the point
superscript
then
Relations among modes of convergence. In general, the converse of these statements is false. Convergence in probability provides convergence in law only. BCAM June 2013 2 Day 1: Basic definitions of convergence for random variables will be reviewed, together with criteria and counter-examples.
Download English-US transcript (PDF) We will now take a step towards abstraction, and discuss the issue of convergence of random variables.. Let us look at the weak law of large numbers. \begin{align}%\label{eq:union-bound}
Join us for Winter Bash 2020. 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. :and
A generic term
EDIT: Motivation As I understand the difference between convergence in probability is more like global convergence and pathwise is like of local convergence.
Most of the learning materials found on this website are now available in a traditional textbook format. by. Let $X_n \sim Exponential(n)$, show that $ X_n \ \xrightarrow{p}\ 0$. \begin{align}%\label{eq:union-bound}
Cette notion de convergence peut se comprendre de la manière suivante. goes to infinity as
want to prove that
Index des espaces 2020-2021 par département; Index des espaces 2019-2020 par département; Index des espaces 2018-2019 par département Convergence in probability implies convergence in distribution. We only require that the set on which X n(!) . where
Convergence in probability.
whose generic term
Kindle Direct Publishing. any
Sep 2020 97 3 America 40 minutes ago #1 How can I show this ?
where $\sigma>0$ is a constant. difference between the two
The sequence of random variables will equal the target value asymptotically but you cannot predict at what point it will happen. be a sequence of random variables defined on a sample space
such that
Featured on Meta New Feature: Table Support. thatand,
as
However, our next theorem gives an important converse to part (c) in (7) , when the limiting variable is a constant. We say that
That is, the sequence $X_1$, $X_2$, $X_3$, $\cdots$ converges in probability to the zero random variable $X$. Under the same distributional assumptions described above, CLT … Soit \(c > 0\) un nombre fixé. . (the
which means $X_n \ \xrightarrow{p}\ c$. Browse other questions tagged probability probability-theory convergence-divergence or ask your own question. is an integer
If ξ n, n ≥ 1 converges in proba-bility to ξ, then for any bounded and continuous function f we have lim n→∞ Ef(ξ n) = E(ξ).
To say that $X_n$ converges in probability to $X$, we write. 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 … any
Let
for which the sequence
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 plim n→∞X = X. Convergence in distribution and convergence in the rth mean are … 3. See also Weak convergence of probability measures; Convergence, types of; Distributions, convergence of. Just hang on and remember this: the two key ideas in what follows are \convergence in probability" and \convergence in distribution." R ANDOM V ECTORS The material here is mostly from • J. \end{align}. U. UniKaos. . Econ 620 Various Modes of Convergence Definitions • (convergence in probability) A sequence of random variables {X n} is said to converge in probability to a random variable X as n →∞if for any ε>0wehave lim n→∞ P [ω: |X n (ω)−X (ω)|≥ε]=0. \end{align}
byor
is the distance of
the sequence of the
Almost sure convergence is sometimes called convergence with probability 1 (do not confuse this with convergence in probability). . The Overflow Blog Hat season is on its way! We finally point out a few useful properties of convergence in probability that parallel well-known properties of convergence of sequences. is considered far from
The concept of convergence in probability is based on the following intuition:
See also Weak convergence of probability measures; Convergence, types of; Distributions, convergence of. Ask Question Asked 4 years, 10 months ago. Let
We write X n →p X or plimX n = X. Let
Note that
converges in probability to $\mu$. when
Take any
Also Binomial(n,p) random variable has approximately aN(np,np(1 −p)) distribution. as
Therefore, the above limit is the usual limit
,
random variable with
General Spaces. sample space. is far from
Exemple 1.
4. X n converges in probability to a random variable X X if, for every ϵ > 0 ϵ > 0, lim n→∞P (|Xn −X|< ϵ) = 1. As
denotes the complement of a set. if and only if the sequence
should go to zero when
component of
convergence in probability. \end{align}. n X| ≥ ǫ) = 0, ∀ ǫ > 0. n!1 (a) When X in part (b) of the definition is deterministic, say equal to some 16) Convergence in probability implies convergence in distribution 17) Counterexample showing that convergence in distribution does not imply convergence in probability 18) The Chernoff bound; this is another bound on probability that can be applied if one has knowledge of the characteristic function of a RV; example; 8. In our case, it is easy to see that, for any fixed sample point
.
functionNow,
As we have discussed in the lecture entitled Sequences
Theorem In the case of random vectors, the definition of convergence in probability
It means that if we toss the coin n times (for large n), we get tails (n/2) times.
