Convergence of random variables explained

In probability theory, there exist several different notions of convergence of sequences of random variables, including convergence in probability, convergence in distribution, and almost sure convergence. The different notions of convergence capture different properties about the sequence, with some notions of convergence being stronger than others. For example, convergence in distribution tells us about the limit distribution of a sequence of random variables. This is a weaker notion than convergence in probability, which tells us about the value a random variable will take, rather than just the distribution.

The concept is important in probability theory, and its applications to statistics and stochastic processes. The same concepts are known in more general mathematics as stochastic convergence and they formalize the idea that certain properties of a sequence of essentially random or unpredictable events can sometimes be expected to settle down into a behavior that is essentially unchanging when items far enough into the sequence are studied. The different possible notions of convergence relate to how such a behavior can be characterized: two readily understood behaviors are that the sequence eventually takes a constant value, and that values in the sequence continue to change but can be described by an unchanging probability distribution.

Background

"Stochastic convergence" formalizes the idea that a sequence of essentially random or unpredictable events can sometimes be expected to settle into a pattern. The pattern may for instance be

• Convergence in the classical sense to a fixed value, perhaps itself coming from a random event
• An increasing similarity of outcomes to what a purely deterministic function would produce
• An increasing preference towards a certain outcome
• An increasing "aversion" against straying far away from a certain outcome
• That the probability distribution describing the next outcome may grow increasingly similar to a certain distribution

Some less obvious, more theoretical patterns could be

• That the series formed by calculating the expected value of the outcome's distance from a particular value may converge to 0
• That the variance of the random variable describing the next event grows smaller and smaller.

These other types of patterns that may arise are reflected in the different types of stochastic convergence that have been studied.

While the above discussion has related to the convergence of a single series to a limiting value, the notion of the convergence of two series towards each other is also important, but this is easily handled by studying the sequence defined as either the difference or the ratio of the two series.

For example, if the average of n independent random variables Y_i, i = 1, ..., n, all having the same finite mean and variance, is given by

X _{n} = (1)/(n) \sum ^{n} _{i =1} Y _{i,}

then as n tends to infinity, converges in probability (see below) to the common mean, μ, of the random variables Y_i. This result is known as the weak law of large numbers. Other forms of convergence are important in other useful theorems, including the central limit theorem.

(\Omega, l{F}, P ) .

Convergence in distribution

Examples of convergence in distribution
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Header1:	Dice factory
Data2:	Suppose a new dice factory has just been built. The first few dice come out quite biased, due to imperfections in the production process. The outcome from tossing any of them will follow a distribution markedly different from the desired uniform distribution. As the factory is improved, the dice become less and less loaded, and the outcomes from tossing a newly produced die will follow the uniform distribution more and more closely.
Header3:	Tossing coins
Data4:	Let be the fraction of heads after tossing up an unbiased coin times. Then has the Bernoulli distribution with expected value and variance . The subsequent random variables will all be distributed binomially. As grows larger, this distribution will gradually start to take shape more and more similar to the bell curve of the normal distribution. If we shift and rescale appropriately, then \scriptstyle Z _{n} = (\sqrt{n)/() }(X_n-\mu) will be converging in distribution to the standard normal, the result that follows from the celebrated central limit theorem.
Header5:	Graphic example
Data6:	Suppose is an iid sequence of uniform random variables. Let \scriptstyle Z _{n} = {\scriptscriptstyle (1)/(\sqrt{n) }}\sum_^n X_i be their (normalized) sums. Then according to the central limit theorem, the distribution of approaches the normal distribution. This convergence is shown in the picture: as grows larger, the shape of the probability density function gets closer and closer to the Gaussian curve.

Loosely, with this mode of convergence, we increasingly expect to see the next outcome in a sequence of random experiments becoming better and better modeled by a given probability distribution. More precisely, the distribution of the associated random variable in the sequence becomes arbitrarily close to a specified fixed distribution.

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. However, convergence in distribution is very frequently used in practice; most often it arises from application of the central limit theorem.

Definition

A sequence

X _{1,} X _{2,} \ldots of real-valued random variables, with cumulative distribution functions F _{1,} F _{2,} \ldots , is said to converge in distribution, or converge weakly, or converge in law to a random variable with cumulative distribution function if\lim _{n\toinfty} F _{n(x)} = F(x),

for every number

x \in R at which is continuous.

