Mat 5:10 Blessed are they which are persecuted for righteousness' sake: for theirs is the kingdom of heaven.
Lucky those who have driven themselves on account of fulfilling the law seeing that theirs is the realm of the skies.
The sense of the Greek word translated as "blessed" shifts its meaning among "happy", "wealthy", and fortunate depending on the context. "Fortunate or lucky seems to work here. This word only means "blessed" in the sense that the lucky are blessed by good fortuned. It is not a religious term.
There is no "are" here. It is added to make a written sentence as opposed to a spoken phrase.
The word translated here as "they which are persecuted" can mean several contradictory things, which need examination. It is a verb meaning is "to chase." It is in the form of an adjective used as a noun, "the ones chasing" or, if passive, "the ones being chased." However, the form of the word is not passive or active, but one where the subject acts on their selves, so "the one chasing themselves." However, it refers to an action completed in the past, so "the ones who have chased themselves." It also means "to drive" or "propel." If we think the following word is being used as a legal term, which doesn't quite fit, it means "prosecute," not "persecute." The form of the verb makes it hard to fit "prosecute" or "persecute" because they are doing it to themselves.
The "for" is a preposition that "on account of."
Even though it is very early in the Gospels, we have already seen the word translated as "righteousness" in both Mat 3:15 and Mat 5:6. In Greek, the word translated as "righteousness" or in many translations, "justice" in the sense of "the fulfillment of the law," but this refers to natural law and the traditions of custom rather than to governmental laws. It does not mean conforming to current social fashions in thinking, which are seen as its opposite. "Righteousness" is the state for which humans were designed. In traditional terms, it means being virtuous, honest, and pure; thinking and acting correctly. Our modern idea of justice, that is, the legal standard giving every person due process, is a much lower standard that this idea of virtue.
As the last "standard" beatitude, this verse repeats the ending of the first, Mat 5:3.
The "for" here is a causal adverb that means "seeing that", "because", or "since."
This brings us to the "theirs." This is the possessive plural pronoun. It refers to the noun "kingdom". Again, the possessive has many uses in Greek. It could describe a producer, "they produce the kingdom"). It could describe an object's nature, "the kingdom consists of them." It could describe an attribute, "the kingdom is like them." It could describe ownership, "the kingdom is owned by them." It could also mean that the poor as a part of the kingdom, one of its groups of citizens.
This brings us to understanding what the "kingdom of heaven" is. Literally, this means the "realm of the skies." This article on The Kingdom of Heaven discusses it further and it is updated regularly.
Contrasting meanings of the negative "being hounded" and the positive "being driven" from the same word.
The Spoken Version:
The speaker moved toward a nearby group of Ascetics. “Lucky!” He continued cheerfully, “those who hound themselves—for the sake of virtue!” He indicated the scrawny, roughly dressed men.”
The crowd, including the Ascetics themselves, chuckled at the characterization.
“Because theirs is—the realm of the skies!”
οἱ δεδιωγμένοι (part pl perf mp masc nom) "They which are persecuted" is from dioko, which means "to pursue", "to chase", "to urge on", "to chase away", "to carry forward", "to be hurried (passive)," or "to drive." It means "persecute" when used as a legal term.
δικαιοσύνης, (noun sg fem gen) "Righteousness" is from dikaiosyne, which means "righteousness", "justice", "fulfillment of the law," and "the business of a judge." It carries the sense of virtue but specifically that of fulfilling legal or social requirements.
αὐτῶν (adj pl masc/fem gen) "Theirs" is from autos, which means "the same," and the reflexive pronouns, "myself", "yourself", "himself", "herself", "itself," or the oblique case of the pronouns, "him", "her," and "it." It also means "one's true self," that is, "the soul" as opposed to the body and "of ones own accord."
τῶν οὐρανῶν. (noun pl masc gen) "Of Heaven" is from the Greek ouranos (ouranos), which means "heaven as in the vault of the sky", "heaven as the seat of the gods", "the sky", "the universe," and "the climate."