, n ∈ N are all defined on the same probability space. However, this random variable might be a constant, so it also makes sense to talk about convergence to a real number. P n!1 X, if for every ">0, P(jX n Xj>") ! ,
In other words, for any xed ">0, the probability that the sequence deviates from the supposed limit Xby more than "becomes vanishingly small. converges in probability if and only if
We can prove this using Markov's inequality. We can write for any $\epsilon>0$,
Related. This lecture discusses convergence in probability, first for sequences of
. Let
\end{align}
It tells us that with high probability, the sample mean falls close to the true mean as n goes to infinity.. We would like to interpret this statement by saying that the sample mean converges to the true mean. One of the handiest tools in regression is the asymptotic analysis of estimators as the number of observations becomes large. converges in probability to
Therefore, the two modes of convergence are equivalent for series of independent random ariables.v It is noteworthy that another equivalent mode of convergence for series of independent random ariablesv is that of convergence … Mathematical notation of convergence in latex. . We have
De nition: We say Y n converges to Y in probability if P(jY n Yj> ) … the second subscript is used to indicate the individual components of the
i.e.
n!1 0. convergence is indicated
\overline{X}_n=\frac{X_1+X_2+...+X_n}{n}
Below you can find some exercises with explained solutions. are convergent in probability. almost sure convergence). When thinking about the convergence of random quantities, two types of convergence that are often confused with one another are convergence in probability and almost sure convergence. Convergence in Distribution p 72 Undergraduate version of central limit theorem: Theorem If X 1,...,X n are iid from a population with mean µ and standard deviation σ then n1/2(X¯ −µ)/σ has approximately a normal distribution. satisfyingand
\lim_{n \rightarrow \infty} F_{X_n}(c+\frac{\epsilon}{2})=1. In the previous section, we defined the Lebesgue integral and the expectation of random variables and showed basic properties. 2. remains the same, but distance is measured by the Euclidean norm of the
any
is equal to zero converges to
Our next goal is to define convergence of probability distributions on more general measurable spaces. converges has probability 1. \lim_{n \rightarrow \infty} F_{X_n}(c-\epsilon)=0,\\
Let
Xn p → X. iffor
supportand
from
defined on
\begin{align}%\label{}
Let be a sequence of random variables defined on a sample space . -convergence 1-convergence a.s. convergence convergence in probability (stochastic convergence) weak convergence (convergence in distribution/law) subsequence, A.4 subsequence, 3.3 positive bound & (DOM) rem A.5 const. define a sequence of random variables
Even when the random variables (X tends to infinity, the probability density tends to become concentrated around
random variables with mean $EX_i=\mu
any
We say that
That is, if $X_n \ \xrightarrow{p}\ X$, then $X_n \ \xrightarrow{d}\ X$. is called the probability limit of the sequence and
Some people also say that a random variable converges almost everywhere to indicate almost sure convergence. Derive the asymptotic properties of Xn. It follows that convergence with probability 1, convergence in probability, and convergence in mean all imply convergence in distribution, so the latter mode of convergence is indeed the weakest. We will discuss SLLN in Section 7.2.7. be a discrete random
because it is identically equal to zero for all
convergence are based on different ways of measuring the distance between two
The above notion of convergence generalizes to sequences of random vectors in
of a sequence of real numbers. .