The requirement that only the continuity points of should be considered is essential. For example, if are distributed uniformly on intervals, then this sequence converges in distribution to the degenerate random variable . Indeed, for all n when, and for all when . However, for this limiting random variable, even though for all . Thus the convergence of cdfs fails at the point where is discontinuous.

Convergence in distribution may be denoted as

where

\scriptstyle l{L} _{X} is the law (probability distribution) of . For example, if is standard normal we can write $X\_n\backslash ,\backslash xrightarrow\backslash ,\backslash mathcal(0,\backslash ,1)$.

For random vectors the convergence in distribution is defined similarly. We say that this sequence converges in distribution to a random -vector if

\lim _{n\toinfty} P (X _{n\in} A) = P (X\in A) for every which is a continuity set of .

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. This is the “weak convergence of laws without laws being defined” — except asymptotically.

In this case the term weak convergence is preferable (see weak convergence of measures), and we say that a sequence of random elements converges weakly to (denoted as) if

E ^{*h(X} _{n)} \to E h(X) for all continuous bounded functions . Here E* denotes the outer expectation, that is the expectation of a “smallest measurable function that dominates ”.

Properties

• Since

F(a) = P (X \le a) , the convergence in distribution means that the probability for to be in a given range is approximately equal to the probability that the value of is in that range, provided is sufficiently large.

• In general, convergence in distribution does not imply that the sequence of corresponding probability density functions will also converge. As an example one may consider random variables with densities . These random variables converge in distribution to a uniform U(0, 1), whereas their densities do not converge at all.
  • However, according to Scheffé’s theorem, convergence of the probability density functions implies convergence in distribution.^[1]
• The portmanteau lemma provides several equivalent definitions of convergence in distribution. Although these definitions are less intuitive, they are used to prove a number of statistical theorems. The lemma states that converges in distribution to if and only if any of the following statements are true:

P (X _{n} \le x) \to P (X \le x) for all continuity points of x\mapsto P (X \le x) ;E f(X _{n)} \to E f(X) for all bounded, continuous functions f (where E denotes the expected value operator);E f(X _{n)} \to E f(X) for all bounded, Lipschitz functions f ;\lim inf E f(X _{n)} \ge E f(X) for all nonnegative, continuous functions f ;\lim inf P (X _{n} \in G) \ge P (X \in G) for every open set G ;\lim\sup P (X _{n} \in F) \le P (X \in F) for every closed set F ;P (X _{n} \in B) \to P (X \in B) for all continuity sets B of random variable X ;\limsup E f(X _{n)} \le E f(X) for every upper semi-continuous function f bounded above;\liminf E f(X _{n)} \ge E f(X) for every lower semi-continuous function f bounded below.

• The continuous mapping theorem states that for a continuous function, if the sequence converges in distribution to, then converges in distribution to .
  • Note however that convergence in distribution of to and to does in general not imply convergence in distribution of to or of to .
• Lévy’s continuity theorem: The sequence converges in distribution to if and only if the sequence of corresponding characteristic functions converges pointwise to the characteristic function of .
• Convergence in distribution is metrizable by the Lévy–Prokhorov metric.
• A natural link to convergence in distribution is the Skorokhod's representation theorem.

Convergence in probability

Examples of convergence in probability
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Header1:	Height of a person
Data2:	Consider the following experiment. First, pick a random person in the street. Let be their height, which is ex ante a random variable. Then ask other people to estimate this height by eye. Let be the average of the first responses. Then (provided there is no systematic error) by the law of large numbers, the sequence will converge in probability to the random variable .
Header3:	Predicting random number generation
Data4:	Suppose that a random number generator generates a pseudorandom floating point number between 0 and 1. Let random variable represent the distribution of possible outputs by the algorithm. Because the pseudorandom number is generated deterministically, its next value is not truly random. Suppose that as you observe a sequence of randomly generated numbers, you can deduce a pattern and make increasingly accurate predictions as to what the next randomly generated number will be. Let be your guess of the value of the next random number after observing the first random numbers. As you learn the pattern and your guesses become more accurate, not only will the distribution of converge to the distribution of, but the outcomes of will converge to the outcomes of .

The basic idea behind this type of convergence is that the probability of an “unusual” outcome becomes smaller and smaller as the sequence progresses.

The concept of convergence in probability is used very often in statistics. For example, an estimator is called consistent if it converges in probability to the quantity being estimated. Convergence in probability is also the type of convergence established by the weak law of large numbers.

