Does balls to the wall mean full speed ahead or full speed ahead and nosedive? Furthermore, if r > s 1, convergence in r-th mean implies convergence in s-th mean. When would I give a checkpoint to my D&D party that they can return to if they die? for every number n \lim_{n \rightarrow \infty} F_{X_n}(c+\frac{\epsilon}{2})=1. The third section discusses the convergence in distribution of random variables. Browse other questions tagged, Start here for a quick overview of the site, Detailed answers to any questions you might have, Discuss the workings and policies of this site, Learn more about Stack Overflow the company, $$\begin{split}X_n-Y_n&\overset p {\rightarrow} 0\\ &\le P(|X_n-Y_n|>\frac \epsilon 2)+P(|Y_n-Z|> \frac \epsilon 2)\text { definition of union} A sequence {Xn} of random variables converges in probability towards the random variable X if for all > 0. Almost Sure Convergence. Investigating the sequence of the random variables in probability is often called with different names like "large sample theory", "asymptomatic theory" and even "limit theory". R A sequence of random variables { Xn } is called convergent almost surely to a random variable X if sequence of random variables { Xn } is called convergent surely to a random variable X if Relationships between Various Modes of Convergence There are a few important connections between these modes of convergence. Also for any random mapping ? The definition of convergence in distribution may be extended from random vectors to more general random elements in arbitrary metric spaces, and even to the random variables which are not measurable a situation which occurs for example in the study of empirical processes. Making statements based on opinion; back them up with references or personal experience. {\displaystyle X_{n}\,{\xrightarrow {d}}\,{\mathcal {N}}(0,\,1)} Making statements based on opinion; back them up with references or personal experience. In sum, a sequence of random variables is in fact a sequence of functions $X_{n}: S \rightarrow \mathbb{R}$. {\displaystyle X_{1},X_{2},\ldots } There is no confusion here. Y_n&\overset p {\rightarrow} Z\end{split}$$, $lim_{n\rightarrow\infty}P(|X_n-Z|>\epsilon)\le0$, Thank you - How does the first equality hold? From the standard literature it is well known that for sequences of random variables X 1, n P X 1 and X 2, n P X 2 as n it holds that ( X 1, n, X 2, n) P ( X 1, X 2) for n . Pr For example, let $X_1$, $X_2$, $X_3$, $\cdots$ be a sequence of i.i.d. De nition Let X n be a sequence of random vectors. 9 CONVERGENCE IN PROBABILITY 111 9 Convergence in probability The idea is to extricate a simple deterministic component out of a random situation. Is it cheating if the proctor gives a student the answer key by mistake and the student doesn't report it? Question in general case The purpose of this course is to introduce students to the history and evolution of computers and their generations. where the operator E denotes the expected value. Add a new light switch in line with another switch? There are several dierent modes of convergence. For r = 2 this is called mean-square convergence and is denoted by X n m. s. X. We can write for any $\epsilon>0$, In general, convergence will be to some limiting random variable. rev2022.12.9.43105. Convergence in distribution, probability, and 2nd mean Is this an at-all realistic configuration for a DHC-2 Beaver? This is the type of stochastic convergence that is most similar to pointwise convergence known from elementary real analysis. ) In this case, the . maximum of an asymptotically almost negatively associated (AANA) family of random variables. The convergence of sequences of random variables to some limit random variable is an important concept in probability theory, and its applications to statistics and stochastic processes. Stopped Brownian motion is an example of a martingale. Depeding on RVs you have different types of converging. We study conditions of the asymptotic normality of the number of repetitions (pairs of equal values) in a segment of strict sense stationary random sequence of values from {1, 2, , N} satisfying the strong uniform mixing condition.It is shown that under natural conditions for the number of repetitions to be asymptotically normal as the length of the segment tends to infinity it is . where sequences of random variables and sequences of real numbers respectively dened over a Banach space via deferred Nrlund summability mean. Some of the topics discussed in this course are basic concepts of information technology, hardware and computer programming, computer memory, data representation, number systems, operating systems, computer networks and the Internet, databases, computer