Example 2.2 (Convergence in probability but not almost surely). Exercise 1.1: Almost sure convergence: omega by omega - Duration: 4:52. herrgrillparzer 3,119 views. The R code for the graph follows (again, skipping labels). by Marco Taboga, PhD. Related. I've encountered these two examples (used to show how a.s. convergence doesn't imply convergence in Rth mean and visa versa). The hierarchy we will show is diagrammed in Fig. For example, the plot below shows the first part of the sequence for $s = 0.78$. For a sequence (Xn: n 2N), almost sure convergence of means that for almost all outcomesw, the difference Xn(w) X(w) gets small and stays small. In some problems, proving almost sure convergence directly can be difficult. By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy. converges has probability 1. In convergence in probability or a.s. convergence w.r.t which measure is the probability? The WLLN also says that we can make the proportion of noodles inside as close to 1 as we like by making the plot sufficiently wide. What's a good way to understand the difference? Because now, a scientific experiment to obtain, say, the speed of light, is justified in taking averages. For convergence in probability, recall that we want to evaluate whether the following limit holds, \begin{align}\lim_{n \rightarrow \infty} P(\lvert X_n(s) - X(s) \rvert < \epsilon) = 1.\end{align}. Can I (should I) change the name of this distribution? Is it appropriate for me to write about the pandemic? Thus, the probability that $\lim_{n \rightarrow \infty} \lvert X_n - X \rvert < \epsilon$ does not go to one as $n \rightarrow \infty$, and we can conclude that the sequence does not converge to $X(s)$ almost surely. Apr 2012 5 0 US Apr 14, 2012 #7 ... Second Moments: Convergence in Probability: Almost sure convergence & convergence in probability: Home. A sequence (Xn: n 2N)of random variables converges in probability to a random variable X, if for any e > 0 lim n Pfw 2W : jXn(w) X(w)j> eg= 0. Almost surely does. How does blood reach skin cells and other closely packed cells? Is it possible for two gases to have different internal energy but equal pressure and temperature? $$\sum_{n=1}^{\infty}I(|S_n - \mu| > \delta)$$ Is there a particularly memorable example where they differ? Shouldn't it be MAY never actually attains 0? answer is that both almost-sure and mean-square convergence imply convergence in probability, which in turn implies convergence in distribution. X. n. k. there exists a subsub-sequence . 2 : X n(!) \frac{S_{n}}{n} = \frac{1}{n}\sum_{i = 1}^{n}X_{i},\quad n=1,2,\ldots. ! Proposition 1. Relations among modes of convergence. Why do Bramha sutras say that Shudras cannot listen to Vedas? From my point of view the difference is important, but largely for philosophical reasons. Welcome to the site, @Tim-Brown, we appreciate your help answering questions here. Here’s the sequence, defined over the interval $[0, 1]$: \begin{align}X_1(s) &= s + I_{[0, 1]}(s) \\ X_2(s) &= s + I_{[0, \frac{1}{2}]}(s) \\ X_3(s) &= s + I_{[\frac{1}{2}, 1]}(s) \\ X_4(s) &= s + I_{[0, \frac{1}{3}]}(s) \\ X_5(s) &= s + I_{[\frac{1}{3}, \frac{2}{3}]}(s) \\ X_6(s) &= s + I_{[\frac{2}{3}, 1]}(s) \\ &\dots \\ \end{align}. Advanced Statistics / Probability. !X 1(!) The wiki has some examples of both which should help clarify the above (in particular see the example of the archer in the context of convergence in prob and the example of the charity in the context of almost sure convergence). X(!)) One thing to note is that it's best to identify other answers by the answerer's username, "this last guy" won't be very effective. X. a.s. n. ks → X. Almost Sure Convergence. Thus, while convergence in probability focuses only on the marginal distribution of jX n Xjas n!1, almost sure convergence puts … This type of convergence is similar to pointwise convergence of a sequence of functions, except that the convergence need not occur on a set with probability 0 (hence the “almost” sure). Convergence in probability means that with probability 1, X = Y. Convergence in probability is a much stronger statement. With the border currently closed, how can I get from the US to Canada with a pet without flying or owning a car? However, personally I am very glad that, for example, the strong law of large numbers exists, as opposed to just the weak law. Almost sure convergence | or convergence with probability one | is the probabilistic version of pointwise convergence known from elementary real analysis. 1, where some famous … rewrite the probability in (4.3) as P(liminf E ε n) = 1, with E n = {|X −X| < ε}. The probability that the sequence of random variables equals the target value is asymptotically decreasing and approaches 0 but never actually attains 0. The example I have right now is Exercise 47 (1.116) from Shao: $ X_n(w) = \begin{cases}1 &... Stack Exchange Network. ˙ = 1: Convergence in probability vs. almost sure convergence: the basics 1. University Math Help. In the following we're talking about a simple random walk, $X_{i}= \pm 1$ with equal probability, and we are calculating running averages, Use MathJax to format equations. $$P(|S_n - \mu| > \delta) \rightarrow 0$$ To subscribe to this RSS feed, copy and paste this URL into your RSS reader. = X(!) Choose some $\delta > 0$ arbitrarily small. 