Definition

A sequence of random variables converges in probability towards the random variable X if for all ε > 0

\lim _{n\toinfty} P ( |X _{n-X|} > \varepsilon ) = 0 .

More explicitly, let P_n(ε) be the probability that X_n is outside the ball of radius ε centered at X. Then is said to converge in probability to X if for any and any δ > 0 there exists a number N (which may depend on ε and δ) such that for all n ≥ N, P_n(ε) < δ (the definition of limit).

Notice that for the condition to be satisfied, it is not possible that for each n the random variables X and X_n 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. At the same time, the case of a deterministic X cannot, whenever the deterministic value is a discontinuity point (not isolated), be handled by convergence in distribution, where discontinuity points have to be explicitly excluded.

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 on a separable metric space, convergence in probability is defined similarly by

\forall\varepsilon> 0 , P ( d(X _{n,X)\geq\varepsilon)} \to 0 .

Properties

• Convergence in probability implies convergence in distribution.^[proof]
• In the opposite direction, convergence in distribution implies convergence in probability when the limiting random variable X is a constant.^[proof]
• Convergence in probability does not imply almost sure convergence.^[proof]
• The continuous mapping theorem states that for every continuous function

g , if $X\_n\; \backslash xrightarrow\; X$, then also&thinsp;

• Convergence in probability defines a topology on the space of random variables over a fixed probability space. This topology is metrizable by the Ky Fan metric: $$d(X,Y)\; =\; \backslash inf\backslash !\backslash big\backslash$$ or alternately by this metric $$d(X,Y)=\backslash mathbb\; E\backslash left[\backslash min(|X-Y|,\; 1)\backslash right].$$

Counterexamples

Not every sequence of random variables which converges to another random variable in distribution also converges in probability to that random variable. As an example, consider a sequence of standard normal random variables

X _{n} and a second sequence Y _{n} = ( -1 ) ^{nX} _{n} . Notice that the distribution of Y _{n} is equal to the distribution of X _{n} for all n , but:$$P(|X\_n\; -\; Y\_n|\; \backslash geq\; \backslash epsilon)\; =\; P(|X\_n|\backslash cdot|(1\; -\; (-1)^n)|\; \backslash geq\; \backslash epsilon)$$

which does not converge to

0 . So we do not have convergence in probability.

Almost sure convergence

Examples of almost sure convergence
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Header1:	Example 1
Data2:	Consider an animal of some short-lived species. We record the amount of food that this animal consumes per day. This sequence of numbers will be unpredictable, but we may be quite certain that one day the number will become zero, and will stay zero forever after.
Header3:	Example 2
Data4:	Consider a man who tosses seven coins every morning. Each afternoon, he donates one pound to a charity for each head that appeared. The first time the result is all tails, however, he will stop permanently. Let X1, X2, … be the daily amounts the charity received from him. We may be almost sure that one day this amount will be zero, and stay zero forever after that. However, when we consider any finite number of days, there is a nonzero probability the terminating condition will not occur.

This is the type of stochastic convergence that is most similar to pointwise convergence known from elementary real analysis.

Definition

To say that the sequence converges almost surely or almost everywhere or with probability 1 or strongly towards X means that$$\backslash mathbb\backslash !\backslash left(\backslash lim\_\backslash !\; X\_n\; =\; X\; \backslash right)\; =\; 1.$$

This means that the values of approach the value of X, in the sense that events for which does not converge to X have probability 0 (see Almost surely). Using the probability space

(\Omega, l{F}, P ) and the concept of the random variable as a function from Ω to R, this is equivalent to the statement$$\backslash mathbb\backslash Bigl(\backslash omega\; \backslash in\; \backslash Omega:\; \backslash lim\_\; X\_n(\backslash omega)\; =\; X(\backslash omega)\; \backslash Bigr)\; =\; 1.$$

Using the notion of the limit superior of a sequence of sets, almost sure convergence can also be defined as follows:$$\backslash mathbb\backslash Bigl(\backslash limsup\_\; \backslash bigl\backslash \; \backslash Bigr)\; =\; 0\; \backslash quad\backslash text\backslash quad\; \backslash varepsilon>0.$$

Almost sure convergence is often denoted by adding the letters a.s. over an arrow indicating convergence:

(S,d) , convergence almost surely is defined similarly:$$\backslash mathbb\backslash Bigl(\backslash omega\backslash in\backslash Omega\backslash colon\backslash ,\; d\backslash big(X\_n(\backslash omega),X(\backslash omega)\backslash big)\backslash ,\backslash underset\backslash ,0\; \backslash Bigr)\; =\; 1$$

Properties

• Almost sure convergence implies convergence in probability (by Fatou's lemma), and hence implies convergence in distribution. It is the notion of convergence used in the strong law of large numbers.
• The concept of almost sure convergence does not come from a topology on the space of random variables. This means there is no topology on the space of random variables such that the almost surely convergent sequences are exactly the converging sequences with respect to that topology. In particular, there is no metric of almost sure convergence.