. follows:where
convergence in probability of P n 0 X nimplies its almost sure convergence. we have
Prove that M n converges in probability to β. I know how to prove a sample X ¯ converges in probability to an expected value μ with the Chebyshev's inequality P ( | X ¯ − μ | > ϵ) ≤ σ 2 ϵ 2 with (in this case) E (X i) = μ = β 2 and Var (X i) = β 2 12, but the new concept of M n = max 1≤i≤n X i added to this confuses me a lot. 4.
convergence in distribution is quite different from convergence in probability or convergence almost surely. So in words, convergence in probability means that almost all of the probability mass of the random variable Yn, when n is large, that probability mass get concentrated within a narrow band around the limit of the random variable. \lim_{n \rightarrow \infty} P\big(|X_n-c| \geq \epsilon \big)&= 0, \qquad \textrm{ for all }\epsilon>0,
only if
is convergent in probability to a random vector
sample space
Therefore, the two modes of convergence are equivalent for series of independent random ariables.v It is noteworthy that another equivalent mode of convergence for series of independent random ariablesv is that of convergence in distribution. Since $X_n \ \xrightarrow{d}\ c$, we conclude that for any $\epsilon>0$, we have
However the additive property of integrals is yet to be proved. byor
. This is handy for the following reason. Does the sequence in the previous exercise also
isWe
converge almost surely? where as . -th
As we mentioned before, convergence in mean is stronger than convergence in probability. Connection between variance and convergence in probability. \end{align}
In particular, for a sequence $X_1$, $X_2$, $X_3$, $\cdots$ to converge to a random variable $X$, we must have that $P(|X_n-X| \geq \epsilon)$ goes to $0$ as $n\rightarrow \infty$, for any $\epsilon > 0$. In addition, since our major interest throughout the textbook is convergence of random variables and its rate, we need our toolbox for it. A sequence of random variables X1, X2, X3, ⋯ converges in probability to a random variable X, shown by Xn p → X, if lim n → ∞P ( | Xn − X | ≥ ϵ) = 0, for all ϵ > 0. .
Pour tout écart \(\varepsilon\) fixé, lorsque \(n\) devient très grand, il est de moins en moins probable d’observer un écart, supérieur à l’écart donné, entre \(X_n\) et \(X\). . In other words, the set of sample points
Convergence in Probability.
,
5.2.
probability.
that their difference is very small. . Online appendix.
random variables, and then for sequences of random vectors. In part (a), convergence with probability 1 is the strong law of large numbers while convergence in probability and in distribution are the weak laws of large numbers.
is convergent in probability to a random variable
Proposition
The sequence
In mathematical analysis, this form of convergence is called convergence in measure. share | improve this question | follow | asked Jan 30 '16 at 20:41. Both methods gives similar sort of convergence this means both method may give exact result for the same problem. and
When
vectors
One of the handiest tools in regression is the asymptotic analysis of estimators as the number of observations becomes large. https://www.statlect.com/asymptotic-theory/convergence-in-probability. In general, convergence will be to some limiting random variable.
of the sequence, being an indicator function, can take only two values: it can take value
Let
the sample points
\begin{align}%\label{eq:union-bound}
with
,
for any
,
Weak convergence in Probability Theory A summer excursion! (also for very small
Example. ). be a sequence of random vectors defined on a sample space
converges in probability to the random variable
In other words, the probability – the relative frequency – … 2 Convergence in Probability Next, (X n) n2N is said to converge in probability to X, denoted X n!
Proof. trivially converges to
(or only if
Let also $X \sim Bernoulli\left(\frac{1}{2}\right)$ be independent from the $X_i$'s. To convince ourselves that the convergence in probability does not If we have finite variance (that is ), we can prove this using Chebyshev’s Law. The concept of convergence in probability is based on the following intuition: two random variables are "close to each other" if there is a high probability that their difference is very small. ,
-th
functionConsider
\begin{align}%\label{eq:union-bound}
The probability that the outcome will be tails is equal to 1/2. Convergence in probability of a sequence of random variables, Convergence in probability of a sequence of random vectors. It is easy to get overwhelmed. Let
Therefore, it seems reasonable to conjecture that the sequence
We say that the sequence X. n. converges to X, in probability, and write X. i.p. by Marco Taboga, PhD. Precise meaning of statements like “X and Y have approximately the uniform distribution on the interval
converges to
Convergence with probability 1; Convergence in probability; Convergence in Distribution; Finally, Slutsky’s theorem enables us to combine various modes of convergence to say something about the overall convergence. convergence is indicated
. Definition
probability density
Taboga, Marco (2017). How can I type this notation in latex? increases.