2.1 Weak laws of large numbers To assess convergence in probability, we look at the limit of the probability value $P(\lvert X_n - X \rvert < \epsilon)$, whereas in almost sure convergence we look at the limit of the quantity $\lvert X_n - X \rvert$ and then compute the probability of this limit being less than $\epsilon$. Note that the weak law gives no such guarantee. Suppose Xn a:s:! … The example comes from the textbook Statistical Inference by Casella and Berger, but I’ll step through the example in more detail. @gung The probability that it equals the target value approaches 1 or the probability that it does not equal the target values approaches 0. Does authentic Italian tiramisu contain large amounts of espresso? Relationship between the multivariate normal, SVD, and Cholesky decomposition. In probability theory one uses various modes of convergence of random variables, many of which are crucial for applications. In contrast, convergence in probability states that "while something is likely to happen" the likelihood of "something not happening" decreases asymptotically but never actually reaches 0. The weak law says (under some assumptions about the $X_n$) that the probability It only takes a minute to sign up. What if we had six note names in notation instead of seven? X (!) This gives you considerable confidence in the value of $S_n$, because it guarantees (i.e. Let us consider a sequence of independent random ariablesv (Z. Here is a result that is sometimes useful when we would like to prove almost sure convergence. Chapter Eleven Convergence Types. For a sequence (Xn: n 2N), almost sure convergence of means that for almost all outcomes w, the difference Xn(w) X(w) gets small and stays small.Convergence in probability is weaker and merely Why couldn't Bo Katan and Din Djarinl mock a fight so that Bo Katan could legitimately gain possession of the Mandalorian blade? The basic idea behind this type of convergence is that the probability of an “unusual” outcome becomes smaller and smaller as the sequence progresses. It's not as cool as an R package. Convergence in probability says that the chance of failure goes to zero as the number of usages goes to infinity. X =)Xn p! As Srikant points out, you don't actually know when you have exhausted all failures, so from a purely practical point of view, there is not much difference between the two modes of convergence. You compute the average The Annals of Mathematical Statistics, 43(4), 1374-1379. Almost sure convergence | or convergence with probability one | is the probabilistic version of pointwise convergence known from elementary real analysis. 2.3K views View 2 Upvoters The binomial model is a simple method for determining the prices of options. As an example, consistency of an estimator is essentially convergence in probability. Thanks for contributing an answer to Cross Validated! ... [0,1]$ with a probability measure that is uniform on this space, i.e., \begin{align}%\label{} P([a,b])=b-a, \qquad \textrm{ for all }0 \leq a \leq b \leq 1. Almost sure convergence vs. convergence in probability: some niceties The goal of this problem is to better understand the subtle links between almost sure convergence and convergence in probabilit.y We prove most of the classical results regarding these two modes of convergence. Stack Exchange network consists of 176 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers. From a practical standpoint, convergence in probability is enough as we do not particularly care about very unlikely events. The reason is that, when [math]n[/math] is very high, the probability of observing [math]X_{n}=1[/math] remains finite (so that the sum of subsequent probabilities diverges), while the probability of observing [math]Y_{n}=1[/math] vanishes to zero (so that the sum of subsequent probabilities converges). Are there cases where you've seen an estimator require convergence almost surely? But, in the case of convergence in probability, there is no direct notion of !since we are looking at a sequence of probabilities converging. Using Lebesgue's dominated convergence theorem, show that if (X. n) n2Nconverges almost surely towards X, then it converges in probability towards X. (a) Xn a:s:! The R code used to generate this graph is below (plot labels omitted for brevity). From then on the device will work perfectly. As you can see, each value in the sequence will either take the value $s$ or $1 + s$, and it will jump between these two forever, but the jumping will become less frequent as $n$ become large. Notice that the probability that as the sequence goes along, the probability that $X_n(s) = X(s) = s$ is increasing. Proposition7.1 Almost-sure convergence implies convergence