**Properties**

- Since
*F*(*a*) = Pr(*X*≤*a*), the convergence in distribution means that the probability for*X*to be in a given range is approximately equal to the probability that the value of_{n}*X*is in that range, provided*n*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
*ƒ*(_{n}*x*) = (1 − cos(2*πnx*))**1**_{{x∈(0,1)}}. These random variables converge in distribution to a uniform*U*(0, 1), whereas their densities do not converge at all. **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 {*X*_{n}} converges in distribution to*X*if and only if any of the following statements are true:- Eƒ(
*X*) → Eƒ(_{n}*X*) for all bounded, continuous functions ƒ; - Eƒ(
*X*) → Eƒ(_{n}*X*) for all bounded, Lipschitz functions ƒ; - limsup{ Eƒ(
*X*) } ≤ Eƒ(_{n}*X*) for every upper semi-continuous function ƒ bounded from above; - liminf{ Eƒ(
*X*) } ≥ Eƒ(_{n}*X*) for every lower semi-continuous function ƒ bounded from below; - limsup{ Pr(
*X*∈_{n}*C*) } ≤ Pr(*X*∈*C*) for all closed sets*C*; - liminf{ Pr(
*X*∈_{n}*U*) } ≥ Pr(*X*∈*U*) for all open sets*U*; - lim{ Pr(
*X*∈_{n}*A*) } = Pr(*X*∈*A*) for all continuity sets*A*of random variable*X*.

- Eƒ(
**Continuous mapping theorem**states that for a continuous function*g*(·), if the sequence {*X*_{n}} converges in distribution to*X*, then so does {*g*(*X*_{n})} converge in distribution to*g*(*X*).**Lévy’s continuity theorem:**the sequence {*X*_{n}} converges in distribution to*X*if and only if the sequence of corresponding characteristic functions {*φ*} converges pointwise to the characteristic function_{n}*φ*of*X*.- Convergence in distribution is metrizable by the Lévy–Prokhorov metric.
- A natural link to convergence in distribution is the Skorokhod's representation theorem.

Read more about this topic: Convergence Of Random Variables, Convergence in Distribution

### Other articles related to "properties":

Zamak 4

... max 4.2 0.4 0.05 0.003 0.002 0.001 0.02 0.001 0.02 0.0005 0.001 Zamak 4

... max 4.2 0.4 0.05 0.003 0.002 0.001 0.02 0.001 0.02 0.0005 0.001 Zamak 4

**properties**Property Metric value English value Mechanical**properties**Ultimate tensile strength 317 MPa 46,000 psi Yield ...Zamak 7

0.002 0.002 0.001 0.075 0.02 - - - †Impurity ‡Alloying element Zamak 7

0.002 0.002 0.001 0.075 0.02 - - - †Impurity ‡Alloying element Zamak 7

**properties**Property Metric value English value Mechanical**properties**Ultimate tensile strength 285 MPa 41,300 ...Geophysics - Physical Phenomena - Mineral Physics

... Further information Mineral physics The physical

... Further information Mineral physics The physical

**properties**of minerals must be understood to infer the composition of the Earth's interior from seismology, the geothermal gradient and other sources ... Mineral physicists study the elastic**properties**of minerals their high-pressure phase diagrams, melting points and equations of state at high pressure and the ... Water is a very complex substance and its unique**properties**are essential for life ...Zamak 5

... max 4.3 1.1 0.06 0.004 0.003 0.0015 0.035 - - - - Zamak 5

... max 4.3 1.1 0.06 0.004 0.003 0.0015 0.035 - - - - Zamak 5

**properties**Property Metric value English value Mechanical**properties**Ultimate tensile strength 331 MPa (270 MPa aged) 48,000 ...Kiawah Island, South Carolina - Real Estate Market

... Many of the Kiawah Island

... Many of the Kiawah Island

**properties**are located directly on the beach or just a short distance away, and there are numerous golf course**properties**and lagoon view**properties**as well ...### Famous quotes containing the word properties:

“The reason why men enter into society, is the preservation of their property; and the end why they choose and authorize a legislative, is, that there may be laws made, and rules set, as guards and fences to the *properties* of all the members of the society: to limit the power, and moderate the dominion, of every part and member of the society.”

—John Locke (1632–1704)

“A drop of water has the *properties* of the sea, but cannot exhibit a storm. There is beauty of a concert, as well as of a flute; strength of a host, as well as of a hero.”

—Ralph Waldo Emerson (1803–1882)