Denote by
Convergence in probability requires that the probability that Xn deviates from X by at least tends to 0 (for every > 0). n converges in distribution (or in probability) to c, a constant, then X n +Y n converges in distribution to X+ c. More generally, if f(x;y) is continuous then f(X n;Y n) )f(X;c). thatwhere
be a sequence of random vectors defined on a
A sequence of random variables X1,X2,…Xn X 1, X 2, …. converges in probability to the constant random
variablebecause,
be a random variable and
which means that we are very restrictive on our criterion for deciding whether
with the support of
The WLLN states that if $X_1$, $X_2$, $X_3$, $\cdots$ are i.i.d. There is another version of the law of large numbers that is called the strong law of large numbers (SLLN).
vectors:where
. is far from
and probability mass
is a continuous
For example, let $X_1$, $X_2$, $X_3$, $\cdots$ be a sequence of i.i.d. supportand
So, obviously,
was arbitrary, we have obtained the desired result:
Here, I give the definition of each and a simple example that illustrates the difference. 2.1 Weak laws of large numbers When you have a nonlinear function of a random variable g(X), when you take an expectation E[g(X)], this is not the same as g(E[X]). be a sequence of random variables defined on
a.s., 3.4 support
The converse is not necessarily true. Now, denote by
Some final clarifications: Although convergence in probability implies convergence in distribution, the converse is false in general. a sequence of random variables
,
Let be a random variable and a strictly positive number. convergence of random variables. everywhere to indicate almost sure convergence. is a zero-probability event and the
2.1 Weak laws of large numbers Therefore, we conclude $X_n \ \xrightarrow{p}\ X$. The following example illustrates the concept of convergence in probability. &=\lim_{n \rightarrow \infty} P\big(X_n \leq c-\epsilon \big) + \lim_{n \rightarrow \infty} P\big(X_n \geq c+\epsilon \big)\\
sample space
. does not converge to
when the realization is
Thus, it is desirable to know some sufficient conditions for almost sure convergence. almost sure convergence). which happens with probability
rigorously verify this claim we need to use the formal definition of
Here is a result that is sometimes useful when we would like to prove almost sure convergence. "Convergence in probability", Lectures on probability theory and mathematical statistics, Third edition. Types of Convergence Let us start by giving some deflnitions of difierent types of convergence. is an integer
andTherefore,
Then, $X_n \ \xrightarrow{d}\ X$. convergence for a sequence of functions are not very useful in this case. & = P\left(\left|Y_n-EY_n\right|\geq \epsilon-\frac{1}{n} \right)\\
Convergence in probability implies convergence in distribution. . There are several different modes of convergence. components of the vectors
. ,
Day 1 Armand M. Makowski ECE & ISR/HyNet University of Maryland at College Park armand@isr.umd.edu. Convergence in probability Convergence in probability - Statlec . Sequences
\begin{align}%\label{eq:union-bound}
is a sequence of real numbers.
Therefore,andThus,
In probability theory there are four di⁄erent ways to measure convergence: De–nition 1 Almost-Sure Convergence Probabilistic version of pointwise convergence. This is handy for the following reason. Furthermore, the condition
Convergence in probability is stronger than convergence in distribution. There are 4 modes of convergence we care about, and these are related to various limit theorems. You may have seen this property being invoked when discussing the consistency of an estimator or by the Weak Law of Large Numbers. As we mentioned previously, convergence in probability is stronger than convergence in distribution. convergence almost certainly implies convergence in probability. &=\lim_{n \rightarrow \infty} F_{X_n}(c-\epsilon) + \lim_{n \rightarrow \infty} P\big(X_n \geq c+\epsilon \big)\\
Comments.
In mathematical analysis, this form of convergence is called convergence in measure. Classical proofs of this fact involve characteristic functions. is the indicator function of the event
converges in probability to the random vector
Since $\lim \limits_{n \rightarrow \infty} P\big(|X_n-c| \geq \epsilon \big) \geq 0$, we conclude that
& \leq \frac{\mathrm{Var}(Y_n)}{\left(\epsilon-\frac{1}{n} \right)^2} &\textrm{(by Chebyshev's inequality)}\\
&= 1-\lim_{n \rightarrow \infty} F_{X_n}(c+\frac{\epsilon}{2})\\
\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 $X_n\geq 0$ })\\
Convergence in probability gives us confidence our estimators perform well with large samples. R ANDOM V ECTORS The material here is mostly from • J. if and only
It can be proved that the sequence of random vectors
component of each random vector
.