in probability. On the other hand, almost-sure and mean-square convergence do not imply each other. X. In this section we shall consider some of the most important of them: convergence in L r, convergence in probability and convergence with probability one (a.k.a. X. De–nition 2 Convergence in Probability a sequence X n converges in probability to X if 8 > 0 and > 0 9 … The following is a convenient characterization, showing that convergence in probability is very closely related to almost sure convergence. the average never fails for $n > n_0$). "Almost sure convergence" always implies "convergence in probability", but the converse is NOT true. In the plot above, you can notice this empirically by the points becoming more clumped at $s$ as $n$ increases. On the other hand, almost-sure and mean-square convergence … We want to know which modes of convergence imply which. At least in theory, after obtaining enough data, you can get arbitrarily close to the true speed of light. Making statements based on opinion; back them up with references or personal experience. This type of convergence is equivalently called: convergence with probability one (written X n!X 1 w.p. Convergence almost surely implies convergence in probability ... Convergence in probability does not imply almost sure convergence in the discrete case. n → X. iff for every subsequence . Almost sure convergence is a stronger condition on the behavior of a sequence of random variables because it states that "something will definitely happen" (we just don't know when). Or am I mixing with integrals. Convergence in probability does not imply almost sure convergence in the discrete case If X n are independent random variables assuming value one with probability 1/ n and zero otherwise, then X n converges to zero in probability but not almost surely. The WLLN (convergence in probability) says that a large proportion of the sample paths will be in the bands on the right-hand side, at time $n$ (for the above it looks like around 48 or 9 out of 50). The notation X n a.s.→ X is often used for al-most sure convergence, while the common notation for convergence in probability is X n →p X or Notice that the $1 + s$ terms are becoming more spaced out as the index $n$ increases. Convergence in probability is weaker and merely requires that the probability of the difference Xn(w) X(w) being non-trivial becomes small. Proof. Thanks, I like the convergence of infinite series point-of-view! This lecture introduces the concept of almost sure convergence. : X n(!) by Marco Taboga, PhD. Attempted editor argues that this should read, "The probability that the sequence of random variables. Thus, the probability that the difference $X_n(s) - X(s)$ is large will become arbitrarily small. For almost sure convergence, we collect all the !’s wherein the convergence happens, and demand that the measure of this set of !’s be 1. Exercise 5.3 | Almost sure convergence Let fX 1;X 2;:::gbe a sequence of r.v. In contrast, convergence in probability states that "while something is likely to happen" the likelihood of "something not happening" decreases asymptotically but never actually reaches 0. To be more accurate, the set of events it happens (Or not) is with measure of zero -> probability of zero to happen. That is, if you count the number of failures as the number of usages goes to infinity, you will get a finite number. Simple example wanted: $ X_n $ converges to $X$ in probability but not almost surely, almost sure convergence and probability of estimator inside a compact set, Countable intersection of almost sure events is also almost sure. Assume you have some device, that improves with time. The current definition is incorrect. To assess convergence in probability, we look at the limit of the probability value $P(\lvert X_n - X \rvert < \epsilon)$, whereas in almost sure convergence we look at the limit of the quantity $\lvert X_n - X \rvert$ and then compute the probability of this limit being less than $\epsilon$. I'm not sure I understand the argument that almost sure gives you "considerable confidence." Almost sure convergence, convergence in probability and asymptotic normality In the previous chapter we considered estimator of several different parameters. Asking for help, clarification, or responding to other answers. Convergence almost surely is a bit stronger. In the previous lectures, we have introduced several notions of convergence of a sequence of random variables (also called modes of convergence).There are several relations among the various modes of convergence, which are discussed below and are summarized by the following diagram (an arrow denotes implication in the arrow's … $\endgroup$ – user75138 Apr 26 '16 at 14:29 Finite doesn't necessarily mean small or practically achievable. As you can see, the difference between the two is whether the limit is inside or outside the probability. Almost Sure Convergence: We say that (X n: n 1) converges almost surely to X 1 if P(A) = 1, where A= f! We only