Sure convergence or pointwise convergence

To say that the sequence of random variables (X_n) defined over the same probability space (i.e., a random process) converges surely or everywhere or pointwise towards X means

$$\backslash forall\; \backslash omega\; \backslash in\; \backslash Omega\; \backslash colon\; \backslash \; \backslash lim\_\; X\_n(\backslash omega)\; =\; X(\backslash omega),$$

where Ω is the sample space of the underlying probability space over which the random variables are defined.

This is the notion of pointwise convergence of a sequence of functions extended to a sequence of random variables. (Note that random variables themselves are functions).

$$\backslash left\backslash \; =\; \backslash Omega.$$

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. The difference between the two only exists on sets with probability zero. This is why the concept of sure convergence of random variables is very rarely used.

Counterexamples

Consider a sequence

\{X _{n\}} of independent random variables such that P(X _{n =1 ) = (1)/(n)} and P(X _{n =0 ) =1- (1)/(n)} . For 0 <\varepsilon< 1 / 2 we have P(|X _{n|\geq} \varepsilon) = (1)/(n) which converges to 0 hence X _{n\to} 0 in probability.

Since

\sum _{n\geq} P(X _{n=1)\toinfty} and the events \{X _{n=1\}} are independent, second Borel Cantelli Lemma ensures that P(\limsup _{n} \{X _{n=1\})=1} hence the sequence \{X _{n\}} does not converge to 0 almost everywhere (in fact the set on which this sequence does not converge to 0 has probability 1 ).

Convergence in mean

Given a real number, we say that the sequence converges in the r-th mean (or in the L^r-norm) towards the random variable X, if the -th absolute moments

E (|X_n|^r ) and E (|X|^r ) of and X exist, and\lim _{n\toinfty} E \left( |X ^{r} _{n - X|} \right) = 0 , where the operator E denotes the expected value. Convergence in -th mean tells us that the expectation of the -th power of the difference between X _{n} and X converges to zero.

This type of convergence is often denoted by adding the letter L^r over an arrow indicating convergence:

The most important cases of convergence in r-th mean are:

• When converges in r-th mean to X for r = 1, we say that converges in mean to X.
• When converges in r-th mean to X for r = 2, we say that converges in mean square (or in quadratic mean) to X.

Convergence in the r-th mean, for r ≥ 1, implies convergence in probability (by Markov's inequality). Furthermore, if r > s ≥ 1, convergence in r-th mean implies convergence in s-th mean. Hence, convergence in mean square implies convergence in mean.

Additionally,

\overset{}{X _{n} \xrightarrow{L ^{r}} X} ⇒ \lim _{n} E [|X ^{r]} _{n|} = E [|X| ^{r].} The converse is not necessarily true, however it is true if \overset{}{X _{n} \xrightarrow{p} X} (by a more general version of Scheffé's lemma).

Properties

Provided the probability space is complete:

• If

X _{n \xrightarrow{\overset{}{p}} X} and X _{n \xrightarrow{\overset{}{p}} Y} , then X = Y almost surely.

• If

X _{n \xrightarrow{\overset{}a.s.} }\ X and X _{n \xrightarrow{\overset{}a.s.} }\ Y, then X = Y almost surely.

• If

X ^{r}} X} _{n \xrightarrow{\overset{}{L} and X ^{r}} Y} _{n \xrightarrow{\overset{}{L} , then X = Y almost surely.

• If

X _{n \xrightarrow{\overset{}{p}} X} and Y _{n \xrightarrow{\overset{}{p}} Y} , then aX _{n+bY} _{n \xrightarrow{\overset{}{p}} aX+bY} (for any real numbers and) and X _{n} Y _{n\xrightarrow{\overset{}{p}} XY} .

• If

X _{n \xrightarrow{\overset{}a.s.} }\ X and Y _{n \xrightarrow{\overset{}a.s.} }\ Y, then aX _{n+bY} _{n \xrightarrow{\overset{}a.s.} }\ aX+bY (for any real numbers and) and X _{n} Y _{n\xrightarrow{\overset{}a.s.} }\ XY.