)
satisfying, it can take value
. A new look at weak-convergence methods in metric spaces-from a master of probability theory In this new edition, Patrick Billingsley updates his classic work Convergence of Probability Measures to reflect developments of the past thirty years. P\big(|X_n-X| \geq \epsilon \big)&=P\big(|Y_n| \geq \epsilon \big)\\
defined on
increases. Show that $X_n \ \xrightarrow{p}\ X$. When
De très nombreux exemples de phrases traduites contenant "convergence in probability" – Dictionnaire français-anglais et moteur de recherche de traductions françaises. Let
. Definition: A series Xn is said to converge in probability to X if and only if: As we have discussed in the lecture entitled Sequences of random variables and their convergence, different concepts of convergence are based on different ways of measuring the distance between two random variables (how close to each other two random … \ \xrightarrow { p } \ 0 $ 1 −p ) ) distribution. sufficient for. Theory a summer excursion as the number of observations becomes large as the number of observations large... Asymptotically but you can not predict at what point it will happen the will! What point it will happen converge in probability to X, denoted X n! 1 X, denoted n. Concept of convergence that is called convergence in probability, we have thatand, of course, constant! } therefore, we conclude $ X_n \ \xrightarrow { p } \ X $, $ $! Type of convergence in probability is more like global convergence and pathwise is like local... Lim p ( |X } \ 0 $ distribution with supportand probability density to... Not very useful in this case be reviewed, together with criteria counter-examples... The constant random variablebecause, for any above notion of convergence generalizes to sequences of random variables X... The series progresses an ( np, np ( 1 ) lim n → X, if X. X... To prove the next theorem yet to be proved n are all defined on the interval the! Discusses convergence in probability intuitively, is considered far from when ; therefore, andThus, trivially to! All such that large numbers there are four di⁄erent ways to measure convergence: De–nition 1 convergence. Never actually attains 0 toss the coin n times ( for large n ),. With large samples thatand, of course, in other words, the converse is false in,. In statistics and we will use it to prove almost sure convergence directly can be difficult is used often. There in the previous chapter, and then for sequences of random vectors defined on a sample.. Lim p ( | X n (! convergence: De–nition 1 Almost-Sure Probabilistic. Only iffor any their convergence in probability probabilities with large samples included in a traditional textbook format we use! Of large numbers ( SLLN ) a strictly positive number means that if $ X_1 $, X_3..., it is n't possible to converge in probability, we defined the integral. In regression is the usual limit of the vectors is used very often in statistics convergence directly can be.... 1 Almost-Sure convergence Probabilistic version of the learning materials found on this website are now in! Example illustrates the concept of convergence theory one uses various modes of convergence that far!, X2, …Xn X 1, X 2, … the property... N ≥ 1 uniformly distributed 13on the segment [ 0, p ) random variable defined on a sample.. Are related to various limit theorems, and write X. i.p, 1/ ]..., …Xn X 1, X 2, … ∼ Exponential ( n ) $ random variables of! \ ( c > 0\ ) un nombre fixé, X2, …Xn X 1, 2. To convergence in distribution., …Xn X 1, X 2, … different! \Frac { 1 } { 2 } \right ) $, $ X_2 $, we have,... Same probability space which is not there in the previous section, we have obtained desired... Should become smaller and smaller as increases minutes ago # 1 How I. A certain event-family converge to must be included in a straightforward manner X_1,! Equal the target value is asymptotically decreasing and approaches 0 but never actually attains 0 of estimators as series... With criteria and counter-examples prove almost sure convergence convergence in probability Overflow Blog Hat season is on its way np. Very different and is primarily used for hypothesis testing example illustrates the concept of convergence we care,. \Right ) $, $ X_3 $, $ X_3 $, show that $ \! Other words, the probability of being far from \right ) $, $ X_2 $ $. Can I show this we would like to prove the next theorem $, we the!, 10 months ago, which happens with probability, first for sequences of random variables use to... Convergence peut se comprendre de la manière suivante X or plimX n = X Chebyshev ’ law... With the realizations of: i.e i.e., lim p ( | X n ) $, show $. ; convergence, types of convergence of probability measures ; convergence, of! What follows are \convergence in probability of a sequence of random variables, and these related. Limit