require that the set on which X n(!) Almost sure convergence is sometimes called convergence with probability 1 (do not confuse this with convergence in probability). ... Convergence in probability vs. almost sure convergence. X. i.p. Thus, while convergence in probability focuses only on the marginal distribution of jX n Xjas n!1, almost sure convergence puts restriction on the joint behavior of all random elements in the sequence I know this question has already been answered (and quite well, in my view), but there was a different question here which had a comment @NRH that mentioned the graphical explanation, and rather than put the pictures there it would seem more fitting to put them here. So, here goes. A sequence of random variables $X_1, X_2, \dots X_n$ converges almost surely to a random variable $X$ if, for every $\epsilon > 0$, \begin{align}P(\lim_{n \rightarrow \infty} \lvert X_n - X \rvert < \epsilon) = 1.\end{align}. Some people also say that a random variable converges almost everywhere to indicate almost sure convergence. While both sequences converge in probability to zero, only [math]Y_{n}[/math] converges almost surely. The answer is that both almost-sure and mean-square convergence imply convergence in probability, which in turn implies convergence in distribution. One thing that helped me to grasp the difference is the following equivalence, $P({\lim_{n\to\infty}|X_n-X|=0})=1 \Leftarrow \Rightarrow \lim_{n\to\infty}({\sup_{m>=n}|X_m-X|>\epsilon })=0$ $ \forall \epsilon > 0$, $\lim_{n\to\infty}P(|X_n-X|>\epsilon) = 0 $ $\forall \epsilon >0$. Now, recall that for almost sure convergence, we’re analyzing the statement. No other relationships hold in general. We can explicitly show that the “waiting times” between $1 + s$ terms is increasing: Now, consider the quantity $X(s) = s$, and let’s look at whether the sequence converges to $X(s)$ in probability and/or almost surely. Sure, I can quote the definition of each and give an example where they differ, but I still don't quite get it. As a bonus, the authors included an R package to facilitate learning. In general, almost sure convergence is stronger than convergence in probability, and a.s. convergence implies convergence in probability. Convergence in distribution di ers from the other modes of convergence in that it is based not on a direct comparison of the random variables X n with Xbut rather on a comparision of the distributions PfX n 2Ag The notation is the following P(! What information should I include for this source citation? Almost sure convergence. Let me clarify what I mean by ''failures (however improbable) in the averaging process''. If almost all members have perfect attendance, then each meeting must be almost full (convergence almost surely implies convergence in probability) X, and let >0. Some people also say that a random variable converges almost everywhere to indicate almost sure convergence. Here, we essentially need to examine whether for every $\epsilon$, we can find a term in the sequence such that all following terms satisfy $\lvert X_n - X \rvert < \epsilon$. Before introducing almost sure convergence let us look at an example. Convergence in probability is a bit like asking whether all meetings were almost full. Thus, it is desirable to know some sufficient conditions for almost sure convergence. If you take a sequence of random variables Xn= 1 with probability 1/n and zero otherwise. In this paper, we focus on almost sure convergence. Remark 1. I've never really grokked the difference between these two measures of convergence. Modes of Convergence in Probability Theory David Mandel November 5, 2015 Below, x a probability space (;F;P) on which all random variables fX ng and X are de ned. as n!1g and write X n!X 1 a.s. as n!1when this convergence holds. Recall that there is a “strong” law of large numbers and a “weak” law of large numbers, each of which basically says that the sample mean will converge to the true population mean as the sample size becomes large. Why is the difference important? Convergence of Sum of Sums of random variables : trivial? 5. When we say closer we mean to converge. Forums. ... = 1: (5.1) In this case we write X n a:s:!X(or X n!Xwith probability 1). There wont be any failures (however improbable) in the averaging process. Almost surely implies convergence in probability, but not the other way around yah? Let’s look at an example of sequence that converges in probability, but not almost surely. (AS convergence vs convergence in pr 1) Almost sure convergence implies convergence in probability. di⁄erent ways to measure convergence: De–nition 1 Almost-Sure Convergence Probabilistic version of pointwise convergence. 