The shocks in a random walk process are permanent. That does not mean it is impossible for a positive shock to be followed by a negative shock. It does mean it is equally likely that a positive shock will be followed by a positive shock as by a negative shock. That's what I mean by "permanent".
Consider two policies:
A. I print $100 and give you $100. Period.
B. I print $100, and give you $100, and at the same time tell you (lead you to expect) that I will take $100 away from you next year, and burn it.
A is an outright gift of newly-printed money. It is helicopter money.
B is not a gift. B is an interest-free loan. The fact that you do not give me an IOU is irrelevant. The fact that I do not buy your IOU is irrelevant. If I know where to find you, and have no problems remembering that I lent you $100, and have no problems forcing you to pay me $100, I do not need little bits of paper with "IOU" written on them to help jog my memory and help me get the $100 back through the courts. But it is exactly as if I printed $100 and bought your one-year IOU for $100 with it. Or bought an IOU from you that someone else had signed. And bonds are IOUs.
B is an Open Market Operation, in which I print $100, buy $100 worth of bonds from you, and reverse that operation one year later. B is not Helicopter Money.
If the central bank had a Price Level Path target, or an NGDP level path target, then helicopter money is equivalent to raising that Price Level path target, or NGDP level path target. Because the stock of money will be permanently higher than it otherwise would be, and so the price level or NGDP level will be permanently higher than it otherwise would be. Or, as I argued in my old post, raising the Price level or NGDP level path target converts some existing money into helicopter money. (As Brad deLong put it) it's helicopter money without the helicopters.