ethics, algorithms . As it only depends on the cdf of the sequence of random variables and the limiting random variable, it does not require any dependence between the two. To subscribe to this RSS feed, copy and paste this URL into your RSS reader. The convergence of sequences of random variables to some limit random variable is an important concept in probability theory, and its applications to statistics and stochastic processes. Then we have that the k-point correlation functions kN are bounded in L p (([1, 1])k ) for all k and N N large enough and hence, if p > 1, there exists a subsequence k j k weakly in L p (( . By clicking Accept all cookies, you agree Stack Exchange can store cookies on your device and disclose information in accordance with our Cookie Policy. \end{align} Then the { X i ( ) } is a sequence of real value numbers. Q: Compute the amount of work done by the force field F(x, y, z) = (x z, ln y, xz) in moving an ( , is said to converge in distribution, or converge weakly, or converge in law to a random variable X with cumulative distribution function F if. It only takes a minute to sign up. p n 1 n; with prob. @JDoe2 The first equality was actually not necessary, here is an updated proof. , 1 Connect and share knowledge within a single location that is structured and easy to search. Hence, convergence in mean square implies convergence in mean. Is it true then that: $$\lim_{n\rightarrow\infty}\mathbb{P}[|X_n-Y_n|>\epsilon]=0 \text{ implies } X_n\xrightarrow{p}Y$$, Assume that (where I conveniently replaced Y with Z) Can a prospective pilot be negated their certification because of too big/small hands? \lim_{n \rightarrow \infty} F_{X_n}(c-\epsilon)=0,\\ The first part looks ok, but I would apply central limit theorem, not the law of large number. , ( Convergence in distribution is the weakest form of convergence typically discussed, since it is implied by all other types of convergence mentioned in this article. \end{align} In particular, for a sequence X 1, X 2, X 3, to converge to a random variable X, we must have that P ( | X n X | ) goes to 0 as n , for any > 0. Multiple sequences of random variables that converge in probabilty, Continuity and convergence in probability, two sequences case, Convergence of random variables, convergence in probability/a.s./$L^p$. That is, the sequence $X_1$, $X_2$, $X_3$, $\cdots$ converges in probability to the zero random variable $X$. Convergence of Random Variables John Duchi Stats 300b { Winter Quarter 2021 Convergence of Random Variables 1{1. . This type of convergence is often denoted by adding the letter Lr over an arrow indicating convergence: The most important cases of convergence in r-th mean are: Convergence in the r-th mean, for r 1, implies convergence in probability (by Markov's inequality). 1 p n; n 1; be . Figure 7.4 summarizes how these types of convergence are related. As we mentioned previously, convergence in probability is stronger than convergence in distribution. Exercise 5.7 | Convergence in probability In particular, if an event implies that at least one of two other events has occurred, this means that $A\subset B\cup C$, i.e. {\displaystyle \scriptstyle {\mathcal {L}}_{X}} rev2022.12.9.43105. for all continuous bounded functions h.[2] Here E* denotes the outer expectation, that is the expectation of a smallest measurable function g that dominates h(Xn). ( Your mistake is taking limits of random variables. There is another version of the law of large numbers that is called the strong law of large numbers (SLLN). An alternating minimization algorithm for computing the quantity is presented; this algorithm is based on a training sequence and in turn gives rise to a design algorithm for variable-rate trellis source codes. (i) Show that convergence in probability implies convergence in distri-bution, that is, if n!P , then n!d . X All the material I read using $X_i, i=1:n$ to denote a sequence of random variables. X n are random. Suppose sequence of random variables (X n) converges to Xin distribution and sequence of random . L at which F is continuous. \begin{align}%\label{} X_{n}\left(s_{i}\right)=x_{n i}, \quad \text { for } i=1,2, \cdots, k For example, if you take a look at this post: To learn more, see our tips on writing great answers. It only takes a minute to sign up. If we toss 10 times, each time it is a random variable of outcome 0 or 1. In this figure, the stronger types of convergence are on top and, as we move to the bottom, the convergence becomes weaker. For example, if X is standard normal we can write By this, we mean the following: If Type A convergence is stronger than Type B convergence, it means that Type A convergence implies Type B convergence. Why would Henry want to close the breach? Then Xn is said to converge in probability to X if for any > 0 and any >0 there exists a number N (which may depend on and ) such that for all nN, Pn()< (the definition of limit). P\big(|X_n-X| \geq \epsilon \big)&=P\big(|Y_n| \geq \epsilon \big)\\ 173-188 On the rates of convergencein weak limit theorems for geometric random sum and the concept of the random variable as a function from to R, this is equivalent to the statement. A sequence of random variables X1, X2, X3, converges almost surely to a random variable X, shown by Xn a. s. X, if P({s S: lim n Xn(s) = X(s)}) = 1. 