2 Convergence in probability Definition 2.1. Accidentally cut the bottom chord of truss. = 1 (1) or also written as P lim n!1 X n = X = 1 (2) or X n a:s:! I've encountered these two examples (used to show how a.s. convergence doesn't imply convergence in Rth mean and visa versa). Example . Proposition 4.2 In each of convergence concepts in definition 4.1 the limit, when it exists, is almost surely unique. "The probability that the sequence of random variables equals the target value is asymptotically decreasing and approaches 0 but never actually attains 0." Lp-Convergence. The SLLN (convergence almost surely) says that we can be 100% sure that this curve stretching off to the right will eventually, at some finite time, fall entirely within the bands forever afterward (to the right). By itself the strong law doesn't seem to tell you when you have reached or when you will reach $n_0$. The sequence of random variables will equal the target value asymptotically but you cannot predict at what point it will happen. $$S_n = \frac{1}{n}\sum_{k=1}^n X_k.$$ By using our site, you acknowledge that you have read and understand our Cookie Policy, Privacy Policy, and our Terms of Service. As he said, probability doesn't care that we might get a one down the road. 3. Almost sure convergence is defined based on the convergence of such sequences. Almost sure convergence. The natural concept of uniqueness here is that of almost sure uniqueness. so almost sure convergence and convergence in rth mean for some r both imply convergence in probability, which in turn implies convergence in distribution to random variable X. Example (Almost sure convergence) Let the sample space S be the closed interval [0,1] with the uniform probability … $$ as n!1); convergence almost certainly (written X n!X 1 a.c. as n!1). BFGS is a second-order optimization method – a close relative of Newton’s method – that approximates the Hessian of the objective function. It is easy to see taking limits that this converges to zero in probability, but fails to converge almost surely. That is, if we define the indicator function $I(|S_n - \mu| > \delta)$ that returns one when $|S_n - \mu| > \delta$ and zero otherwise, then ... this proof is omitted, but we include a proof that shows pointwise convergence =)almost sure convergence, and hence uniform convergence =)almost sure convergence. (Or, in fact, any of the different types of convergence, but I mention these two in particular because of the Weak and Strong Laws of Large Numbers.). In conclusion, we walked through an example of a sequence that converges in probability but does not converge almost surely. In order to understand this lecture, you should first understand the concepts of almost sure property and almost sure event, explained in the lecture entitled Zero-probability events, and the concept of pointwise convergence of a sequence of random variables, explained in the lecture entitled … An important application where the distinction between these two types of convergence is important is the law of large numbers. When we say closer we mean to converge. We have seen that almost sure convergence is stronger, which is the reason for the naming of these two LLNs. The strong law says that the number of times that $|S_n - \mu|$ is larger than $\delta$ is finite (with probability 1). De nition 5.2 | Almost sure convergence (Karr, 1993, p. 135; Rohatgi, 1976, p. In one case we have a random variable Xn = n with probability $=\frac{1}{n}$ and zero otherwise (so with probability 1-$\frac{1}{n}$).In another case same deal with only difference being Xn=1, not n with probability $=\frac{1}{n}$. Eg, the list will be re-ordered over time as people vote. J. jjacobs. I think you meant countable and not necessarily finite, am I wrong? MathJax reference. Proof Assume the almost sure convergence of to on (see the section ( Operations on sets and logical ... We can make such choice because the convergence in probability is given. For almost sure convergence, convergence in probability and convergence in distribution, if X n converges to Xand if gis a continuous then g(X n) converges to g(X). Thus, when using a consistent estimate, we implicitly acknowledge the fact that in large samples there is a very small probability that our estimate is far from the true value. Why do real estate agents always ask me whether I am buying property to live-in or as an investment? As we obtain more data ($n$ increases) we can compute $S_n$ for each $n = 1,2,\dots$. site design / logo © 2020 Stack Exchange Inc; user contributions licensed under cc by-sa. ... Convergence in Probability and in the Mean Part 1 - Duration: 13:37. If you enjoy visual explanations, there was a nice 'Teacher's Corner' article on this subject in the American Statistician (cite below). (something $\equiv$ a sequence of random variables converging to a particular value). as $n$ goes to $\infty$. To learn more, see our tips on writing great answers. If X n are independent random variables assuming value one with probability 1/n and zero otherwise, then X n converges to zero in probability but not almost surely. Just because $n_0$ exists doesn't tell you if you reached it yet. For another idea, you may want to see Wikipedia's claim that convergence in probability does not imply almost sure convergence and its proof using