• If

X ^{r}} X} _{n \xrightarrow{\overset{}{L} and Y ^{r}} Y} _{n \xrightarrow{\overset{}{L} , then aX _{n+bY} ^{r}} aX + bY} ^{n \xrightarrow{\overset{}{L} (for any real numbers and).

• None of the above statements are true for convergence in distribution.

The chain of implications between the various notions of convergence are noted in their respective sections. They are, using the arrow notation:

\begin{matrix} \xrightarrow{\overset{}{L ^{s}}} & \underset{s>r\geq 1 }{ ⇒ } & \xrightarrow{\overset{}{L ^{r}}} & & \\ & & \Downarrow & & \\ \xrightarrow{ a.s. } & \Rightarrow & \xrightarrow & \Rightarrow & \xrightarrow \end

These properties, together with a number of other special cases, are summarized in the following list:

• Almost sure convergence implies convergence in probability:^[proof]

X _{n \xrightarrow{a.s.} }\ X \quad\Rightarrow\quad X_n\ \xrightarrow\ X

• Convergence in probability implies there exists a sub-sequence

(n _{k)} which almost surely converges:^[2] X _{n \xrightarrow{\overset{}{p}} X} ⇒ X _{n _{k}} \xrightarrow{ a.s. }\ X

• Convergence in probability implies convergence in distribution:^[proof]

X _{n \xrightarrow{\overset{}{p}} X} ⇒ X _{n \xrightarrow{\overset{}{d}} X}

• Convergence in r-th order mean implies convergence in probability:

X ^{r}} X} _{n \xrightarrow{\overset{}{L} ⇒ X _{n \xrightarrow{\overset{}{p}} X}

• Convergence in r-th order mean implies convergence in lower order mean, assuming that both orders are greater than or equal to one:

X ^{r}} X} _{n \xrightarrow{\overset{}{L} ⇒ X ^{s}} X,} _{n \xrightarrow{\overset{}{L} provided r ≥ s ≥ 1.

• If X_n converges in distribution to a constant c, then X_n converges in probability to c:^[proof]

X _{n \xrightarrow{\overset{}{d}} c} ⇒ X _{n \xrightarrow{\overset{}{p}} c,} provided c is a constant.

• If converges in distribution to X and the difference between X_n and Y_n converges in probability to zero, then Y_n also converges in distribution to X:^[proof]

X _{n \xrightarrow{\overset{}{d}} X, |X} _{n-Y} _{n| \xrightarrow{\overset{}{p}} 0} ⇒ Y _{n \xrightarrow{\overset{}{d}} X}

• If converges in distribution to X and Y_n converges in distribution to a constant c, then the joint vector converges in distribution to :^[proof]

X _{n \xrightarrow{\overset{}{d}} X, Y} _{n \xrightarrow{\overset{}{d}} c ⇒} (X _{n,Y} _{n) \xrightarrow{\overset{}{d}} (X,c)} provided c is a constant.

Note that the condition that converges to a constant is important, if it were to converge to a random variable Y then we wouldn't be able to conclude that converges to .

• If X_n converges in probability to X and Y_n converges in probability to Y, then the joint vector converges in probability to :^[proof]

X _{n \xrightarrow{\overset{}{p}} X, Y} _{n \xrightarrow{\overset{}{p}} Y ⇒} (X _{n,Y} _{n) \xrightarrow{\overset{}{p}} (X,Y)}

• If converges in probability to X, and if for all n and some b, then converges in rth mean to X for all . In other words, if converges in probability to X and all random variables are almost surely bounded above and below, then converges to X also in any rth mean.
• Almost sure representation. Usually, convergence in distribution does not imply convergence almost surely. However, for a given sequence which converges in distribution to X₀ it is always possible to find a new probability space (Ω, F, P) and random variables defined on it such that Y_n is equal in distribution to for each, and Y_n converges to Y₀ almost surely.
• If for all ε > 0,

\sum _{n} P \left(|X _{n} - X| > \varepsilon\right) < inf ty,

then we say that converges almost completely, or almost in probability towards X. When converges almost completely towards X then it also converges almost surely to X. In other words, if converges in probability to X sufficiently quickly (i.e. the above sequence of tail probabilities is summable for all), then also converges almost surely to X. This is a direct implication from the Borel–Cantelli lemma.