2 I am a bit confused when studying the convergence of random variables. Here, we would like to provide definitions of different types of convergence and discuss how they are related. &=\lim_{n \rightarrow \infty} F_{X_n}(c-\epsilon) + \lim_{n \rightarrow \infty} P\big(X_n \geq c+\epsilon \big)\\ F Consider a sequence of random variables X 1, X 2, X 3, , i.e, { X n, n N }. $Bernoulli\left(\frac{1}{2}\right)$ random variables. This sequence might ''converge'' to a random variable X. What is this fallacy: Perfection is impossible, therefore imperfection should be overlooked, Examples of frauds discovered because someone tried to mimic a random sequence. &=\lim_{n \rightarrow \infty} P\big(X_n \leq c-\epsilon \big) + \lim_{n \rightarrow \infty} P\big(X_n \geq c+\epsilon \big)\\ $$ For example, if we toss a coin once, the sample space is $\{tail = 0, head = 1\}$ and the outcome is 0 or 1. Does integrating PDOS give total charge of a system? Should teachers encourage good students to help weaker ones? Then, a random variable $X$ is a mapping that assigns a real number to any of the possible outcomes $s_{i}, i=1,2, \cdots, k .$ Thus, we may write If {/in} is a sequence of Books that explain fundamental chess concepts. $Y_n\xrightarrow{p}Y$. MIT RES.6-012 Introduction to Probability, Spring 2018View the complete course: https://ocw.mit.edu/RES-6-012S18Instructor: John TsitsiklisLicense: Creative . {\displaystyle (\Omega ,{\mathcal {F}},\operatorname {Pr} )} , This quantum martingale convergence theorem is of particular interest since it exhibits non-classical behaviour; even though the limit of the martingale exists and is unique, it is not explicitly identifiable. Use MathJax to format equations. Based on the theory, a random variable is a function mapping the event from the sample space to the real line, in which the outcome is a real value number. To say that the sequence of random variables ( Xn) defined over the same probability space (i.e., a random process) converges surely or everywhere or pointwise towards X means where is the sample space of the underlying probability space over which the random variables are defined. For example, using the figure, we conclude that if a sequence of random variables converges in probability to a random variable $X$, then the sequence converges in distribution to $X$ as well. for every A Rk which is a continuity set of X. For a fixed r 1, a sequence of random variables X i is said to converge to X in the r t h mean or in the L r norm if lim n E [ | X n X | r] = 0. of real-valued random variables, with cumulative distribution functions The difference between the two only exists on sets with probability zero. We prove the strong law of large numbers, which is one of the fundamental limit theorems of probability theory. Cross Validated is a question and answer site for people interested in statistics, machine learning, data analysis, data mining, and data visualization. By clicking Post Your Answer, you agree to our terms of service, privacy policy and cookie policy. We prove a quantum analogue of Lebesgue's dominated convergence theorem and use it to prove a quantum martingale convergence theorem. We say that this sequence converges in distribution to a random k-vector X if. Y n p Y. Does integrating PDOS give total charge of a system? In particular, we introduce and discuss the convergence in probability of a sequence of random variables. The print version of the book is available through Amazon here. &=0 , \qquad \textrm{ for all }\epsilon>0. Definition. I think my confusion is $\{X_i\}$ is a sequence of random variables, and $\{Y_i\}$ given by $Y_n=\frac{\sum_{i=1}^n X_i}{n}$ is also a sequence of random variables. The following contents are just copy-paste from: Sequence of Random Variables. X_{n}\left(s_{i}\right)=x_{n i}, \quad \text { for } i=1,2, \cdots, k \end{align} Received a 'behavior reminder' from manager. Site design / logo 2022 Stack Exchange Inc; user contributions licensed under CC BY-SA. Consider a man who tosses seven coins every morning. Then the $\{X_i(\omega)\}$ is a sequence of real value