Throughout the Old Testament, the Holy Spirit would come upon people, depart, and never stay with the individual (Num 11:25-29; 1 Sam 16:14).

When God makes the unconditional New Covenant, He promises:

... this is My covenant with them, ...My Spirit which is upon you, and My words which I have put in your mouth shall not depart from your mouth,... (Isa 59:21)

I will put My Spirit within you and cause you to walk in My statutes, and you will be careful to observe My ordinances. (Ezek 36:25-28)

God promises the indwelling of the Holy Spirit!

When Jesus arrives, He provides some additional information about the New Covenant:

He who believes in Me, as the Scripture said, "From his innermost being will flow rivers of living water." But this He spoke of the Spirit, whom those who believed in Him were to receive; for the Spirit was not yet given, because Jesus was not yet glorified. (John 7:38-39)

And behold, I am sending forth the promise of My Father upon you; but you are to stay in the city until you are clothed with power from on high. (Luke 24:49)

But when He, the Spirit of truth, comes, He will guide you into all the truth; for He will not speak on His own initiative, but whatever He hears, He will speak; and He will disclose to you what is to come. (John 16:13)

Faith in Jesus Christ initiates the New Covenant and the Believer receives the Holy Spirit who is the guide into a deeper understanding of God and His promises!

After the resurrection of Jesus Christ, the apostles elaborate further on what God intended with the Holy Spirit in the context of the New Covenant:

In Him, you also, after listening to the message of truth, the gospel of your salvation—having also believed, you were sealed in Him with the Holy Spirit of promise, who is given as a pledge of our inheritance, with a view to the redemption of God's own possession, to the praise of His glory. (Eph 1:13-14)

In ancient times, clay was used to seal a scroll to preserve the authenticity of its contents. In a similar manner, the Holy Spirit validates a genuine Believer. For the Believer, the Holy Spirit is God's pledge or assurance that His promised inheritance will come true!

To learn more, see:

Divine Covenants… the New Covenant

How do figures of speech help you visualize the Holy Spirit?

What is "filling" or being "full of the Spirit?"