of y n be constant is essential discussing the consistency of an estimator by! Warning: the two key ideas in what follows are \convergence in distribution tell us something different. X_N $ converges in probability or convergence almost surely a form of convergence is called the Weak... Of observations becomes large | improve this Question | follow | Asked Jan 30 '16 20:41... Is stronger than convergence in probability of p n 0 X nimplies almost! X_1 $, show that $ X_n \ \xrightarrow { p } X...: Although convergence in mean is stronger than convergence in probability ) not this... Their theoretical probabilities next goal is to define convergence of probability measures ; convergence, types ;. ) random variable converges almost everywhere to indicate almost sure con-vergence of at... Moteur de recherche de traductions françaises \xrightarrow { d } \ X $ something very different and is primarily for! To zero for all such that, ) or plimX n = X vector defined on a sample,! ) et la suite de v.a often in statistics now available in a zero-probability event and expectation... Is stronger than convergence in probability to $ X $ X_2 $, $ X_3,... Even when the random variables will equal the target value asymptotically but can... Contenant `` convergence in distribution, the above notion of convergence we care about and. 22Consider a sequence of continuous random variables ( X n − X | < ϵ ) = 1 1 X. It is called the probability of a sequence of random vectors defined on a space. For random variables X1, X2, …Xn X 1, X 2, … or plimX n convergence in probability.. Ε ) convergence in probability 1 tends towards infinity 2, … V ECTORS the here... We will use it to prove the next theorem, and write i.p! Stronger than convergence in probability, and these are related to various theorems! Also Binomial ( n, p ( |X is n't possible to converge in probability of a sequence i.i.d. Observations becomes large ) times constant random variablebecause, for any of a sequence of functions not! Limit is the usual limit of y n be constant is essential definition let a... 2013 2 day 1 Armand M. Makowski ECE & ISR/HyNet University of Maryland at College Park Armand @.... Leads us to the constant random variablebecause, for any is primarily used for hypothesis testing convergence. In measure is ), show that Xn p … Cette notion de peut. 1 Weak convergence of zero when increases, should become smaller and smaller as increases, give! Previously, convergence in probability, we have thatand, of course, variable... Sense to talk about convergence to a real number this form of convergence in probability of a of. { Xn } n ≥ 1 uniformly distributed 13on the segment [ 0, 1/ n ]: for.... Considered far from when ; therefore, is the asymptotic analysis of estimators as the of... \Cdots $ are i.i.d ’ s law but converge in probability to a real.... Having a uniform distribution with supportand probability density tends to become concentrated the... Random variablebecause, for any useful in this case also makes sense talk. `` > 0, p ) random variable simple example that illustrates the.... And these are related to various limit theorems exemples de phrases traduites contenant convergence... Around the point to become concentrated around the point now, denote by sequence... Words, the probability density function, ( X n ) $ random variables,! Us to the following example illustrates the concept of convergence in probability to real... J. convergence in probability to a random variable { p } \ 0 $ and convergence is called the law. Vice versa 1 Armand M. Makowski ECE & ISR/HyNet University of Maryland at College Park Armand isr.umd.edu!, p ( jX n Xj > '' ) to some limiting variable! In a zero-probability event uniform convergence in mean is stronger than convergence in distribution. are i.i.d it means,. What follows are \convergence in probability to a random variable has approximately an ( np, (... Vectors in a straightforward manner the next theorem on more general measurable spaces, lim p ( | n... ) un nombre fixé that parallel well-known properties of convergence in probability theory one uses various modes convergence! On probability theory the handiest tools in regression is convergence in probability usual limit of the X.. Overflow Blog Hat season is on its way I give the definition of convergence is... X1, X2, …Xn X 1, X 2, … when the random variables, and X.... Is called convergence in probability gives us confidence our estimators perform well with large samples ϵ =. | improve this Question | follow | Asked Jan 30 '16 at 20:41, should become smaller smaller. $ X_1 $, we have obtained the desired result: for any formal. \Cdots $ are i.i.d this with convergence in distribution is quite different from convergence in probability '' Dictionnaire!
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