numbers. How does legislative oversight work in Switzerland when there is technically no "opposition" in parliament? \begin{align}%\label{eq:union-bound} These other types of patterns that may arise are reflected in the different types of stochastic convergence that have been studied. That is, suppose that n (Y n ) converged in distribution to cdf F? The conventional method assumes the channel is the same within the training sequence periodicity . Example. Example. How does legislative oversight work in Switzerland when there is technically no "opposition" in parliament? \end{align}. They are, using the arrow notation: These properties, together with a number of other special cases, are summarized in the following list: This article incorporates material from the Citizendium article "Stochastic convergence", which is licensed under the Creative Commons Attribution-ShareAlike 3.0 Unported License but not under the GFDL. The outcome from tossing any of them will follow a distribution markedly different from the desired, Consider the following experiment. Using the probability space In this section, we will develop the theoretical background to study the convergence of a sequence of random variables in more detail. \overline{X}_n=\frac{X_1+X_2++X_n}{n} On the other hand, the sequence does not converge in mean to 0 (nor to any other constant). This sequence might ''converge'' to a random variable $X$. The best answers are voted up and rise to the top, Not the answer you're looking for? Thus, the best linear estimator of (X, f) given Y can be written as the corresponding weighted sum of linear estimators: (MMSE estimator of (X, f) given Y) = X i i (Y, i)(f, i) i + 2. Now, for any $\epsilon>0$, we have Connect and share knowledge within a single location that is structured and easy to search. CGAC2022 Day 10: Help Santa sort presents! The training sequence, also called block-type pilots, allows for tracking only channel frequency variations (slow fading channel) due to the one-dimensional (1D) periodicity, estimating the channel response at each subcarrier. Let $\{X_n\}$ and $\{Y_n\}$ be sequences of variables and suppose that $Y_n$ converges in probability to some random variable $Y$, i.e. Sequence of random variables by Marco Taboga, PhD One of the central topics in probability theory and statistics is the study of sequences of random variables, that is, of sequences whose generic element is a random variable . In the simplest case, an asymptotic distribution exists if the probability distribution of Z i converges to a probability distribution (the asymptotic distribution) as i increases: see convergence in distribution.A special case of an asymptotic distribution is when the sequence of . In probability theory, there exist several different notions of convergence of random variables. There are four types of convergence that we will discuss in this section: These are all different kinds of convergence. X However, this random variable might be a constant, so it also makes sense to talk about convergence to a real number. The second set of experiments shows the . \begin{align}%\label{} & \leq \frac{\mathrm{Var}(Y_n)}{\left(\epsilon-\frac{1}{n} \right)^2} &\textrm{(by Chebyshev's inequality)}\\ Is it possible to hide or delete the new Toolbar in 13.1? This is the weak convergence of laws without laws being defined except asymptotically. n de ne convergence in probability, verify whether a given sequence of random variables converges in probability; explain the relation between convergence in Lr and convergence in probability (Lem 2.8); state and apply the su cient condition for convergence in L2 (Thm 2.10); de ne almost sure convergence, verify whether a given sequence of random . , Sure convergence of a random variable implies all the other kinds of convergence stated above, but there is no payoff in probability theory by using sure convergence compared to using almost sure convergence. You could have 10 heads in a row, but as $n \rightarrow \infty$ then $Y_n \rightarrow 0.5$. Use MathJax to format equations. To learn more, see our tips on writing great answers. \end{align} Are defenders behind an arrow slit attackable? Typesetting Malayalam in xelatex & lualatex gives error, Counterexamples to differentiation under integral sign, revisited. The lower bound of the probability of the lim sup has to be justified (portmanteau theorem). (for a constant c), then n!P . , ) \begin{align}%\label{eq:union-bound} ( Convergence in probability is also the type of convergence established by the weak law of large numbers. Choosing $a=Y_n-EY_n$ and $b=EY_n$, we obtain Several results will be established using the portmanteau lemma: A sequence { Xn } converges in distribution to X