• If is a sum of n real independent random variables:

S _{n} = X _{1+ … +X} _{n}

then converges almost surely if and only if converges in probability.

• The dominated convergence theorem gives sufficient conditions for almost sure convergence to imply L¹-convergence:

}

• A necessary and sufficient condition for L¹ convergence is

X _{n\xrightarrow{\overset{}{P}}} X and the sequence (X_n) is uniformly integrable.

• If

X _{n \xrightarrow{\overset{}{p}} X} , the followings are equivalent ^[3] X ^{r}} X} _{n \xrightarrow{\overset{}{L} ,E [|X ^{r]} _{n|} → E [|X| ^{r]} < inf ty ,\{|X ^{r\}} _{n|} is uniformly integrable.

• If

X _{n} are discrete and independent, then X _{n} \stackrel{p}{ → } X implies that X _{n} \stackrel{a.s.}{ → } X . This is a consequence of the second Borel–Cantelli lemma.

See also

• Proofs of convergence of random variables
• Convergence of measures
• Convergence in measure
• Continuous stochastic process: the question of continuity of a stochastic process is essentially a question of convergence, and many of the same concepts and relationships used above apply to the continuity question.
• Asymptotic distribution
• Big O in probability notation
• Skorokhod's representation theorem
• The Tweedie convergence theorem
• Slutsky's theorem
• Continuous mapping theorem

References

• Book: Bickel, Peter J.. Klaassen. Chris A.J.. Ritov. Ya’acov. Wellner. Jon A.. 1998. Efficient and adaptive estimation for semiparametric models. Springer-Verlag. New York. 978-0-387-98473-5.
• Book: Billingsley, Patrick. Probability and Measure. 1986. 2nd. Wiley Series in Probability and Mathematical Statistics. Wiley.
• Book: Billingsley, Patrick. 1999. Convergence of probability measures. registration. John Wiley & Sons. 2nd. 1–28. 978-0-471-19745-4.
• Book: Dudley, R.M.. 2002. Real analysis and probability. Cambridge University Press. Cambridge, UK. 978-0-521-80972-6.
• Book: Fristedt, Bert. A Modern Approach to Probability Theory. Gray. Lawrence. Springer Science+Business Media. 1997. 978-1-4899-2837-5. New York. 10.1007/978-1-4899-2837-5.
• Book: Grimmett, G.R.. Stirzaker. D.R.. 1992. Probability and random processes. 2nd. Clarendon Press, Oxford. 271–285. 978-0-19-853665-9.
• Book: Jacobsen, M.. 1992. Videregående Sandsynlighedsregning (Advanced Probability Theory). 3rd. HCØ-tryk, Copenhagen. 18–20. 978-87-91180-71-2.
• Book: Ledoux. Michel. Talagrand. Michel. Michel Talagrand. Probability in Banach spaces. Springer-Verlag. Berlin. 1991. xii+480. 978-3-540-52013-9. 1102015.
• Book: Romano, Joseph P.. Siegel. Andrew F.. 1985. Counterexamples in Probability and Statistics . Counterexamples in Probability and Statistics . Chapman & Hall. Great Britain. 978-0-412-98901-8.
• Book: Grimmett. Geoffrey R.. Stirzaker. David R.. Probability and Random Processes. 2020. Oxford University Press. 4th. 978-0-198-84760-1.
• Book: van der Vaart, Aad W.. Aad van der Vaart. Wellner. Jon A.. 1996. Weak convergence and empirical processes. Springer-Verlag. New York. 978-0-387-94640-5.
• Book: van der Vaart, Aad W.. Asymptotic statistics. 1998. Cambridge University Press. New York. 978-0-521-49603-2.
• Book: Williams, D.. Probability with Martingales. Cambridge University Press. 1991. 978-0-521-40605-5.
• Book: Wong, E.. Hájek. B.. Stochastic Processes in Engineering Systems. Springer–Verlag. New York. 1985.
• Web site: Lecture 7: Weak Convergence . Gordan . Zitkovic . November 17, 2013.

Notes and References

1. Book: Durrett. Rick. Probability: Theory and Examples. 2010. 84.
2. Book: Gut, Allan. Probability: A graduate course. 2005. Springer. Theorem 3.4. 978-0-387-22833-4.
3. Web site: real analysis - Generalizing Scheffe's Lemma using only Convergence in Probability . 2022-03-12 . Mathematics Stack Exchange.