fm and such that X 0 = 0. The basic inequality for the method of bounded martingale differences is Azuma Hoeffding inequality (c.f. [1]) IPfX k xg e Gammax 2 =2ka 2 8x 0: 1) In the special case of i.i.d. differences IPfXm Gamma Xm Gamma1 = ag = 1 Gamma IPfXm Gamma Xm Gamma1 = Gammaffla= 1 Gamma ffl)g = ffl 2 (0; 1) it is easy to see that IPfX k xg exp[ GammakH(ffl (1 Gamma ffl)x= ak)jffl) where H(qjp) ....

K. Azuma (1967): Weighted sums of certain dependent random variables, Tohoku Math. J. 3, 357--367.

....variety of problems solved using this and similar techniques. We note that our result requires the application of a recent result of the authors[11] that enables us to develop algorithms from applications of the Lovasz Local Lemma( 5] see also [1] It also requires the use of Azuma s inequality[2] and a relatively new concentration inequality due to Talleygrand[15] However, in this short summary, we do not discuss these aspects. Rather we focus on a structural decomposition used in the proof and point out why p is (approximately) the boundary at which things become di cult. 2 A ....

K. Azuma, Weighted sums of certain dependent random variables. Tokuku Math. Journal 19 (1967), 357 - 367.

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K. Azuma. Weighted Sums of Certain Dependent Random Variables. Tohoku Math. Journ. 19 (1967) pp. 357-367.

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K. Azuma, "Weighted sums of certain dependent random variables," To hoku Math. J., vol. 19, pp. 357--367, 1967.

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K. Azuma. Weighted sums of certain dependent random variables. Tohoku Math. J., 19(3):357--367, 1967.

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K. Azuma. Weighted sums of certain dependent random variables. Tohoku Math. Journal, 19:357--367, 1967.

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K. Azuma, Weighted Sum of Certain Dependent random Variables, Tohoku Math.J. 3 (1967), 357-367.

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K. Azuma. Weighted sums of certain dependent random variables. Tohoku Mathematical Journal, 19:357--367, 1967.

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Azuma, K.: Weighted sums of certain dependent random variables. Tohoku Math. J. Vol. 2(19) (1967) 357--367

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K. Azuma, Weighted Sum of Certain Dependent random Variables, Tohoku Math.J. 3 (1967), 357-367.

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K. Azuma. Weighted sum of certain dependent random variables. T^ohoku Math. J. 3:357-367, 1967.

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K. Azuma,Weighted sums of certain dependent random variables, T^ohoku Math. J. 19, 357-367.

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K. Azuma. Weighted sums of certain dependent random variables. Tohoku Mathematical Journal, 68:357--367, 1967.

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K. Azuma. Weighted sum of certain dependent random variables. Tohoku Math. J. 3:357--367, 1967.

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K. Azuma. Weighted sums of certain dependent random variables. Tohoku Mathematical Journal, 19:357--367, 1967.

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K. Azuma. Weighted sums of certain dependent random variables. Tohoku Math. Journal, (19):357-367, 1967.

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K. Azuma, Weighted sums of certain dependent random variables, Tohoku Mathe- matical Journal, 37 (1967) 357-367.

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K. Azuma. Weighted sums of certain dependent random variables. Tohoku Mathematical Journal, 19:357--367, 1967.

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K. Azuma, Weighted sums of certain dependent random variables, Tohoku Mathematical Journal, 37 (1967) 357-367.

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K. Azuma, Weighted Sums of Certain Dependent Random Variables. Tokuku Math. Journal 19 (1967), 357 - 367.

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K. Azuma, Weighted sums of certain dependent random variables. Tokuku Math. Journal 19 (1967), 357 - 367.

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K. Azuma, "Weighted sums of certain dependent random variables," Tohoku Mathematical Journal, vol. 37, pp. 357--367, 1967. 14

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K. Azuma. Weighted sums of certain dependent random variables. Tohoko Mathematics Journal, 19(3), 1967. 357--367.

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X. Azuma, Weighted sums of certain dependent random variables, Tohuku Math. Journal, 19 (1967), pp. 357--367.

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X. Azuma, Weighted sums of certain dependent random variables,Tohuku Math. Journal, 19 #1967#, pp. 357#367.

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