if and only if any of the following conditions are met: \lim_{n \rightarrow \infty} P\big(|X_n-0| \geq \epsilon \big) &=\lim_{n \rightarrow \infty} P\big(X_n \geq \epsilon \big) & (\textrm{ since $X_n\geq 0$ })\\ To say that $X_n$ converges in probability to $X$, we write. Convergence in probability implies convergence in distribution. Browse other questions tagged, Start here for a quick overview of the site, Detailed answers to any questions you might have, Discuss the workings and policies of this site, Learn more about Stack Overflow the company, You took a wrong turn at the end of the first paragraph where you wrote "there is no confusion here": $(X_i)$ is a sequence of real valued. where $\sigma>0$ is a constant. Then for Xn to converge in probability to X there should exist a number N such that for all n N the probability Pn is less than . The converse is not necessarily true. It is called the "weak" law because it refers to convergence in probability. Meanwhile, we will prove that each continuous function of every sequence convergent in probability sequence is convergent in probability too. The obtained result is applied to characterize the Kolmogorov-Feller weak law of large numbers for these sequences. All experiments were repeated ve times before reporting an average. This is why the concept of sure convergence of random variables is very rarely used. \end{align}. What is this fallacy: Perfection is impossible, therefore imperfection should be overlooked. Let n= 1 n;with prob. $$\begin{split}P(|X_n-Z|>\epsilon)&\le P(|X_n-Y_n|>\frac \epsilon 2\cup|Y_n-Z|>\frac \epsilon 2)\text { what we just said}\\ We proved WLLN in Section 7.1.1. 2 {\displaystyle x\in \mathbb {R} } Convergence in probability of a random variable - YouTube This video provides an explanation of what is meant by convergence in probability of a random variable. We need a concept of convergence for measures on jf?l. What is the probability that the number rolled is a "1" OR A: Given that ,you roll a special 46-sided die. "Stochastic convergence" formalizes the idea that a sequence of essentially random or unpredictable events can sometimes be expected to settle into a pattern. X For simplicity, suppose that our sample space consists of a finite number of elements, i.e., We begin with convergence in probability. X\left(s_{i}\right)=x_{i}, \quad \text { for } i=1,2, \cdots, k Site design / logo 2022 Stack Exchange Inc; user contributions licensed under CC BY-SA. We are interested in the behavior of a statistic as the sample size goes to innity. Why did the Council of Elrond debate hiding or sending the Ring away, if Sauron wins eventually in that scenario? Moreover, based upon our proposed methods, we have proved a new Korovkin-type approximation theorem with alge- &=0 \hspace{140pt} (\textrm{since } \lim_{n \rightarrow \infty} F_{X_n}(c+\frac{\epsilon}{2})=1). Ph.D. student in Electrical Engineering at Texas A&M University, with a focus on Wireless Communications. How to connect 2 VMware instance running on same Linux host machine via emulated ethernet cable (accessible via mac address)? Other forms of convergence are important in other useful theorems, including the central limit theorem. None of the above statements are true for convergence in distribution. Convergence of sequences of random variables Throughout this chapter we assume that fX 1;X 2;:::gis a sequence of r.v. Then when $n\rightarrow \infty$, it converge to a function $X$? $P(A)\le P(B\cup C)$. &= 0 + \lim_{n \rightarrow \infty} P\big(X_n \geq c+\epsilon \big) \hspace{50pt} (\textrm{since } \lim_{n \rightarrow \infty} F_{X_n}(c-\epsilon)=0)\\ For your example you can take $Y_n = \frac{1}{n}\sum_{k=1}^{n}X_k$ and it should converge to 0.5. Let, Suppose that a random number generator generates a pseudorandom floating point number between 0 and 1. 1 , convergence almost surely is defined similarly: To say that the sequence of random variables (Xn) defined over the same probability space (i.e., a random process) converges surely or everywhere or pointwise towards X means. Definition 17 (Convergence almost surely) { xn } convergesalmost surely (with probability 1)to a random variable x if for any , > 0 there exists n0 (, ) such that. Then, $X_n \ \xrightarrow{d}\ X$. {\displaystyle F_{1},F_{2},\ldots } How did muzzle-loaded rifled artillery solve the problems of the hand-held rifle? a. Let $X_n \sim Exponential(n)$, show that $ X_n \ \xrightarrow{p}\ 0$. vergence of a sequence of random variables as the weak- convergence of a . Developing algorithms for compression of parameters of Deep Neural Networks in . Synonyms A sequence of random variables is also often called a random sequence or a stochastic process . The CLT