Borel–Cantelli lemma. But it 's self-contained and does n't care that we might get a one down the road infinite series!! N! 1 ) almost sure convergence lecture introduces the concept of almost sure convergence which in implies! $ exists does n't necessarily mean small or practically achievable there cases where you 've seen estimator! The list will be re-ordered over time as people vote for this source citation 1/n... Views View 2 Upvoters ( as convergence vs convergence in probability and asymptotic normality in the discrete case,... That the $ 1 + s $ terms are becoming more spaced out the... Take a sequence that converges in probability says that it will converge in probability... convergence in probability Sums...: convergence with probability one | is the probability 3,119 views arbitrarily close the. Mandalorian blade to be unique in an appropriate sense two Types of convergence decreasing. Gives you `` considerable confidence in the value of $ S_n $, because it guarantees (.! An R package to facilitate learning hand, almost-sure and mean-square convergence imply which variables the. We considered estimator of several different parameters are often required to be unique in appropriate! Difference $ X_n ( s ) - X ( s ) - X ( s ) X. That requires strong consistency where some famous … chapter Eleven convergence Types model is a optimization! Of the objective function reach $ n_0 $ of espresso n't Bo Katan could legitimately gain possession of the function! Like the convergence of such sequences failing is less than before when comparing the right side of the Mandalorian?! The binomial model is a convenient characterization, showing that convergence in probability a. Sequence converges in probability almost sure convergence vs convergence in probability RSS feed, copy and paste this into! Using lower upper bound constraints and using a big M constraints attempted editor that. Through an example, the definition of a `` consistent '' estimator only requires convergence in probability but! Sample size increases the estimator should get ‘ closer ’ to the true speed of light, is in... Meant countable and not necessarily finite, am I wrong structured fuzzing is. Different internal energy but equal pressure and temperature is almost surely, While the weak LLN says that will!: 4:52. herrgrillparzer 3,119 views is it possible for two gases to have different internal but... How a.s. convergence does n't imply convergence in probability and asymptotic normality in the almost sure convergence vs convergence in probability! Sure uniqueness, almost-sure and mean-square convergence … in some problems, proving almost sure convergence convergence. `` the probability that the weak law gives no such guarantee see our tips on writing great.... Using lower upper bound constraints and using a big M constraints require convergence almost everywhere to almost! A comparison to OLS a Statistical application that requires strong consistency other closely packed cells s = $! The definition of each and a simple method for determining the prices of options can conclude that the number..., see our tips on writing great answers a.s. as n! 1 ) ; convergence almost (! Some problems, proving almost sure convergence converging to a particular value ) the parameter interest... Surely unique probability vs. almost sure convergence everywhere ( written X n 1when... 2.1 weak laws of large numbers S_n $, because it guarantees ( i.e probability one written. Review of shrinkage in ridge regression and a comparison to OLS surely ) almost! Sufficient conditions for almost sure convergence, we appreciate your help answering questions here views View 2 Upvoters ( convergence. Requires strong consistency reached or when you have some device, that improves with time method – that the. In this paper, we focus on almost sure convergence in distribution mean. See, the difference between these two Types of convergence is stronger than convergence in mean... ), 1374-1379 policy and cookie policy it is easy to see taking limits this... Why could n't Bo Katan could legitimately gain possession of the sequence of random variables converging a!, the difference becomes clearer I think you meant countable and not necessarily finite, am I?... Is almost surely possession of the upper equivlance with the border currently closed, how can I get from us! And approaches 0 but never actually attains 0 walked through an example as sample... Light, is almost surely know which modes of convergence in probability to zero almost sure convergence vs convergence in probability. Almost sure convergence, While the weak law gives no such guarantee showing... Writing great answers necessarily mean small or practically achievable how can I from!: gbe a sequence of random variables to indicate almost sure convergence let fX ;. Tips on writing great answers convergence of Sum of Sums of random variables very closely related to almost sure.. Difference between the two is whether the limit, when it exists, is justified taking!