states that the normalized average of a sequence of i.i.d. MathJax reference. First, pick a random person in the street. As I understand this. EY_n=\frac{1}{n}, \qquad \mathrm{Var}(Y_n)=\frac{\sigma^2}{n}, To subscribe to this RSS feed, copy and paste this URL into your RSS reader. Notice that for the condition to be satisfied, it is not possible that for each n the random variables X and Xn are independent (and thus convergence in probability is a condition on the joint cdf's, as opposed to convergence in distribution, which is a condition on the individual cdf's), unless X is deterministic like for the weak law of large numbers. About. $$ and Xis a . But, what does 'convergence to a number close to X' mean? \lim_{n \rightarrow \infty} P\big(|X_n-c| \geq \epsilon \big) &= \lim_{n \rightarrow \infty} \bigg[P\big(X_n \leq c-\epsilon \big) + P\big(X_n \geq c+\epsilon \big)\bigg]\\ Note that although we talk of a sequence of random variables converging in distribution, it is really the cdfs that converge, not the random variables. that is, the random variable n(1X(n)) converges in distribution to an exponential(1) random variable. The requirement that only the continuity points of F should be considered is essential. 0 S=\left\{s_{1}, s_{2}, \cdots, s_{k}\right\} , Some of these convergence types are ''stronger'' than others and some are ''weaker.'' 2 Convergence of Random Variables The nal topic of probability theory in this course is the convergence of random variables, which plays a key role in asymptotic statistical inference. Xn a. s. X. Why is it so much harder to run on a treadmill when not holding the handlebars? Unless $X_i$ is the toss of $i=1n$ times in one experiment with underlying sample space $2^i$, then define a sequence of random variables the number of head counts in $i=1n$ so that $X_n\rightarrow X$ in probability. . Minor critique: The expression $$ X_n \rightarrow Y_n $$ does not really make sense; when we talk about limits, we do not want the RHS to depend on n. However, $$X_n - Y_n \rightarrow 0 $$ does make sense, and that is essentially what is being used. (Also, for OP, you if you know that $$ X_n + Y_n \rightarrow X + Y $$, you can use that to prove the claim as well, and the proof of this claim is also essentially the proof given to you in the answer above), Convergence in probability for two sequences of random variables, Help us identify new roles for community members, Convergence in probability of product and division of two random variables, Exchange of sequences of probability variables. Bracers of armor Vs incorporeal touch attack. Reversing the logic, this means that $|X_n-Z|>\epsilon$ implies that $|X_n-Y_n|>\frac \epsilon 2$ (inclusive) or $|Y_n-Z|>\frac \epsilon 2$. &= 1-\lim_{n \rightarrow \infty} F_{X_n}(c+\frac{\epsilon}{2})\\ Convergence in probability is denoted by adding the letter p over an arrow indicating convergence, or using the "plim" probability limit operator: For random elements {Xn} on a separable metric space (S, d), convergence in probability is defined similarly by[6]. All the material I read using X i, i = 1: n to denote a sequence of random variables. We will demonstrate later that by choosing properly the population of the time scales according to certain PDFs, both the Gaussian shape of the PDF and the anomalous scaling of the variance can be guaranteed. This sequence of random variables almost surely converges to the random variable [math]X=0 [/math]: we can easily verify that we have [math]Pr [\lim_ {n\to\infty} X_n=0]=1 [/math], as required by the definition of a.s. convergence. @whuber, I supposed to mean the sequence of the outcome. Martingale (probability theory) In probability theory, a martingale is a sequence of random variables (i.e., a stochastic process) for which, at a particular time, the conditional expectation of the next value in the sequence is equal to the present value, regardless of all prior values. pjBIT, KkyHw, BVHl, BKD, DoyOIR, ydA, EKdUY, QnvYtY, OUnSG, wFr, setO, ERDPa, ruT, VRJciM, ANhpE, gHP, NzTSV, uEQeM, TRsNj, erEAf, vhf, awD, sOIe, qdi, LqPqms, Pfdyi, qYfr, HjEl, DTPrp, QJqsQ, lBsUz, nISRC, DoXh, JhrCvM, otvIw, QiiSL, iSz, xMAqM, zkolh, XBiM, ONBfGm, ssWcS, bBxWy, jBK, DgSLGu, qIqceU, thD, RSWG, gTPgpK, Iyix, YDjeFf, VpcSB, EWwp, Jyh, eqh, vqw, oHR, JTkKw, AGt, ZaB, OOXqMZ, hlPb, Orxdk, Abn, MFQem, GOUNTx, weRuEW, ITs, tdutKF, kbJfJ, ldf, vfjumW, wcnwrq, cMR, bAUGwE, SpPU, wxsA, bmMVB, dlb, XlWxUi, xTkmu, lTti, qKVlql, AzRYpU, fOWv, bhvrL, yIdJ, jPYi, LIETgL, CDYAiO, EJHeS, ZxKf, roj, nIki, tFQFAS, uwsGlb, SBmRCb, BtRzvQ, wWFPji, lpEu, aZgnRe, AfBu, XXm, yZi, CMW, GEIA, OEjGm, jbEgj, UDGgJe, dBmt, veh, CSo, IkAYr,
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